1. science
2. mathematics
3. algebra

arXiv:1411.6814v1 [math.DS] 25 Nov 2014
HILBERT’S 16TH PROBLEM. WHEN VARIATIONAL PRINCIPLES MEET
DIFFERENTIAL SYSTEMS
JAUME LLIBRE1 AND PABLO PEDREGAL2
Abstract. We provide an upper bound for the number of limit cycles that polynomial
differential systems of a given degree may have. The bound turns out to be a polynomial of
degree four in the degree of the system. Under suitable conditions the technique is applicable
to general smooth differential systems as well. It brings together those two areas of Analysis
implied in the title, by transforming the task of counting limit cycles into counting critical
points for a certain smooth, non-negative functional for which limit cycles are zeroes. We
thus solve the second part of Hilbert’s 16th problem providing a uniform upper bound for
the number of limit cycles which only depends on the degree of the polynomial differential
system.
Contents
1. Introduction
1.1. On the number of limit cycles
1.2. Statement of main results
2. A non-technical motivation for the guiding principle
3. Counting limit cycles
4. Hilbert’s 16th problem. Polynomial systems.
5. The driving principle
5.1. The general strategy
6. Some initial considerations
7. The steps to be covered
8. Reparameterizations and natural manifolds
8.1. The natural manifold in the limit
9. The perturbed situation
10. Some abstract results
11. The first derivative
12. The second derivative
13. A general result for critical points
14. Perturbation
15. Bifurcation and multiplicity
16. Singular systems
17. Appendix I. Morse inequalities
18. Appendix II. Variational problems in dimension one.
18.1. The derivative of a functional
18.2. The direct method of the Calculus of Variations for vector fields problems
19. Appendix III. Some calculations.
20. Appendix IV. A direct treatment of singular systems
20.1. A refined result
20.2. Counting critical paths for a singular system
2
3
4
5
9
10
11
12
13
15
16
18
18
19
22
23
24
26
28
31
32
32
33
33
33
35
37
38
2010 Mathematics Subject Classification. Primary 34C07.
Key words and phrases. limit cycles, polynomial vector fields, polynomial differential systems, Hilbert’s 16th
problem.
1
2
J. LLIBRE AND P. PEDREGAL
21. Appendix V. More on Hilbert’s 16th problem
21.1. On the configuration of the limit cycles of the polynomial differential systems
21.2. The 16-th Hilbert problem restricted to algebraic limit cycles
21.3. Limit cycles and the inverse integrating factor
Acknowledgements
References
39
39
40
41
41
42
1. Introduction
We deal with polynomial differential systems in R2 of the form
dx
dy
= x0 = P (x, y),
= y 0 = Q(x, y),
dt
dt
where the maximum degree of the polynomials P and Q is n. This n is called the degree of the
polynomial differential system (1).
We recall that a limit cycle of the differential system (1) is a periodic orbit of this system
isolated in the set of all periodic orbits of system (1). As far as we knwon the notion of limit
cycle appeared in the years 1891 and 1897 in the works of Poincar´e [40]. Moreover, he proved
that a polynomial differential equation (1) without saddle connections has finitely many limit
cycles, see [40].
In the Second International Congress of Mathematicians, celebrated in Paris in 1900, Hilbert
[24] proposed a list of 23 relevant problems to be solved during the XX century. The 16–th
problem of this list reads:
Problem of the topology of algebraic curves and surfaces
The maximum number of closed and separate branches which a plane algebraic curve of the
nth order can have has been determined by Harnack. There arises the further question as to
the relative position of the branches in the plane. As to curves of the 6th order, I have satisfied
myself–by a complicated process, it is true–that of the eleven branches which they can have
according to Harnack, by no means all can lie external to one another, but that one branch
must exist in whose interior one branch and in whose exterior nine branches lie, or inversely.
A thorough investigation of the relative position of the separate branches when their number
is the maximum seems to me to be of very great interest, and not less so the corresponding
investigation as to the number, form, and position of the sheets of an algebraic surface in space.
Till now, indeed, it is not even known what is the maximum number of sheets which a surface
of the 4th order in three dimensional space can really have.
In connection with this purely algebraic problem, I wish to bring forward a question which,
it seems to me, may be attacked by the same method of continuous variation of coefficients,
and whose answer is of corresponding value for the topology of families of curves defined by
differential systems. This is the question as to the maximum number and position of Poincar´e’s
boundary cycles (cycles limites) for a differential system of the first order and degree of the
form
dy
Y
= ,
dx
X
where X and Y are rational integral functions of the nth degree in x and y. Written homogeneously, this is
dz
dy
dx
dz
dy
dx
−z
−x
−y
X y
+Y z
+Z x
= 0,
dt
dt
dt
dt
dt
dt
(1)
where X, Y , and Z are rational integral homogeneous functions of the nth degree in x, y, z,
and the latter are to be determined as functions of the parameter t.
HILBERT’S 16TH
3
Clearly the 16–th Hilbert problem is formulated in two parts. The first part is about the
mutual disposition of the maximal number of separate branches of an algebraic curve, and its
extension to nonsingular real algebraic varieties. The second part is dedicated to limit cycles
of polynomial differential systems in R2 , where Hilbert asked for the maximal number and
relative position of limit cycles of polynomial differential systems (1). Usually the first part
of the 16–th Hilbert problem is studied in real algebraic geometry, while the second part is
considered in the theory of differential systems. Hilbert also pointed out that there might exist
possible connections between these two parts. Some of these connections are described in the
survey about the 16–th Hilbert problem written by Jibin Li, see [30].
From now on, when we talk about the 16–th Hilbert problem, we always mean the second
part of the 16–th Hilbert problem.
In 1988 Noel Lloyd [36] observed with respect to the 16–th Hilbert problem that the striking
aspect is that the hypothesis is algebraic, while the conclusion is topological.
Arnold in 1977 and 1983 (see [2] and [3], respectively) stated the weakened, infinitesimal or
tangential 16–th Hilbert problem which we do not consider here, but there are excellent surveys
for this modified problem. See for instance the survey of Ilyashenko [27] on the 16–th Hilbert
problem, the already mentioned survey of Jibin Li, the book of Colin Christopher and Chengzhi
Li [11], the survey due to Kaloshin [29], the one of Yakovenko [48], or more recently the work
of Binyamini, Novikov and Yakovenko [6].
According to Smale [43], except for the Riemann hypothesis, the second part of the 16–th
Hilbert problem seems to be the most elusive of Hilbert’s problems. Smale states the following
stronger version of the second half of 16–th Hilbert problem with respect to the number of limit
cycles:
Consider the polynomial differential system (1) in R2 . Is there a bound K on the number of
limit cycles of the form K ≤ nq where n is the maximum of the degrees of P and Q, and q is a
universal constant ?
The topological configurations or possible distribution of limit cycles mentioned as position
by Hilbert has also attracted the attention of many authors. Coleman in his work [13] on the
16–Hilbert problem said: For n > 2 the maximal number of eyes is not known, nor is it known
just which complex patterns of eyes within eyes, or eyes enclosing more than a single critical
point, can exist. Here “eye” means a nest of limit cycles. We shall see later that some of
the questions on the possible topological configurations of limit cycles realized by polynomial
differential systems have been solved.
1.1. On the number of limit cycles. Dulac [18] claimed in 1923 that any polynomial differential system (1) always has finitely many limit cycles. Ilyashenko [25] found an error in Dulac’s
paper in 1985. Later, two long works have appeared, independently, providing proofs of Dulac’s
´
assertion: one due to Ecalle
[19] in 1992, and the other to Ilyashenko [26] in 1991. As Smale
mentioned in [43], these two books have yet to be thoroughly digested by the mathematical
´
community. We remark that in no case the results of Ecalle
and Ilyashenko prove that there
exists a uniform upper bound for the maximum number of limit cycles of all the polynomial
differential systems of a given degree.
Bamon [4] proved in 1986 that any polynomial differential system of degree 2 has finitely
many limit cycles. His result uses previous results of Ilyashenko.
From the work of Dulac [18], it follows that if a polynomial differential system (1) has all
its saddle connections forming a simple homoclinic or heteroclinic loop, then the system also
has finitely many limit cycles, see the nice work of Sotomayor [44] for more details. Here a
homoclinic or heteroclinic loop is formed by k = 1 or k > 1 saddles (eventually some saddles
can be repeated), and k different separatrices connecting these saddles and forming a loop
(eventually some points of this loop can be identified in a repeated saddle) in such a way that
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J. LLIBRE AND P. PEDREGAL
at least in one of the two sides of the loop a Poincar´e return map is defined. Let µi < 0 < λi
be the eigenvalues of these saddles. If
k
Y
λi
6= 1,
µ
i=1 i
then the loop is called simple. There are many results providing upper bounds for the maximum
number of limit cycles which can accumulate or bifurcate from different kinds of homoclinic
and heteroclinic loops.
In 1957, Petrovskii and Landis [38] claimed that the polynomial differential systems of degree
n = 2 have at most 3 limit cycles. Soon (in 1959) a gap was found in the arguments of Petrovskii
and Landis, see [39]. Later, Lan Sun Chen and Ming Shu Wang [8] in 1979, and Songling Shi
[42] in 1982, provided the first polynomial differential systems of degree 2 having 4 limit cycles,
and up to now 4 is the maximum number of limit cycles known for a polynomial differential
system of degree 2.
Lower bounds for the maximum number of limit cycles that a polynomial differential system
of degree n can have have been given mainly by Christopher and Lloyd [12], Jibin Li [30], and
more recently by Maoan Han and Jibin Li [23].
1.2. Statement of main results. The following theorem provides an upper bound for the
maximum number of limit cycles that a polynomial differential system of degree n can have.
So it provides an answer to the second part of the 16-th Hilbert problem, and an answer to the
stronger version of it stated by Smale.
Theorem 1. An upper bound for the maximum number of limit cycles that a polynomial differential system of degree n can have is
1
(n − 1)(4n3 − 11n2 + 13n − 2).
2
This upper bound yields a universal exponent q = 4 for the stronger version due to Smale.
It is interesting to indicate, as we shall see later, that our methods under convenient assumptions apply to smooth differential systems too. As a matter of fact, Theorem 1 will be a direct
consequence of the following more general result valid for smooth differential systems.
Theorem 2. Consider the smooth differential system (1), and assume that it only has isolated
equilibria. Suppose that N is the finite number of real solutions (counting multiplicity) of the
non-linear system
(2)
P (Pxx + Qyx ) + Q(Pxy + Qyy ) = 0,
Px + Qy = 0,
and that m is the finite number of connected components of the curve Px + Qy = 0. Then an
upper bound for the number of limit cycles that the differential system (1) may have is
N +2
+ m − 1.
2
The paper is dedicated to proving these two Theorems 1 and 2. In particular, it is of the
utmost importance to understand the role played by these two pieces of information:
(1) the divergence curve Div(x, y) ≡ Px (x, y) + Qy (x, y) = 0; and
(2) the roots of the system
P (Pxx + Qyx ) + Q(Pxy + Qyy ) = 0,
Px + Qy = 0,
built with the components P and Q of the vector field determining the differential
system.
HILBERT’S 16TH
5
Before getting into a serious discussion for a rigorous proof of these two theorems, it is
instructive to have an overall, intuitive description of the principle that will guide us, as it is
quite different from the typical techniques utilized in differential systems. For this same reason,
we have included a number of final appendices to at least state various standard results coming
from variational techniques.
Most likely, the bound in Theorem 1 is not sharp at all. Especially for specific examples
or classes of differential systems, where more information about the number of zeroes of (2)
and/or the number or distribution of the components of the curve Div = 0 is available, one
might find a tighter bound for that particular example or class.
2. A non-technical motivation for the guiding principle
The main driving idea is easy to explain, and comprehend. Suppose that instead of having
to count limit cycles of a certain differential system, we would have to count isolated zeroes of
a given smooth, non-negative function f : R → R, like the one in Figure 1. How can we do
that? We learnt from Calculus courses that one possibility is to focus on the critical points of
f , where the derivative vanishes f 0 = 0 with strictly positive value of f . As a matter of fact,
it is pretty clear that, regardless of whether there are non-isolated zeroes, if we find all those
critical points, and it turns out to be a finite number N , then f cannot have, in any way, more
than N + 1 isolated zeroes. It could have less zeroes, it could even have none, it could have a
whole continuum of zeroes somewhere, but it cannot have more than N + 1 isolated zeroes.
The behavior of f at infinity may interfere with this general picture, as we could have a
“critical point at infinity”, Figure 2. Though in finite dimension, we could still manage to
understand the role of these points, in infinite dimension it is much more delicate, as one would
have to introduce a small perturbation of f , depending on a small parameter ε > 0, with a
strictly convex standard function g like g(x) = |x|p (with p > 1, and possibly large), analyze the
ε-situation, and explore very carefully the passage as ε & 0: non-degeneracy of critical points,
convergence of critical points, collapsing of different critical points to a unique limit point, etc.
Figure 1.
We would like to use this idea of finding an upper estimate for the limit cycles of a differential
system by counting critical points of a certain functional whose zeroes contain those isolated
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J. LLIBRE AND P. PEDREGAL
limit cycles of a given differential system like (1). It is not difficult to find such a functional
Z
1 1
(3)
E(x, y) =
(P (x(t), y(t))y 0 (t) − Q(x(t), y(t))x0 (t))2 dt,
2 0
where pairs (x, y) : [0, 1] → R2 represent smooth, periodic paths in R2 . It is clear that E is
non-negative, vanishes for closed, integral curves of the differential system (1) (provided they
can be parameterized in [0, 1]), and it is smooth. Other possibilities come to mind like
Z 1
1 0
[(x (t) − P (x(t), y(t)))2 + (y 0 (t) − Q(x(t), y(t)))2 ] dt.
(4)
0 2
However, this second choice does not permit enough flexibility to have integral curves parameterized in a fixed interval. We will stick to (3), though functionals like (4) have been proposed
for an alternative treatment of ODE’s, see for instance [1].
Figure 2.
Our guiding principle is then:
Limit cycles of the differential system (1), as isolated periodic integral curves,
cannot exist without generating critical paths with strictly positive value of E,
much in the same way as with functions in finite dimension. If we can establish
an upper bound for the number of these critical paths, and check that all of
this formalism can be applied in our situation, we will have an upper bound for
the number of limit cycles as well in the form
#{isolated zeroes of E} ≤ #{critical points of E with positive value} + 1,
that is to say
#{limit cycles of (1)} ≤ #{critical points of E with positive value} + 1.
Although it is not important for our main objective here, one can even refine the bound to
(5)
#{limit cycles of (1)} ≤ #{critical points of E with positive value},
because there is a continuum of zeroes of E at constant paths that count as a “single, isolated”
zero of E.
To cover fully and completely this program in infinite dimension many steps are to be examined.
HILBERT’S 16TH
7
Before proceeding any further, we now have a look at the equations for critical paths of E.
The Euler-Lagrange equation (system) for a functional like (3) is
(ZQ)0 + Z(Px y 0 − Qx x0 ) = 0,
−(ZP )0 + Z(Py y 0 − Qy x0 ) = 0,
where Z = P (x, y)y 0 − Q(x, y)x0 is exactly the integrand for E, except for the square. Assuming
that we can manipulate freely derivatives and equations, ignoring potential singular points where
there is no derivative, we would arrive at
Z 0 Q + Z Div y 0 = 0,
Z 0 P + Z Div x0 = 0,
where Div = Px + Qy is the divergence of the vector field (P, Q). Recalling that Z = P y 0 − Qx0 ,
multiplying the first equation by P , the second by Q, and subtracting them, we conclude that,
for a critical path, we should have
Z 2 Div = 0,
and then
ZZ 0 P = 0,
ZZ 0 Q = 0.
If we neglect equilibria where both P and Q vanish, we would have to conclude that
Z 2 Div = 0,
ZZ 0 = 0.
But the second equation implies that Z is constant in [0, 1]. If we are interested in critical
points with strictly positive value of E, then Z cannot be identically null for those, and so
we conclude, heuristically, that the critical paths we are interested in should have their image
contained entirely in the curve Div = 0. Since this curve is a one-dimensional object, and
critical paths have to be periodic, one main point in our analysis refers to finding the points of
the curve Div = 0 where critical paths may turn around.
This simple, straightforward argument most definitely inform us about the central role played
by the curve Div = 0. It does not furnish any insight about how many such critical points we
might have. But this is not surprising, because we would have to translate this simple idea of
counting critical points for a real function, to infinite dimension. This big jump requires to deal
with many issues, before one could state and prove results like Theorems 1 and 2.
For the sake of notation, write
(6)
u0 = F(u),
u = (x, y),
F = (P (x, y), Q(x, y)).
All of our concern focuses in proving the following result as it furnishes the key tool to count
critical points with positive value of E.
Theorem 3. Let u = (x, y) : [0, 1] → R2 be a critical periodic path for the functional E in (3)
(in a precise sense to be specified in the sequel) with a strictly positive value of E.
(a) Div(u(t)) = 0 for every t ∈ [0, 1], except possibly for t = 0 (t = 1), t = 1/2 when
u → ∞ at those times.
(b) The derivative u0 never vanishes, and its discontinuities can only take place at the two
times t = 0 (t = 1), and t = 1/2.
(c) Possible finite jumps on the derivative u0 can only take place at those points of the curve
Div = 0 where the vector field F is tangent to it (contact points).
Once the above theorem has been proved, and the formalism of counting zeroes through
critical points has been shown to be valid in our situation, it is a matter of counting how many
critical periodic paths complying with those conditions one may find, because this number is
exactly the upper bound for limit cycles of the system according to (5). For a general polynomial
system, this is accomplished roughly with the help of two classical results: Harnack’s theorem
to count the number of connected components of plane algebraic curves, and Bezout’s theorem
to count the number of zeroes that a polynomial system may have. Combining these results,
one can find the maximum number of critical periodic paths written in Theorem 1. Note how
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J. LLIBRE AND P. PEDREGAL
system (2) expresses the condition of the vector field F = (P, Q) being tangent to the curve
Div = 0: condition (c) in Theorem 3.
For the sake of illustration, consider a quadratic polynomial system in which both P and Q
are at most polynomials of degree two with no common factors. Suppose, in addition, that the
function Div = Px + Qy is a linear function which is proper (non-constant). In this case:
(i) Div = 0 is a straight line;
(ii) the first equation in system (2) is quadratic, and so the system itself can have none,
one, or at most two contact points.
Then Theorem 3 informs us that critical periodic paths are determined by two points (for
t = 0(= 1) and t = 1/2) taken from the set which is the union of the zeroes of (2) together with
two points at infinity in the straight line Div = 0. See Figure 3.
Figure 3.
The conclusion is:
(1) if there are two distinct zeroes for (2), the quadratic polynomial differential system
cannot have more than six limit cycles;
(2) if there is just one zero for (2), it can have at most three limit cycles;
(3) if the system (2) does not have real zeroes, then there is just one critical path running
through the line Div = 0 from one point at infinity to the other one, and back to the
first.
This result for quadratic systems is worth stating explicitly.
Theorem 4. A planar quadratic differential system cannot have more than six limit cycles.
Before turning to cover all of the necessary steps for a rigorous justification of Theorem 3
and the bound (5) in subsequent sections, we see first how one can prove Theorems 1 and 2
based on those two fundamental facts: Theorem 3, and the bound (5).
HILBERT’S 16TH
9
3. Counting limit cycles
As just indicated, we would like to prove Theorem 2 based on Theorem 3, the bound (5),
and some additional technical results that will be proved below.
Sketch of the proof of Theorem 2 (to be completed below). According to our discussion above,
and just as in the quadratic case, the question we would like to answer now is: how many
paths u, as in the statement of Theorem 3, there can possibly be? We know that their image
should be entirely contained in the curve Div = 0, and their derivatives will have jumps in two
points (including points at infinity), unless a smooth full curve can be embedded into the set
Div = 0. If we identify one such smooth curve (an oval) fully contained in Div = 0, with a
curve joining two points at infinity in case one connected component is homeomorphic to a line,
we can certainly use those two points of jump (including points at infinity) of the derivatives
to count them all. We need to consider two classes of points:
(1) Points at infinity. Because Div = 0 is a curve (possibly with various connected components), u can escape to infinity at both end-points of each unbounded connected
component. As a matter of fact, if one connected component is an oval, according to
our agreement above, we will identify it with a curve joining two points at infinity.
(2) Finite jumps can only take place where the vector field F is tangent to Div = 0 (condition (c) in Theorem 3). This condition leads precisely to system (2), but there should
be a clear discussion about singular points of the curve Div = 0 as special solutions of
(2).
There is also the case of weak equilibria:
P = Q = 0,
Px + Qy = 0.
By assumption, these points are isolated. There is nothing special about them, from our
perspective.
Much more relevant is the discussion about singular points of the curve Div = 0, which are
the solutions of the overdetermined system
(7)
Pxx + Qyx = Pxy + Qyy = 0,
Px + Qy = 0.
We have found two alternatives to treat this singular situation. One takes advantage of the
fact that differential systems for which (7) has no roots are generic, i.e. if a particular system
is such that (7) has roots, then a small perturbation (even a linear perturbation) destroys that
set of roots. A typical perturbation result (see Section 16 below for a specific discussion) for
the associated error functional E ensures that, in the limit for the singular system1, critical
periodic paths might be destroyed but not created, in such a way that the upper bound for
limit cycles cannot deteriorate for these singular systems. We have chosen this way of doing
things because it is more general, and is does not require any further hypotheses. However,
one looses the intuition about how the counting of limit cycles takes place for these systems.
For this reason, we have also included the details of that second form of dealing with them in
Appendix 20. This direct treatment of singular systems requires some further adjustments, and
refinements of results which we have found quite interesting.
Based on those perturbation arguments in Section 16, we can assume without loss of generality that we do not have singular points solutions of (7), and so connected components of the
curve Div = 0 are homeomorphic either to an oval, or to a line. In this case, the number of solutions of system (2), regardless of where they come from, may be used, according to inequality
(5), to provide an upper bound of the number of limit cycles of the system. Note that, because
paths u in Theorem 3 can escape to infinity through the curve Div = 0, each unbounded branch
1We are just using this term here for the sake of notation though these systems are not really singular. We
just mean that the corresponding curve Div = 0 has some singular points.
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J. LLIBRE AND P. PEDREGAL
of this curve Div = 0 must be endowed with two additional points at infinity, that somehow
must be regarded as special solutions for the master system (2).
Suppose that system (2) has N solutions (counting multiplicity if appropriate). They have to
be divided into m groups for each connected component of the curve Div = 0. Any such branch,
as pointed out above, must be endowed with two points at infinity including the bounded
branches. Paths u in Theorem 3 are determined by choosing two different points from the
same group (the same connected component, and including the points at infinity if necessary)
corresponding to times t = 0 (t = 1), and t = 1/2 where u0 will have a jump, finite or infinite.
This leads to solving the combinatorial problem:
m
m X
X
xi + 2
m
Maximize in (xi ) ∈ N :
subject to
xi = N.
2
i=1
i=1
It is elementary to realize that the maximum is reached for xi = N , and xj = 0, j 6= i, and the
value of the maximum is the one in the statement of Theorem 2 plus the possible m − 1 ovals
of the other components of the curve Div(x, y) = 0.
There is the further ambiguity of the starting point u(0). This ambiguity would also have to
be taken into account when counting limit cycles exactly in the same way: the same limit cycle
may correspond to several zeroes of the functional E0 depending on the starting point u(0) as
long as Div(u(0)) = 0. But again, since our critical paths are generated through two jumps in
the derivative, and limit cycles do intersect the curve Div = 0 at least twice (see comments in
next section related to the Bendixson criterium), the number of our critical periodic paths will
still be an upper bound for the number of limit cycles.
For a differential system for which the number of possible solutions of (2) can be determined
in terms of some features of the underlying vector field, our program would go through, and we
will have an upper bound (most likely a conservative one) for the number of limit cycles. This
is the case for a polynomial differential system.
4. Hilbert’s 16th problem. Polynomial systems.
Suppose both components, P and Q, of F are polynomials of degree at most n, with no
common factors, so that equilibria are isolated. By the previous remarks about singular systems, we can also assume that the connected components of the algebraic curve Div = 0 are
homeomorphic to an oval or to a line, because singular systems cannot yield a bigger upper
bound for limit cycles.
The discussion in the preceding section asks for caring about the number of connected components of the algebraic curve Div(x, y) = 0, as well as the number of solutions of system (2).
The polynomial Div(x, y) = Px (x, y) + Qy (x, y) has degree at most n − 1. An upper bound will
be the result of combining two classic theorems.
Theorem 5. Let the differential system (6) be a polynomial system of degree n. An upper
bound on the possible number of limit cycles it may have is given by
1
(n − 2)(n − 3)
2(n − 1)2 + 2
− 1 = (n − 1)(4n3 − 11n2 + 13n − 2),
+ (n − 1) 1 +
2
2
2
which has degree four in the degree of the polynomial differential system.
Proof. Once we have Lemma 37, we just need to find an upper bound for the number N of
solutions that system (2) may have when P and Q are polynomials of, at most, degree n, and
also an upper bound for the number of connected components of the algebraic curve Div = 0
of degree m = n − 1. To this end, we recall and state two classic results.
Theorem 6 (Harnack). The maximum number of connected components of an algebraic curve
of degree m in PR2 is
HILBERT’S 16TH
11
(a) 1 + (m − 1)(m − 2)/2 if m is even,
(b) (m − 1)(m − 2)/2 if m is odd.
In this statement PR2 is the real projective plane.
Theorem 7 (Bezout). Let R(x, y) and S(x, y) be two real polynomials. If both polynomials do
not share a non-trivial common factor, then the algebraic system of equations
R(x, y) = S(x, y) = 0
has at most degree(R)degree(S) solutions in PR2 .
Concerning the first, we need to translate that result in an upper bound for the number of
connected components that such a curve may have in R2 . Bearing in mind that the projective
plane PR2 essentially adds the straight line at infinity of R2 , an upper bound for the number
of components of the curve Div = 0 is
1
m 1 + (m − 1)(m − 2) ,
2
assuming that each connected component in PR2 could intersect the straight line at infinity in
the maximum number m of points. Note that a component of Div = 0 of degree m in PR2 cut
by the line at infinity can produce in the affine plane at most m components.
With respect to the maximum number of zeroes of system (2), we conclude, through Bezout’s
theorem above, that it is 2(n − 1)2 provided we discard the possibility of both equations having
a non-trivial common factor. If this were the case, then at least one connected component of
the curve Div = 0 would be an integral curve of the system because all points in such connected
component would have the vector field F tangent to it. Recall that solutions of system (2) are
precisely those points at which the vector field F is tangent to Div = 0 (Theorem 3). Limit
cycles could not, then, intersect that particular component, though that curve itself would be
a limit cycle if it is a true oval. The bigger upper bound occurs when there is no such common
factor.
The final number in the statement of the theorem is then a direct consequence of Theorem
2.
As already pointed out earlier, the upper bound should be the number in the statement plus
one. However, given that we also have the whole continuum of constant paths contained in
Div = 0 for which E0 vanishes, it is not hard to argue that they count as a unique, isolated
zero, if one passes to an appropriate quotient space. Indeed, all those zeroes would belong to
the same connected component: there would be no way to separate them. It might be helpful
to go back to Figure 1 to realize how the full segment of zeroes to the left of the picture actually
counts as a single zero for our counting purposes. This discussion is however irrelevant to the
main accomplishment of this paper, since we would have to add one to our upper bounds in
Theorems 1 and 2.
We now turn to treat a rigorous proof of all the ingredients necessary for the justification of
Theorem 3, and the additional statements used in the two last sections.
5. The driving principle
2
2
Let F : R 7→ R be the polynomial vector field with components F(x, y) = (P (x, y), Q(x, y)),
i.e. the polynomial vector field associated with the polynomial differential system (1). We
usually shall write this polynomial differential system in the vectorial form
(8)
u0 = F(u),
where
u = (x, y).
We assume that equilibria of system (8) are isolated, i.e. we suppose that there is no common
factor for P and Q.
12
J. LLIBRE AND P. PEDREGAL
We are interested in the limit cycles of system (8), and, in particular, we would like to bound
its number. The integral curves of (8) are analytic, and, in particular, C ∞ . In what follows,
we shall usually work with the differential system (8) using only that this system is C ∞ , except
when we mention explicitly that it is polynomial. This is due to the fact that most of our
results hold for C ∞ differential systems (8).
Put Div(x, y) = div F(x, y) = Px (x, y) + Qy (x, y). The set Div = 0 will play a major
role in our analysis. We can, without loss of generality, assume that such a set is a curve,
possibly with various connected components, some of which may eventually become a single
point. Otherwise, Div(x, y) is a constant, either vanishing or non-vanishing. In the first case,
the differential system would have a Hamiltonian structure, and as such, it could not have limit
cycles. In the second, by the Bendixson criterium (see for instance Theorem 7.10 of [16]) the
system has no periodic solutions, and in particular no limit cycles. Moreover, again by the
Bendixson criterium, the limit cycles of (8) must intersect the curve Div = 0. In the event that
a connected component of Div = 0 turns out to be an integral curve, then limit cycles cannot
intersect that particular component, and so such a component will be ignored altogether as if
it were not part of the curve Div = 0.
5.1. The general strategy. Our way of dealing with limit cycles u(t) of the differential system
(8) is by noticing that they are zeroes of the functional
Z
1 1 ⊥
(F (u(t)) · u0 (t))2 dt,
(9)
E0 (u) =
2 0
where F⊥ (u) = (−Q(u), P (u)), provided they can be reparameterized in the interval [0, 1]. We
shall see that each limit cycle can indeed be reparameterized in such a way that its period is
one.
The functional E0 may be regarded as a defect (or error) measure of how far a certain closed
path u is from being a periodic orbit. Limit cycles, as a special class of periodic orbits, are also
zeroes of E0 . However, from our viewpoint, limit cycles should correspond to “isolated” zeroes
of E0 .
A procedure to bound the number of limit cycles of system (8) consists in finding an upper
bound for the number of possible isolated zeroes of the functional E0 .
The basic driving idea is to perform this bounding through the number of
critical points with positive E0 . This can be typically done with the help of
the first Morse inequality. Roughly speaking, every N isolated zeroes of E0 ,
i.e. every N limit cycles, will give rise to N − 1 critical points with positive E0
(under important hypotheses to be written and checked), in such a way that
˜ of such critical points, then the
if we prove that there cannot be more than N
˜
differential system cannot have more than N +1 limit cycles. At the same time,
the system might or might not have non-isolated periodic orbits. We are not
interested in those.
To explain further the strategy, and envision how our objective might be accomplished, we
stick to an abstract situation in which we have a functional E : H → R, defined in a (real)
Hilbert space H, enjoying the following fundamental properties:
(i) E is non-negative, and coercive in the sense E(u) → ∞ whenever u → ∞;
(ii) E is smooth;
(iii) E complies with the Palais-Smale condition: if E(uj ) ≤ M , for all j, and a fixed positive
constant M , and E 0 (uj ) → 0 as j → ∞, then a certain subsequence converges in H;
(iv) all critical points of E are non-degenerate in the sense that E 00 (u) : H → H is a
non-singular, linear operator when E 0 (u) = 0;
(v) E is known to have, at least, a certain finite number of isolated zeroes: E(u(k) ) = 0 for
k = 1, 2, . . . , N .
HILBERT’S 16TH
13
Notice that (v) does not exclude the possibility that the differential system could have, at the
same time, a continuum of periodic orbits somewhere. Indeed, constant paths are also zeroes
of E0 .2 If the system turns out not to have limit cycles, there is nothing to be done.
Theorem 8. Under hypotheses (i)–(v), the functional E admits, at least, N − 1 critical points
with a strictly positive value for E.
This statement is essentially the first Morse inequality. Find additional information about
it in Appendix 17 (in particular Corollary 30). Though this is a highly technical area reserved
for experts, and there are various good sources (like [9], [14], [20], or the original contributions
by Palais and Smale themselves), we have found the textbook [5] particularly well-suited and
accessible for our purposes, not only in what relates to the first Morse inequality, but also for
many other issues of interest in this paper, as we will see later.
We can reinterpret this statement by writing
#{isolated zeroes of E} ≤ #{critical points of E with positive value} + 1,
and so the number of those critical points become an upper bound for the number of isolated
zeroes of E. If we can manage to have an upper bound for the number of those critical values,
then we will have an upper bound for the number of isolated zeroes. From Theorem 8, it follows
immediately the next corollary.
Corollary 9. Suppose that the functional E : H → R is non-negative, coercive, smooth, and
enjoys the Palais-Smale property. Suppose, in addition, that it has a finite number M of nondegenerate, critical points with positive value for E. Then E cannot have more than M + 1
isolated zeroes.
Again, we are claiming nothing about the possibility that the functional could have nonisolated zeroes. We simply do not care about them as their presence cannot spoil an upper
bound for the number of isolated zeroes. At most, those additional zeroes could lead to the
fact that our bound might not be sharp, but it will be an upper bound nevertheless.
Corollary 9 will be our guiding principle. If for the functional E0 in (9), we can cover
the various assumptions and conditions of this corollary, we will have an upper bound for the
number of limit cycles of the differential system (8).
This rough, general scheme, though expressed in such innocent terms in Corollary 9, requires
many steps to be covered, and many difficulties to be overcome. The goal of this contribution is
to explore that this strategy can indeed be successful in finding an upper bound for the number
of limit cycles for a polynomial system in terms of its degree.
6. Some initial considerations
There are two main initial sources of concern to begin with.
Our first discussion relates to the following statement.
Proposition 10. Suppose a smooth, closed path u : [0, 1] → R2 is such that F⊥ (u(t))·u0 (t) ≡ 0,
and u0 is zero nowhere. Then u is a reparameterization of a unique periodic integral curve (a
limit cycle if such a periodic integral curve is isolated) of (8).
2Periodic paths made up of pieces of integral curves are also zeroes of E . As a matter of fact, because
0
integral curves are unique through each point, the only possibility is to run forward along a piece of integral
curve, and then backwards, even several times, and finally go back to the starting point, always running through
a unique integral curve. These piecewise integral curves can be discarded if our ambient space of paths demands
more regularity. See Appendix 20. Even if they are allowed, their presence, as we are emphasizing, is not
relevant for our purposes.
14
J. LLIBRE AND P. PEDREGAL
Proof. The condition F⊥ (u(t)) · u0 (t) ≡ 0 on the path u implies that there is an 1-periodic,
smooth function µ(t) such that u0 (t) = µ(t)F(u(t)) for all t ∈ [0, 1], and that µ(t) never vanishes,
so that its sign does not change in [0, 1].
Define U(s) by putting
Z t
u(t) = U (s) where s =
µ(τ ) dτ.
0
Note that the map t 7→ s is one-to-one under the assumptions, either µ > 0, or µ < 0. Then, it
is clear that U0 (s) = F(U(s)), and that
Z 1
µ(τ ) dτ,
u(0) = U(0), U(S) = U(0), S =
0
so S is the period of the integral curve represented by U. Note that in general the function µ
may depend on each particular u.
Proposition 10 asserts that zeroes of E0 are not isolated as they can be continuously reparameterized to obtain different paths (all of them parameterized in [0, 1]), though the same
underlying curve. We need to establish a clear criterium to select one such unique parameterization representing each underlying curve. In other words, we can establish an equivalence
relation among periodic paths through their images in R2 : two paths are equivalent if their
images, as subsets of R2 , are identical; and we need to clearly and coherently select one unique
representative of each equivalence class.
There are many more situations for which E0 vanishes, other than on periodic solutions of
system (8). We accept that paths u are smooth, as limit cycles that we want to study are. The
condition
(10)
F⊥ (u(t)) · u0 (t) ≡ 0
can happen in more situations than those envisioned in Proposition 10. For instance, if u is
identically constant. What is correct is that (10) always implies the existence of a smooth
function µ(t), t ∈ [0, 1], such that u0 (t) = µ(t)F(u(t)). But µ may vanish, even in a nonnegligible set, and might then change sign. We can classify all possibilities in the following
way.
Suppose u(t) is a smooth curve of period unity. The class of smooth functions
{ν(s), s ∈ [0, 1] : ν(0) = 0, ν(1) ∈ Z \ {0}}
represents all possible reparameterizations of the same underlying periodic curve by putting
u(ν(s)), s ∈ [0, 1]. Zero is excluded as a possible value for ν(1) because the image of ν must
cover, at least once, the full period [0, 1]. We will have to devise a scheme to select just one
unique such parameterization for each periodic curve.
Another main, initial trouble refers directly to our functional E0 . It is not coercive in
whatever reasonable space of paths we may regard it, because after all, integral curves may
escape to infinity. This prevents the application of our general philosophy directly, and pushes us
to introduce a certain perturbation to remedy this fact. We, therefore, focus on the functionals
Eε = E0 + εE, where
Z
1 1 ⊥
E0 (u) =
(F (u(t)) · u0 (t))2 dt,
2 0
Z
1
1 1 00 2
2
E(u) = kukHper
[|u (t)| + |u0 (t)|2 + |u(t)|2 ] dt.
2 ([0,1];R2 ) =
2
2 0
Because integral curves of (8) are smooth, we can certainly choose as our ambient space
2
Hper
([0, 1]; R2 ) of periodic curves with square-integrable weak derivatives up to order two.
HILBERT’S 16TH
15
Since E0 depends (in a non-coercive way) in first derivatives, and as we shall see, we only need
second derivatives to remedy this lack of coercivity, we can replace the functional E by
Z
1 1 00 2
|u (t)| dt,
2 0
taking into account only second derivatives. The combination E0 + εE enjoys the same fundamental properties for either choice for E. We will therefore use one or the other as convenient,
without further notice.
But we can do more if it is convenient for our purposes. Since integral curves are smooth,
as pointed out above, we can select a functional analytical setting of as smooth curves as
convenient. In particular, we can decide to choose
Z
1 1 p) 2
|u (t)| dt
E(u) =
2 0
for p as high as we may need it to be. In this case the underlying space of curves would be
p
([0, 1]; R2 ), the class of maps with derivatives of order up to p which are square integrable in
Hper
[0, 1]. We could decide on the value of p depending on some structural features of the differential
system (8), like the degree in the case of a polynomial system. Formally, the specific value of
p will not change in the least our analysis, and so for definiteness, we will stick to the value
p = 2, and invoke the possibility of having a higher value of p whenever necessary. As a matter
of fact, we will (Appendix 20).
These initial considerations lead to a more involved framework in which our strategy has to
be developed. The one concerning different descriptions of curves asks for a way to select, in a
consistent unique way, representations of curves. This requires to work in infinite dimensional
manifolds immersed in our ambient space, and use versions of our Corollary 9 valid for functionals defined on manifolds. Secondly, we need to treat the case of small, but fixed ε, and pay
close attention to the many issues associated with the limit passage as ε & 0.
7. The steps to be covered
There might be different scenarios valid for the same general strategy. The one we have
chosen is always dictated by the final objective of providing an upper bound for the number
of limit cycles of a polynomial differential system in terms of the degree of the components P ,
and Q.
It might be convenient, to know where we are at each stage, to describe in some detail the
various steps to be covered for this program to be valid and complete.
(a) We need to take care of reparameterizations and representations of the same underlying
curve as described earlier. All of these different representations of the same underlying
curve have to be factored out somehow. This will lead us to working in the natural
manifold associated with E0 (or rather with Eε ) with respect to all those reparameterizations.
(b) There is a further restriction to be taken into account because, as pointed out earlier,
limit cycles ought to intersect the curve Div = 0. This fact may be taken to limit the
starting point for feasible paths u in our ambient space by demanding that they start
out (for t = 0) from the curve Div = 0. There is though some ambiguity as there might
be various (even infinite) such points in that curve. It is, however, a finite number of
points for a polynomial system.
(c) Deal with the situation for small, but positive and fixed ε. We want to be able to apply
here the first Morse inequality to bound limit cycles through the number of possible
critical points of Eε , restricted to the appropriate ambient space. This important task
requires:
(c.1) Palais-Smale condition.
16
J. LLIBRE AND P. PEDREGAL
(c.2) Non-degeneracy of critical points.
(d) Analysis of critical points for Eε , for ε fixed.
(e) Passage to the limit. We need to comprehend how such critical points evolve as ε & 0,
and how to relate each such branch with some limit path described directly in terms of
the properties of F. More precisely, two main issues are to be examined:
(e.1) Analyze the resulting singular perturbation problem.
(e.2) Explore the multiplicity issue, i.e. how many branches may coalesce into the same
limit.
(f) Perform the counting of critical points for a polynomial differential system depending
on its degree, and establish an upper bound for the number of limit cycles.
The last step (f) has already been anticipated in Sections 3, and 4. The rest of this paper
is organized to provide proofs and justifications for all other steps, and finally have a full proof
of Theorems 1 and 2.
8. Reparameterizations and natural manifolds
There is the issue of reparameterizations. We would like to restrict attention to those paths
2
([0, 1]; R2 ) which are parameterized in an optimal way with respect
in our ambient space Hper
2
([0, 1]; R2 ), we will only consider the
to Eε , ε > 0, fixed. By this we mean that for each u ∈ Hper
reparameterization of it which renders the value of Eε as small as possible among all feasible
representations of a given curve.
2
More precisely, let u(t) ∈ Hper
([0, 1]; R2 ) represent a smooth, non-constant, closed curve
2
in R . After a standard perturbation argument to avoid the zero set of u0 , we can always
find, without loss of generality, one parameterization of it, still denoted u(t), with u0 vanishing
nowhere. For instance, once the underlying curve is smooth, one can always parameterized it
by arc-length. Put
(11)
U(s) = u(t(s)),
s ∈ [0, 1],
for an arbitrary reparameterization t(s) of it, where t(0) = 0, t(1) ∈ Z. We allow that the
path u passes through u(0) several times, even in a smooth fashion. Likewise, the function t(s)
does not have to be monotone, so that the family of maps U, given in (11), for t(s) varying,
also cover the possibility of reparameterizations turning backward and forward many times over
arbitrary portions of the underlying curve. If we set F(u, z, Z) for the full integrand for Eε ,
omitting the parameter ε so as to argue in general, we would write
Z 1
I(t) ≡ Eε (U(s)) =
F(U(s), U0 (s), U00 (s)) ds,
0
for the reparameterization U of u associated with the feasible function t(s). This integral equals
Z 1
I(t) =
F(u(t), u0 (t)t0 , u00 (t)(t0 )2 + u0 (t)t00 ) ds.
0
Regarded as an integral functional for feasible functions t(s), we can write
Z 1
I(t) =
F(t(s), t0 (s), t00 (s)) ds,
0
where
(12)
F(t, ξ, η) = F(u(t), u0 (t)ξ, u00 (t)ξ 2 + u0 (t)η).
Keep in mind that u is given, and fixed for this selection process: we are looking for the optimal
t(s), for each periodic curve given.
It is interesting to realize how functions t(s) for which t(1) is as small as possible, so that
u(t(1)) is the first time that the path u(t) goes back to u(0) in a smooth way, yield a smaller
value of the integrals I(t) as compared to those choices for which t(1) is bigger. Indeed, if
HILBERT’S 16TH
17
u(t0 ) = u(0), and t0 ∈ (0, 1) is the minimum of such values of t with this property, by setting
t(s) = t0 s, and U(s) = u(t0 s), s ∈ [0, 1], we would have
Z 1
I(t) =
F(U(s), U0 (s), U00 (s)) ds
0
Z
1 t0
F(u(t), t0 u0 (t), t20 u00 (t)) dt
=
t0 0
Z
Z
1 ε t0 4 00 2
t 2 t0 ⊥
=
t0 |u (t)| dt + 0
(F (u(t)) · u0 (t))2 dt
t0 2 0
2 0
Z 1
Z 1
ε
1
|u00 (t)|2 dt +
(F⊥ (u(t)) · u0 (t))2 dt
<
2 0
2 0
= Eε (u).
Note how this argument is not valid to discard the ambiguity clockwise-counterclockwise for
each path, i.e. t(1) = ±t0 .
Theorem 31 and Proposition 32 in Appendix II, see Section 18, show that the existence of a
unique optimal reparameterization depends smoothly on the curve itself.
2
Proposition 11. Let u ∈ Hper
([0, 1]; R2 ) have a nowhere vanishing derivative, and let tu ∈
2
(0, 1] be the first time u goes back to u(0) in a smooth way (so that u ∈ Hper
([0, tu ]; R2 ) or
p
([0, tu ]; R2 )). There is a unique, optimal, smooth t(s) minimizing I(t) among all those
u ∈ Hper
complying with t(0) = 0, t(1) = tu . Moreover, the dependence of t(s) upon u(t) is smooth.
Proof. Just notice that for
ε 2 1 ⊥
|Z| + (F (u) · z)2 ,
2
2
F(t, ξ, η) given in (12) satisfies the hypotheses of Theorem 31 for each particular path u(t). F(u, z, Z) =
Using Proposition 11, we define
˙ ε = {u ∈ H 2 ([0, 1]; R2 ), parameterized optimally with respect to Eε },
M
per
2
([0, 1]; R2 ), non-constant, parameterized optimally with respect to Eε },
Mε = {u ∈ Hper
and regard the error functional
Eε = E0 + εE
where
E0 (u) =
1
2
Z
0
1
(F⊥ (u(t)) · u0 (t))2 dt and E(u) =
1
2
Z
1
|u00 (t)|2 dt,
0
˙ ε is a manifold immersed in the ambient Hilbert space
defined over these two manifolds. M
2
Hper
([0, 1]; R2 ), and Mε is a “punctured” manifold. We refer to [5], pages 356-358 for a brief,
right-to-the-point definition of infinite dimensional manifolds modeled on a Hilbert or Banach
space. The next lemma follows the ideas of the discussion in [5], pages 331-332, and stress the
concept of natural manifold.
Lemma 12. The following statements hold.
˙ ε is a continuous, connected, infinite-dimensional manifold.
(a) M
(b) Mε is a smooth, connected, infinite-dimensional manifold (with boundary).
(c) Mε is the natural manifold for Eε , and so the set of critical values of Eε |Mε is exactly
the set of critical values of (the unrestricted functional) Eε that lie on Mε .
˙ ε is exactly the natural manifold corresponding to Eε with
Proof.
(a) The definition of M
respect to reparameterizations.
˙ ε because
(b) The smoothness of Eε ensures the smoothness of Mε but possibly not of M
Eε remains null for reparameterizations of constant paths. The second derivative Eε00
restricted to the submanifold of reparameterizations of constant paths is therefore singular, and the hypothesis (c) of Theorem (6.3.13) in [5] is violated.
18
J. LLIBRE AND P. PEDREGAL
(c) This statement is one general main consequence associated with natural manifolds.
Sometimes it is summed up by saying that natural manifolds do not give rise to multipliers.
8.1. The natural manifold in the limit. It is interesting to know the properties of optimal
reparameterizations in the limit situation when ε & 0. Because second-order terms, corresponding to second derivatives, for our functional are associated with the small perturbation
parameter ε, as indicated earlier, we can neglect those terms asymptotically, and conclude that
the optimal reparameterizations tend to the optimal t(s) minimizing the functional
Z 1
1 ⊥
(F (u(t(s))) · u0 (t(s)))2 t0 (s)2 ds
2
0
corresponding to E0 . In this case, optimal reparameterizations can be found explicitly through
optimality conditions (Euler-Lagrange equation). In fact, in this form the optimal t(s) is found
through an elementary fact in the Calculus of Variations (Proposition 33 in Appendix II, see
Section 18), namely, if the integrand F(t, ξ) of the functional does not depend explicitly in s,
then the optimal t(s) is the one complying with
F(t, t0 ) − t0 Fξ (t, t0 ) = constant.
In our particular situation, for
F(t, ξ) =
1
(F(u(t)) · u0 (t))2 ξ 2 ,
2
it is immediate to conclude that the optimal parameterization corresponds to having the integrand constant independent of time i.e.
(13)
(F⊥ (u(t)) · u0 (t))2 = constant.
We will come back to this condition later.
9. The perturbed situation
We briefly describe our guiding principle based on the first Morse inequality, as discussed
earlier in Section 5, for the perturbed situation corresponding to the functional Eε , for each
ε > 0, small but fixed. The limiting process will take place afterwards.
Suppose that we have found N isolated limit cycles uj (t) for the differential system (8). We
are not sure if we have found all, not even if there is a finite number of them, or if there is
a whole continuum of zeroes somewhere else. We have E0 (uj ) = 0, j = 1, 2, . . . , M . Each
underlying curve, according to our discussion above, admits a unique representative uj,ε in Mε .
Each path uj,ε is just a suitable reparameterization of uj itself, optimally chosen with respect
to Eε . The values Eε (uj,ε ) will not vanish, but for all ε sufficiently small, and suitable r > 0
(not necessarily small), we will have
(14)
Eε (uj,ε ) <
inf
v∈Mε ,kv−uk,ε k=r
Eε (v)
for all 1 ≤ j, k ≤ M . Indeed, note that the left-hand side is arbitrarily small for ε small because
Eε is a small perturbation of E0 , while r is chosen so that the right-hand side is not small. This
is just a small perturbation version of the fact that limit cycles are isolated, and so the error
functional E0 is positive a bit away from them. By Theorem 31, and bearing in mind that balls
ku − uj,ε k ≤ r are weakly closed sets, we will have at least one local minimum uj,ε in each such
ball. Assuming that our functionals and our perturbation comply with the necessary properties
HILBERT’S 16TH
19
for the first Morse inequality to be applied, we would conclude (Appendix I) that there must
be at least M − 1 critical points for Eε with positive Morse index
M ≤ ] critical points for Eε with positive Morse index Eε + 1
≤ ] critical points for Eε with positive value of Eε + 1.
Knowing that ε is doomed to converge to zero, if we can bound the number of these last critical
points, independently of M and of ε, and without any reference to the number of zeroes of
E0 , but depending only on some features of the underlying vector field F, like the degree in
case of a polynomial differential system, we will have the bound sought. This is our objective,
according to the plan explicitly described in Section 7.
˙ ε at constant paths. This
There is also the issue of the lack of smoothness of the manifold M
can be easily solved for each ε > 0 fixed, by smoothing out in any way such manifold a bit
without interfering with (14). Alternatively, to avoid details about that smoothing-out process
and bearing in mind that constant paths furnish absolute minimizers for the functional E0 , and
so hE00 (u), ui < 0 for every path u sufficiently close to a constant path, one can have the validity
of the Morse inequalities once these small sets around constant paths have been removed, just
as in Corollary (6.5.11) of [5].
We can therefore assume, henceforth, that we are dealing with a single smooth manifold
Mε which is a natural manifold for Eε with respect to reparameterizations without any further
comment.
10. Some abstract results
We start now to focus on the important properties of the functional Eε itself, for fixed ε > 0.
The structure of Eε as a small perturbation of E0 through a multiple of the square of the norm
of the ambient space lead us to consider a general scenario as follows.
The first statement is a standard result whose proof is included for the sake of completeness.
Lemma 13. Let H be a (real) Hilbert space, and consider the functional Eε : H → R of the
form
ε
Eε (u) = kuk2H + E0 (u)
2
where ε > 0, and E0 : H → R complies with
(i) it is non-negative, and smooth;
(ii) E00 : H → H is a (non-linear) compact operator;
(iii) E000 (u) : H → H is a linear, compact operator for every u ∈ H.
In addition, we assume that for every small ε, Eε has a finite number of critical points independently of ε. Then
(a) Eε is coercive;
(b) Eε complies with the Palais-Smale condition;
(c) there are arbitrarily small values of ε for which all critical points of Eε are nondegenerate.
Proof. The fact that Eε is coercive is immediate from the non-negativity of E0 , and the strict
positivity of ε.
Concerning the Palais-Smale property, just notice that Eε0 = ε1 + E00 , where 1 : H → H is
the identity operator. From the compactness of E00 , we directly deduce, if {uj } is uniformly
bounded and Eε0 (uj ) → 0, that
εuj = Eε0 (uj ) − E00 (uj ) → −u
if E00 (uj ) → u. Hence {uj } converges strongly in H. This is the Palais-Smale property.
It is well-known (see for instance Theorem 6.8 in [7]), due to the compactness assumed on
E000 (u), that the spectrum of such a linear mapping is at most countable, and may accumulate
20
J. LLIBRE AND P. PEDREGAL
only around zero. Because of the explicit hypothesis about the number of possible critical values
of Eε uniformly for small ε, we conclude that the union of all eigenvalues of the operators E000 (u)
for u, a critical point of Eε , can only accumulate around zero, and so there are arbitrarily small
values of ε for which the only solution of the equation εU + E000 (u)U = 0 is U ≡ 0, provided u
is taken from the (finite) set of critical points of Eε . This directly implies the non-degeneracy
of critical points of Eε , for arbitrarily small values of ε, because Eε00 (u) = ε1 + E000 (u).
There is a straightforward version of this fact when the functional is restricted to a natural
manifold for Eε . The proof is exactly the same. The important remark is that, because Mε is
a natural manifold for Eε , the directions transversal to Mε cannot spoil Lemma 13 precisely
because the derivative of Eε vanishes in those directions.
Lemma 14. Under the same assumptions than in Lemma 13, let Mε be the corresponding
natural manifold of Eε with respect to reparameterizations. Then
(a) Eε |Mε is coercive;
(b) Eε |Mε complies with the Palais-Smale condition;
(c) there are arbitrarily small values of ε such that all critical points of Eε |Mε are nondegenerate.
Proof. For the sake of notation, set E ε = Eε |Mε .
There is nothing to say about coercivity.
Assume that for a certain sequence {uj } ⊂ Mε , we have, simultaneously, that
0 ≤ E ε (uj ) ≤ M,
0
E ε (uj ) → 0,
with M independent of j (recall ε is fixed). The condition on the derivatives means that
directional derivatives tangent to Mε tend to zero. But transversal derivatives to Mε actually
do vanish for every uj precisely because Mε is a natural manifold: Eε is minimized with respect
to reparameterizations, and so derivatives tangential to reparameterizations (i.e. transversal to
Mε ) should vanish. This discussion ensures that, as a matter of fact, Eε0 (uj ) → 0. Conclude that
the Palais-Smale condition is valid by Lemma 13. As already pointed out earlier, this situation
is often times summed up by saying that natural manifolds do not give rise to multipliers.
Concerning the non-degeneracy of critical points, note that
00
(15)
Eε |Mε (u) = Eε00 (u)|Tu ,
if Tu is the tangent space of Mε at u ∈ Mε . Hence, the non-singularity of Eε00 (u) over H
implies the non-singularity of the restriction in (15), and, hence, the non-degeneracy of critical
points for the restricted functional.
An additional constraint has to be incorporated into the manifolds Mε . From now on we
will take elements in this manifold to comply also with the condition
Div(u(0)) = 0,
so that feasible paths should depart from the curve Div = 0. There is another version of
Lemma 13 when the functional is restricted to a certain class of submanifolds M immersed into
H, provided we adapt the hypotheses appropriately.
Lemma 15. Let I : H → R be a smooth, weakly continuous functional over the Hilbert space
H, whose derivative I 0 : H → H is compact: I(uj ) → I(u), I 0 (uj ) → I 0 (u) whenever uj * u.
Consider the sub-manifold
(16)
M = {u ∈ H : I(u) = 0} ⊂ H,
and assume further that for every u ∈ M, I 0 (u) never vanishes. Then M is a smooth manifold,
the tangent space Tu to M at u ∈ M is given by
Tu = {v ∈ H : hI 0 (u), vi = 0},
HILBERT’S 16TH
21
and the projection Πu onto Tu is
I 0 (u)
Πu (v) = v − hI (u), vi 0
=
kI (u)k2
0
I 0 (u)
I 0 (u)
1− 0
⊗
v.
kI (u)k kI 0 (u)k
Recall that 1 designates the identity operator. The proof of Lemma 15, as in the finitedimensional situation, is a direct consequence of the implicit function theorem. Hence, even a
more general statement (without the compactness properties) than Lemma 15 is valid. For a
version of the implicit function theorem in a Banach space setting, see (3.1.10) in [5]. For a
similar situation written also in the form of Lemma 15, see Chapter 9 in [28].
The lemma we will be invoking later is the final outcome of all previous statements. The
functional Eε : H → R will be as in Lemma 13, and we will consider the restriction Eε |M of
Eε to the manifold M in (16) associated with a functional I as in Lemma 15.
Lemma 16. Let H be a (real) Hilbert space, and consider the functional Eε : H → R of the
form
ε
Eε (u) = kuk2H + E0 (u)
2
where ε > 0, and E0 : H → R complies with
(i) it is non-negative, and smooth;
(ii) E00 : H → H is a (non-linear) compact operator;
(iii) E000 (u) : H → H is a linear, compact operator for every u ∈ H.
Moreover let a functional I : H → R be as in Lemma 15, with corresponding manifold M as
in (16), and such that the projection of the derivative I 0 (u) onto the tangent space of Mε at u
never vanishes. Assume that Eε |M has a finite number of critical points (in M) independently
of ε. Then Mε ∩ M is a smooth manifold, and
(a) Eε |Mε ∩M is coercive;
(b) Eε |Mε ∩M complies with the Palais-Smale condition;
(c) there are arbitrarily small values of ε such that all critical points of Eε |Mε ∩M are nondegenerate.
Proof. The proof can be settled exactly as that for Lemmas 13 and 14. After all
Eε |Mε ∩M = ( Eε |M )|Mε ,
and derivatives and second derivatives of Eε |M are exactly the composition (once or twice, for
the first and second derivatives, respectively) with the operator
1−
I 0 (u)
I 0 (u)
⊗ 0
0
kI (u)k kI (u)k
which, by our assumption on I 0 , is a compact perturbation of the identity, and I 0 (u), restricted
to the tangent space of Mε , never vanishes.
Note that in our specific case, we take
I(u) = Div(u(0)),
Div(x, y) = Px (x, y) + Qy (x, y),
and
(17)
hI 0 (u), vi = ∇ Div(u(0)) · v(0),
for paths u, and v optimally parameterized with respect to Eε . The derivative I 0 (u) is thus the
2
constant path ∇ Div(u(0)). Because weak convergence in our ambient space Hper
([0, 1]; R2 ) (or
p
Hper
([0, 1]; R2 )) implies strong convergence for u, both I and I 0 comply with the compactness
necessary in Lemma 16. The non-singularity condition ∇ Div 6= 0 is, however, violated at
singular points of the curve Div = 0. Note that constant paths over the curve Div = 0, where
the derivative I 0 (u) is orthogonal to Mε , have already been excluded. Recall the comments at
the end of Section 9. In the polynomial case, which is the main target of our attention in this
22
J. LLIBRE AND P. PEDREGAL
paper, the curve Div = 0 is an algebraic curve, and, as such, it may not be C ∞ in all its points.
Lack of smoothness takes place in singular points of the curve Div = 0 where the gradient in
(17) vanishes. But as has already been indicated, our perturbation results in Section 16, permit
us to focus here on the non-singular case, where all of the connected components of the curve
Div = 0 are smooth curves.
Lemma 16 is a clear statement of the important conditions that we must check to ensure
that the first Morse inequality is valid, and can be used for Eε |Mε ∩M :
(i) E00 : H → H is a (non-linear) compact operator;
(ii) E000 (u) : H → H is a linear, compact operator for every u ∈ H.
(iii) Eε |M has a finite number of critical points (in M) independently of ε.
We next turn to examine the first two conditions in the following two sections, respectively,
while the third one will be a a posteriori consequence of our analysis for the limit process, as
ε & 0, in subsequent sections.
11. The first derivative
We would like to compute the derivative E00 of the functional
Z
1 1 ⊥
(F (u(t)) · u0 (t))2 dt
E0 (u) =
2 0
2
([0, 1]; R2 ). There is no question about the smoothness (Fr´echet differentiability)
for u ∈ Hper
of the functional because all of the functions involved are smooth. To that end, let U ∈
2
Hper
([0, 1]; R2 ) be arbitrary, and we differentiate with respect to δ the section
Z
1 1 ⊥
E0 (u + δU) =
[F (u(t) + δU(t)) · (u0 (t) + δU0 (t))]2 dt,
2 0
and then set δ = 0. This operation would furnish the directional derivative of the functional at
u in the direction U. It is immediate to have
Z 1
0
(18)
hE0 (u), Ui =
(F⊥ (u(t)) · u0 (t))[∇F⊥ (u(t))U(t) · u0 (t) + F⊥ (u(t)) · U0 (t)] dt.
0
By Lemma 34, the vector v = E00 (u) will be the minimizer (with respect to U) of the augmented
functional
Z 1
1
1
1
[ |U00 (t)|2 + |U0 (t)|2 + |U(t)|2 −(F⊥ (u(t))·u0 (t))[∇F⊥ (u(t))U(t)·u0 (t)+F⊥ (u(t))·U0 (t)]] dt.
2
2
0 2
2
Therefore the equation for v ∈ Hper
([0, 1]; R2 ) will be (Theorem 35)
(19)
[v00 ]00 − [v0 + (F⊥ (u) · u0 )F⊥ (u)]0 + v + (F⊥ (u) · u0 )u0T ∇F⊥ (u) = 0 in (0, 1).
2
Lemma 17. Let {uj } be a bounded sequence in Hper
([0, 1]; R2 ), and {vj } the sequence of
0
corresponding derivatives vj = E0 (uj ) which are solutions of (19) for u = uj . Then {vj } is
2
relatively compact in Hper
([0, 1]; R2 ).
Proof. Put
Gj ≡ (F⊥ (uj ) · u0j )F⊥ (uj ),
⊥
Hj ≡ (F⊥ (uj ) · u0j )u0T
j ∇F (uj ),
so that
(20)
(vj00 )00 − (vj0 + Gj )0 + vj + Hj = 0 in [0, 1].
2
Because a certain subsequence of {uj } converges weakly to some u in Hper
([0, 1]; R2 ), and
this means weak convergence up to second derivatives, since the vector fields Gj and Hj only
depend upon at most first derivatives, they do converge strongly (in L2per ([0, 1]; R2 )) to
G ≡ (F⊥ (u) · u0 )F⊥ (u),
H ≡ (F⊥ (u) · u0 )u0T ∇F⊥ (u),
HILBERT’S 16TH
23
respectively. This is nothing but the classical Rellich–Kondrachov theorem which implies that
the injection W 2,p ⊂ W 1,p is always compact. See Theorem 9.16 in [7] for the case with first
derivatives W 1,p ⊂ Lp . By multiplying (20) by vj , and integrating by parts (weak formulation),
we find that
Z 1
[|vj00 |2 + (vj0 + Gj ) · vj0 + (vj + Hj ) · vj ] dt = 0.
0
Hence
kvj k2Hper
2 ([0,1];R2 )
Z
=−
1
(Gj · vj0 + Hj · vj ) dt.
0
In particular, we have
2 ([0,1];R2 ) ≤ kGj kL2 ([0,1];R2 ) + kHj kL2 ([0,1];R2 ) ,
kvj kHper
and {vj } (or an appropriate subsequence) converges weakly. On the other hand, let v = E00 (u)
which must be a solution of (20) corresponding to G and H. As before, one can conclude that
(vj00 − v00 )00 − (vj0 − v0 + Gj − G)0 + vj − v + Hj − H = 0 in [0, 1],
and, similarly,
kvj − vk2Hper
2 ([0,1];R2 ) = −
Z
1
[(Gj − G) · (vj0 − v0 ) + (Hj − H) · (vj − v)] dt,
0
2 ([0,1];R2 ) ≤ kGj − GkL2 ([0,1];R2 ) + kHj − HkL2 ([0,1];R2 ) .
kvj − vkHper
This implies the desired strong convergence vj → v, because Gj − G → 0, and Hj − H → 0 as
well.
As a direct consequence of Lemma 17 we have:
2
2
([0, 1]; R2 ) is compact.
([0, 1]; R2 ) → Hper
Corollary 18. The map E00 : Hper
12. The second derivative
If we have a certain smooth functional E : H → R over a Hilbert space H, with (continuous)
derivative E 0 : H → H, there are various ways to deal with the second derivative, but probably
the best suited for our purposes is to consider the derivative
d 00
hE (u), (U, U)i =
hE 0 (u + δU), Ui,
dδ δ=0
where both vector fields U and U belong to H. In view of (18) we have
Z 1
d 00
hE0 (u), (U, U)i =
(F⊥ (u + δU) · (u0 + δU0 ))[∇F⊥ (u + δU)U · (u0 + δU0 )
dδ δ=0 0
0
+F⊥ (u + δU) · U ] dt
Z 1
=
(F⊥ (u) · u0 )[∇2 F⊥ (u) : (U, U, u0 ) + ∇F⊥ (u) : (U, U0 )
0
0
+∇F⊥ (u) : (U, U )] dt
Z
+
1
0
[∇F⊥ (u)U · u0 + F⊥ (u) · U0 ][∇F⊥ (u)U · u0 + F⊥ (u) · U ] dt
0
Through this long formula, we can understand, for such a u given and fixed, the linear operator
2
2
2
E000 (u) : Hper
([0, 1]; R2 ) → Hper
([0, 1]; R2 ). Let U ∈ Hper
([0, 1]; R2 ) be given. The image
00
2
2
V = E0 (u)U ∈ Hper ([0, 1]; R ) is determined through
(21)
hE000 (u), (U, U)i = hE000 (u)U, Ui = hV, Ui
24
J. LLIBRE AND P. PEDREGAL
2
for all U ∈ Hper
([0, 1]; R2 ). The element V defined through (21) is the solution of a standard
quadratic variational problem, for which the weak form of its optimality condition is precisely
(21). This form is especially suited to show the compactness we are after. Set
F = (F⊥ (u) · u0 )∇2 F⊥ (u) : u0 + ∇F⊥ (u)u0 ⊗ ∇F⊥ (u)u0 ,
G = ∇F⊥ (u) + F⊥ (u) ⊗ ∇F⊥ (u)u0 ,
H = ∇F⊥ (u) + ∇F⊥ (u)u0 ⊗ F⊥ (u),
J = F⊥ (u) ⊗ F⊥ (u).
This choice is dictated so that
Z 1
0
0
[F((U, U) + G(U0 , U) + H(U, U ) + J(U0 , U )] dt.
hE000 (u), (U, U)i =
0
2
Lemma 19. Let F, G, H, J ∈ L2 ((0, 1); R2×2 ), and U ∈ Hper
([0, 1]; R2 ). Consider the quadratic variational problem
Z
1 1 00 2
2
Minimize in V ∈ Hper
([0, 1]; R2 ) :
[|V | +|V0 +U0T J+UT H|2 +|V+U0T G+UT F|2 ] dt.
2 0
There is a unique solution V which is characterized by the identity
Z 1
00
0
(22)
[V00 U + (V0 + U0T J + UT H)U + (V + U0T G + UT F)U] dt = 0 in (0, 1),
0
2
for every U ∈ Hper
([0, 1]; R2 ).
The existence of such a solution is a standard fact in the Calculus of Variations (see Theorem
35 below). Note again how the choice of the vector fields F, G, H, J has been tailored exactly
so that (22) reproduces (21). Exactly as in Lemma 17, one can show the following.
2
([0, 1]; R2 ), the operator U 7→ V = E000 (u)U is selfLemma 20. For fixed, given u ∈ Hper
adjoint and compact.
13. A general result for critical points
We turn next to examine the critical points for Eε . To this goal, we first focus on a general
result that we will apply to our particular situation in the forthcoming section.
The following lemma is a basic but crucial result for us. Once again, we can assume that the
(connected components of the) curve are smooth so the gradient ∇ Div never vanishes when
Div = 0.
Lemma 21. Let F (u, z, Z) : R2 × R2 × R2 → R be a smooth integrand, and suppose that the
functional
Z 1
F (u(t), u0 (t), u00 (t)) dt
0
admits a critical point u : [0, 1] → R2 over the class of feasible paths
A = {v(t) ∈ H 2 ([0, 1]; R2 ) : v(0) = v(1), v0 (0) = v0 (1), φ(v(0)) = 0}
for a given smooth function φ : R2 → R with no critical point on its zero set φ = 0. Assume
that Fu (v, v0 , v00 ) and
Z t
0
00
Fz (v, v , v ) −
Fu (v, v0 , v00 ) ds
0
belong to L1 ((0, 1); R2 ) for every feasible v ∈ A. Then:
HILBERT’S 16TH
25
(a) the vector field
(23)
(24)
d
FZ (u, u0 , u00 ) − Fz (u, u0 , u00 )
dt
is absolutely continuous in (0, 1), and
d
d
FZ (u, u0 , u00 ) − Fz (u, u0 , u00 ) + Fu (u, u0 , u00 ) = 0 a.e. t in (0, 1).
dt dt
(b) The jump in the vector field in (23) at 0 must be parallel to ∇φ(u(0)).
Notice that the integrability demanded on those combinations of partial derivatives of F in
the statement is equivalent to having Fu (v, v0 , v00 ) and Fz (v, v0 , v00 ) integrable for every feasible
v ∈ A. We have however decided to keep the statement as it is for that is exactly the form in
which those combinations of partial derivatives will occur in the proof.
Proof of Lemma 21. Suppose U(δ, t) : [0, δ0 ) × [0, 1] → R2 is a feasible variation for u so that
u(t) + U(δ, t) is admissible for every δ < δ0 , and U(0, ·) ≡ 0. In particular,
(25)
U0 (δ, 0) = U0 (δ, 1),
U(δ, 0) = U(δ, 1),
φ(u(0) + U(δ, 0)) = 0
for all δ. We should have
Z 1
d F (u(t) + U(δ, t), u0 (t) + U0 (δ, t), u00 (t) + U00 (δ, t)) dt = 0.
dδ δ=0
0
This derivative has the form
Z 1
(26)
[Fu (u, u0 , u00 ) · Uδ (0, t) + Fz (u, u0 , u00 ) · U0δ (0, t) + FZ (u, u0 , u00 ) · U00δ (0, t)] dt.
0
Let v(t) ∈ L1 (0, 1; R2 ) be an arbitrary curve with
Z 1
Z 1
(27)
v(t) dt =
tv(t) dt = 0,
0
0
and put
Z
t
(t − s)v(s) ds,
V(t) =
0
Z
V (t) =
0
t
v(s) ds.
0
Then U(δ, t) = δV(t) is an admissible variation according to (25), and bearing in mind (27).
Therefore we should have in (26) for this U(δ, t) (V00 = v) ,
Z 1
(28)
[Fu (u, u0 , u00 ) · V(t) + Fz (u, u0 , u00 ) · V0 (t) + FZ (u, u0 , u00 ) · v(t)] dt = 0.
0
If we also put
Z t
Ψ(t) =
Fu (u(s), u0 (s), u00 (s)) ds,
0
Z
Φ(t) =
t
[−Ψ(s) + Fz (u(s), u0 (s), u00 (s))] ds,
0
two continuous, bounded functions by hypothesis, then an integration by parts in the first term
of (28) yields
Z 1
[−Ψ(t) · V0 (t) + Fz (u, u0 , u00 ) · V0 (t) + FZ (u, u0 , u00 ) · v(t)] dt = 0.
0
A second integration by parts leads to
Z 1
[−Φ(t) + FZ (u, u0 , u00 )] · v(t) dt = 0
0
for all such v. Note that the boundary terms drop out in all those integrations by parts, again
by (27). Because of the arbitrariness of v, except for the restrictions (27), we conclude that
(29)
FZ (u, u0 , u00 ) − Φ = c + Ct in (0, 1),
26
J. LLIBRE AND P. PEDREGAL
with c and C constants. In particular, since Φ is absolutely continuous (it belongs to W 1,1 ((0, 1);
R2 )), we know that FZ (u, u0 , u00 ) must be absolutely continuous too in (0, 1), and as such, it
cannot have jumps in (0, 1), though it can possibly have at the end-points. By differentiating
once in (29) with respect to t,
d
FZ (u, u0 , u00 ) − Fz (u, u0 , u00 ) + Ψ = C in (0, 1),
dt
and even further
d
d
FZ (u, u0 , u00 ) − Fz (u, u0 , u00 ) + Fu (u, u0 , u00 ) = 0 in (0, 1).
dt dt
If we now take this information back to (26) for an arbitrary variation U(δ, t), perform several
suitable integrations by parts on the various terms, and recall the periodicity conditions for U
in (25), we are only left with the contributions on the boundary points, and obtain
d
FZ (u, u0 , u00 ) − Fz (u, u0 , u00 )]t=0 · Uδ (0, 0) − [FZ (u, u0 , u00 )]t=0 · U0δ (0, 0) = 0,
dt
where brackets indicate jumps at the given time. Since the vector U0δ (0, 0) can be chosen
arbitrarily, and independently of Uδ (0, 0), we conclude that
[
[FZ (u, u0 , u00 )]t=0 = 0,
and
d
FZ (u, u0 , u00 ) − Fz (u, u0 , u00 )]t=0 · Uδ (0, 0) = 0.
dt
On the other hand, by differentiating with respect to δ in (25) and putting δ = 0, we know that
(30)
[
∇φ(u(0)) · Uδ (0, 0) = 0.
(31)
Comparing (30) with (31) leads to the conclusion that this vector ∇φ(u(0)) must be parallel
to the jump
(32)
[
d
FZ (u, u0 , u00 ) − Fz (u, u0 , u00 )]t=0 .
dt
14. Perturbation
We would like to apply Lemma 21 to our particular situation in which we take φ ≡ Div,
F (u, z, Z) = F 0 (u, z) + εF (Z)
for small ε, and
2
1 2
1 ⊥
|Z| , F 0 (u, z) =
F (u) · z .
2
2
The partial derivatives involve in optimality conditions are
F (Z) =
Fu = (F⊥ (u) · z)∇F⊥ (u)z,
Fz = (F⊥ (u) · z)F⊥ (u),
FZ = εZ.
The integrability required in Lemma 21 is easy to check, because vector fields in
A = {v(t) ∈ H 2 ([0, 1]; R2 ) : v(0) = v(1), v0 (0) = v0 (1), φ(v(0)) = 0}
are smooth. There is no difficulty here.
The combination of partial derivatives in the optimality system (23) is
(33)
εu000 − (F⊥ (u) · u0 )F⊥ (u).
HILBERT’S 16TH
27
Notice that the term on the right is absolutely continuous in all of [0, 1] for u ∈ A, and so the
condition about the absolute continuity of that combination in Lemma 21 implies that u000 is
also absolutely continuous. Thus u0000 exists for a.e. t ∈ (0, 1). Then equation (24) becomes
d
εu0000 −
(F⊥ (u) · u0 )F⊥ (u) + (F⊥ (u) · u0 )∇F⊥ (u)u0 = 0
dt
for a.e. t ∈ (0, 1). Concerning the jump condition, keep in mind again that the terms in (33)
are continuous except u000 . Hence, according to Lemma 21,
(34)
[u000 ]t=0 k ∇ Div(u(0)).
Since we would like to understand the limit as ε & 0, we put explicitly the ε-dependence
into the possible critical paths. Let uε be a critical point of our perturbed functional, so that
d
(F⊥ (uε ) · u0ε )F⊥ (uε ) + (F⊥ (uε ) · u0ε )∇F⊥ (uε )u0ε = 0 for a.e. t ∈ (0, 1),
(35)
εu0000
ε −
dt
and
⊥
0
⊥
[εu000
ε − (F (uε ) · uε )F (uε )]t=0 k ∇ Div(uε (0)).
We have also concluded above that [u000
ε ]t=0 should also be parallel to ∇ Div(uε (0)), and so we
can certainly put
(36)
[(F⊥ (uε ) · u0ε )F⊥ (uε )]t=0 k ∇ Div(uε (0)).
In particular, we will keep in mind the possibility that the jump at t = 0 may vanish, but we
will not write down explicitly this possibility for the sake of neatness in the explanation.
We are facing a singular perturbation problem. It is well-known (see for instance [37]) that
the solutions of (35) will locally converge, as ε & 0, to the solutions of the unperturbed system,
and only in negligible intervals the perturbed solutions may either escape to infinity, or else
exhibit localized concentration for the derivatives (layers). We shall try to understand how
these two possibilities can occur.
Suppose that in a certain fixed subinterval J of [0, 1] all the derivatives up to order two
are uniformly bounded with respect to ε (in J). Then, from (35) in such a subset J, uε will
converge to a solution u of the unperturbed system
d
(37)
− (F⊥ (u) · u0 )F⊥ (u) + (F⊥ (u) · u0 )∇F⊥ (u)u0 = 0 for a.e. t ∈ J.
dt
We have a look at this system in a more careful way than we did earlier in Section 2. To this
aim, and for the sake of transparency, we will use the notation
F=
u=
Z ≡ F⊥ (u) · u0 =
W ≡ F(u) · u0 =
(P, Q),
(x, y),
P (x, y)y 0 − Q(x, y)x0 ,
P (x, y)x0 + Q(x, y)y 0 .
Note that Z 2 is precisely the integrand for E0 . System (37) is recast as
(ZQ)0 + Z(−Qx x0 + Px y 0 ) = 0,
(ZP )0 + Z(Qy x0 − Py y 0 ) = 0.
After some elementary, convenient manipulation of these two equations, system (37) becomes
Z 0 (P 2 + Q2 ) = −ZW Div,
Z 2 Div = 0.
By multiplying the first equation by Z, we can conclude that Z 2 is constant (because equilibria
of the initial differential system (8) are assumed to be isolated), and so we have two possibilities:
(i) Z 2 = 0: in this case the critical path is part of an integral curve of the differential
system (recall that Z 2 is precisely the integrand of the error functional E0 );
(ii) Z 2 6= 0: in this case, the second equation ensures that this portion of the critical path
should be entirely contained in the manifold Div = 0.
28
J. LLIBRE AND P. PEDREGAL
But since the manifold Div = 0 cannot contain a non-negligible part of an integral curve,
because solutions of system (2) correspond to contact points and we have assumed that there is
a finite number of them, we conclude that these two possibilities cannot coexist in a connected
interval J where we have the uniform convergence assumed earlier. On the other hand, if we
recall the discussion in Subsection 8.1, we see that globally on the pairwise disjoint union of
all possible such subintervals J where uniform convergence of paths and their derivatives take
place, the value of Z 2 should be constant, and the same constant for all of them. This means
that the image of a critical path with strictly positive error should be entirely contained in the
manifold Div = 0, and Z 2 does never vanish. Remember that Z 2 cannot vanish either due to
the fact that u is a constant path, because those were excluded in the manifold Mε .
We now turn to the situation in which second derivatives {u00ε } concentrate at a certain time
t = t0 , so that first derivatives {u0ε } show a jump at t = t0 , but the jump of Z = F⊥ (u) · u0
remains bounded according to the discussion in the preceding paragraph. Both at the left and
right of t0 , we can apply the previous argument for the case in which derivatives are uniformly
bounded, and so by continuity of u, we should have Div(u(t0 )) = 0. We can, in addition,
perform a translation in time and assume, without loss of generality, that in fact t0 = 0, so that
the jump on the first derivative at any time is actually governed by the limit of (36) as ε & 0.
Since Z = F⊥ (u) · u0 is bounded at t = 0, though discontinuous, we conclude from (36) that
[(F⊥ (u) · u0 )F⊥ (u)]t=0 k ∇ Div(u(0)),
if uε → u around 0. Because of the continuity of u, this last condition translates into
(38)
F⊥ (u(0))k ∇ Div(u(0)) ⇔ F(u(0)) · ∇ Div(u(0)) = 0.
We therefore see that the possible jumps for critical paths can only take place at contact points
of the manifold Div = 0 where the vector field F is tangent to it.
Finally, we claim that discontinuities of u0 , if they occur, can only take place at t = 0, and
t = 1/2. This is so because Z 2 = (F⊥ (u) · u0 )2 must be constant over the curve Div = 0, and
u is periodic. Since u is continuous, and so is F⊥ (u), the derivative u0 is in charge of taking u
forward and backwards, in a similar time interval.
We can summarize our discussion in the following result. It is just a more precise statement
of Theorem 3.
Theorem 22. Let {uε } be a sequence of solutions to (35) together with (36), and put uε → u
pointwise. Then u is locally absolutely continuous, and the following statements hold:
(a) Div(u(t)) = 0 for every t ∈ [0, 1], except possibly for t = 0 (t = 1), t = 1/2 when
u → ∞.
(b) The derivative u0 never vanishes, and its discontinuities can only take place at the two
times t = 0 (t = 1), and t = 1/2.
(c) Possible finite jumps on the derivative u0 can only take place at points of the curve
Div = 0 where the vector field F is tangent to it.
15. Bifurcation and multiplicity
We are concerned in this section about the possibility that various branches of the set of
critical paths, for ε positive, may coalesce into the same limit u. This issue is also crucial for
our strategy.
We will be using a classical, bifurcation result for simple multiplicity that we have taken
from [5], Theorem 4.1.12, appropriately restated for our situation.
Theorem 23. Let X and Y be Banach spaces, and F(v, ε) : X × [0, ε0 ) → Y, a C 2 -mapping
of all its arguments. Suppose that:
(i) F(vε , ε) = 0 for all ε ∈ [0, ε0 ).
(ii) Fv (v0 , 0) : X → Y is a Fredholm operator of index zero.
HILBERT’S 16TH
29
(iii) The dimension of the kernel of Fv (v0 , 0) is one.
(iv) For the kernel K, and range R, of Fv (v0 , 0), it is true that Fv,ε (v0 , 0)K ∩ R = {0}.
Then the pair (v0 , 0) is a bifurcation point for the equation F(v, ε) = 0, and there is exactly
one continuous curve of non-trivial solutions (in addition to (vε , ε)) bifurcating from (v0 , 0).
In the particular situation when there is only one space involved H = X = Y, and it is
a Hilbert space, for each ε; the operator Fv (·, ε) is self-adjoint; and Fv,ε (·, ε) is the identity
operator, then the hypotheses of the above theorem essentially reduce to ensuring that the
dimension of the kernel of Fv (v0 , 0) is one.
Corollary 24. Let F(v, ε) : H × [0, ε0 ) → H be a C 2 -mapping of all its arguments such that
(i) F(vε , ε) = 0 for all ε ∈ [0, ε0 ),
(ii) Fv (·, ε) is self-adjoint for all ε,
(iii) Fv,ε (·, ε) is the identity operator for every ε, and
(iv) the dimension of the kernel of Fv (v0 , 0) is one.
Then the pair (v0 , 0) is a bifurcation point for the equation F(v, ε) = 0, and there is exactly
one continuous curve of non-trivial solutions (in addition to (vε , ε)) bifurcating from (v0 , 0).
Note that under the hypotheses assumed, it is a standard result in Functional Analysis that
K = R⊥ , the orthogonal complement of R (Corollary 2.18 in [7]). The subspaces K, and R
are given in the statement of Theorem 23. This fact easily implies the hypothesis of Fv (v0 , 0)
being a Fredholm operator of index zero, as well as the fact that
Fv,ε (v0 , 0)K ∩ R = {0},
if Fv,ε (v0 , 0) is the identity operator.
In our situation the operator F(v, ε) is the derivative Eε0 (v) restricted to an appropriate
tangent space corresponding to the intersection of the manifolds Mε , and M (recall the notation
in Section 10). Because Mε is a natural manifold (and it does not generate multipliers), and
the projection onto the tangent space of M is a compact perturbation of the identity, a version
of Corollary 24 is valid in our setting. This is similar to Lemma 16. Note that Fv is a second
derivative of a smooth functional, and as such it is a self-adjoint operator, and that Fv,ε (·, ε) is
the identity operator because the perturbation of the functional E0 involves the square of the
norm in the underlying space.
There is a further point that deserves some comment. Because our situation is a singular
perturbation case, the limit space for ε = 0 is not the same. Said differently, it is a situation of
bifurcation at infinity. However, the situation can be easily solved by working in subintervals
1
J ⊂ [0, 1] exhausting the unit interval, where the ambient space can be taken to be Hper
(J; R2 ).
Alternatively, one can argue by contradiction assuming that there are various branches emanating from a limit path u, and reaching a contradiction with the fact that the dimension of
the appropriate kernel would be greater than one.
In short, we see that the main point in treating the multiplicity issue is to ensure that the
dimension of the kernel of the second derivative E000 (u) at one of the limit paths u described in the
1
preceding section is one, taking for granted, without loss of generality, that u ∈ Hper
([0, 1]; R2 ).
2
Proposition 25. Let u ∈ Hper
([0, 1]; R2 ) be one of the limit paths described in Lemma 22 of
Section 11. The dimension of the kernel of the linear operator E000 (u) is one.
Proof. We go back to the expression
Z 1
0
hE000 (u), (U, U)i =
(F⊥ (u) · u0 )[∇2 F⊥ (u) : (U, U, u0 ) + ∇F⊥ (u) : (U, U0 ) + ∇F⊥ (u) : (U, U )] dt
0
Z
(39)
+
0
1
0
[∇F⊥ (u)U · u0 + F⊥ (u) · U0 ][∇F⊥ (u)U · u0 + F⊥ (u) · U ] dt
30
J. LLIBRE AND P. PEDREGAL
for the second derivative of E0 at one limit u of critical paths, as in the preceding section. This
identity was computed at the beginning of Section 12. Both vector fields U and U are periodic,
and regarded as perturbations of u.
According to our conclusions in Section 14, we have that u complies with
Div(u) ≡ 0,
(40)
[F⊥ (u) · u0 ]2 ≡ c.
Note that this is also coherent with the conclusion at the end of Subsection 8.1. But even more
is true. Because the perturbations U and U must respect belonging to the natural manifold
Mε in the limit when ε & 0, according to (13) we must have
Div(u(0) + δU(0)) = 0,
⊥
[F (u + δU) · (u0 + δU0 )]2 = c(δ),
for δ small, and the same for U, with a different function c(δ). By differentiating with respect
to δ, we arrive at
∇ Div(u(0)) · U(0) = 0,
(41)
2(F⊥ (u) · u0 )[∇F⊥ (u)U · u0 + F⊥ (u) · U0 ] = c0 (0).
The second identity implies that the two factors occurring in the second integral of the second
derivative above (39) is actually a constant, and can be, without loss of generality, assumed to
be zero for our purposes (see Appendix III in Section 19).
We can therefore write
Z 1
00
(F⊥ (u) · u0 )·
hE0 (u), (U, U)i =
(42)
0
0
[∇2 F⊥ (u) : (U, U, u0 ) + ∇F⊥ (u) : (U, U0 ) + ∇F⊥ (u) : (U, U )] dt.
Suppose U as above is such that (42) vanishes for all U. We then claim the following.
Claim 26. U must be parallel to u0 .
The proof of this claim is a consequence of some long and tedious computations that have
been reproduced in Appendix III to avoid any interruption at this stage.
Put, hence, U(t) = α(t)u0 (t) for some scalar α(t). Lemma 22 informed us that the jump
discontinuities of u0 can only happen at t = 0, and t = 1/2, and, because of the continuity
of U, this is so for α(t) as well. Therefore in the two subintervals (0, 1/2) and (1/2, 1) all the
functions involved are smooth, and, moreover, from (41) and (40) we have
∇F⊥ (u)U · u0 + F⊥ (u) · U0 ≡ c,
(43)
for some constant c. But
U = αu0 =⇒ U0 = α0 u0 + αu00 ,
and
(44)
F⊥ (u) · u0 = c =⇒ ∇F⊥ (u)u0 · u0 + F⊥ (u) · u00 ≡ 0.
Substituting those expressions of U, and U0 into (43), and comparing with (44), we conclude
that
α0 F⊥ (u) · u0 ≡ c.
The factor (F⊥ (u) · u0 )2 is the integrand for E0 that, once again, must be a non-vanishing
constant for a critical point that is not an absolute minimizer. We therefore conclude that
(α0 )2 ≡ c. If we take into account the additional restriction expressed in the first identity of
(41), and from (40) ∇ Div(u) · u0 = 0 for all t, we see that the value of α(0) is the unique free
parameter that remains for U. This show that the dimension of the kernel of E000 (u) is one, as
desired.
HILBERT’S 16TH
31
This proposition enables us to apply Corollary 24, and conclude that at most two critical
points of Eε may coalesce into a single limit as ε & 0. This ambiguity corresponds precisely
to the way in which critical paths uε are run: either clockwise, or counter-clockwise. Since this
same ambiguity has also to be taken into account exactly in the same way, as already remarked
earlier, for limit cycles, it can be ignored altogether when performing the counting of critical
paths.
16. Singular systems
In this section, we focus on the treatment of singular differential systems for which we allow
the curve Div = 0 to have singular points so that the overdetermined system (7)
(45)
Pxx + Qyx = Pxy + Qyy = 0,
Px + Qy = 0
has some (finite number of) solutions. Our argument revolves around the idea that one such
system can be uniformly approximated by a sequence of non-singular systems. One possibility
is to perturb such singular system in the form
Fδ (x, y) = (P (x, y), Q(x, y)) − δ(x, y)
for δ > 0 and small. Note that F and Fδ share the set of singular points for their respective
divergences, while Divδ = 0 is the curve Div = 2δ. For δ small, we therefore destroy the
solutions of (45). But one is not in the obligation of using such linear perturbation, as the
result we invoke below is much more general. We just write Fδ for one such perturbation of F.
We can definitely apply our previous results to the family of functionals
Z
Z
1 1 ⊥
ε 1 00 2
0
2
(46)
Eδ (u) =
(Fδ (u(t)) · u (t)) dt +
|u (t)| dt.
2 0
2 0
We have not used explicitly the subindex ε in the functional to emphasize that we regard ε as
given and fixed in the process of moving δ.
The issue is: can new critical paths be generated in the limit as δ & 0? Since the degree
of Fδ does not depend on δ, in the case F is a polynomial system of a certain degree, by our
previous results for non-singular systems, we would have a uniform (in δ) upper bound for the
number of critical points of Eδ . Because the perturbation is, this time, a regular perturbation
affecting only to lower order terms (δ does not affect second derivatives), the straightforward
conclusion is that, in the limit, that upper bound cannot be increased, see Proposition 28.
More specifically, we can resort to Theorem 6.2 in [9]. Right after this result, in that same
reference, we find the following assertion as a consequence of that theorem.
Proposition 27. The Morse type numbers are lower semicontinuous under C 0 -perturbations,
provided all of the mappings involved are coercive and comply with the Palais-Smale property.
This Morse type numbers count the number of critical points of a certain Morse index k. So
they are exactly numbers Mk in Appendix 17 below.
Proposition 28. Let ε > 0 be fixed, and let Fδ be a C 0 -perturbation of F
(47)
kFδ − Fk → 0 as δ & 0
over arbitrary compact subsets of R2 . Then if Mδ , and M0 are the (finite) number of critical
paths for Eδ in (46), and E0 with positive value, respectively, then M0 ≤ lim inf δ→0 Mδ .
Proof. The proof is a direct application of Proposition 27. Note that under (47), Eδ is indeed a
C 0 -perturbation of E0 as functionals, as required by Proposition 27, because in our perturbations
Eδ , we do not change the higher-order term depending on second derivatives (recall that ε is
kept constant here).
32
J. LLIBRE AND P. PEDREGAL
This proposition ensures that the first Morse inequality is preserved as δ & 0 in the sense
M00 ≤ M0 + 1 ≤ lim inf Mδ + 1,
(48)
δ→0
if M00 stands for the number of zeroes of E0 (the number of limit cycles for the differential
system determined by F). If we have a uniform upper bound for Mδ , that same bound is also
valid for the number of limit cycles in the limit according to (48). We can then proceed with
the analysis of the limit passage when ε & 0 as in Sections 14, and 15.
17. Appendix I. Morse inequalities
The statement of the Morse inequalities involves the notion of Morse index for a nondegenerate, critical point of a smooth functional. Suppose E : H → R is a non-negative,
coercive, C 2 -functional. Let u ∈ H be a critical point E 0 (u) = 0.
Definition 17.1. We say that the critical point u is non-degenerate if the linear mapping
E 00 (u) : H → H is non-singular. If the number of negative eigenvalues of E 00 (u) is finite, such
number is called the (Morse) index of u.
Notice that local minima have index zero. Corollary (6.5.10) of [5] reads as follows.
Corollary 29. Let E be as above, and enjoy the Palais-Smale property. Suppose E has a finite
number Mk of critical points of each fixed index k. Then
∞
X
M0 ≥ 1, M1 − M0 ≥ −1, M2 − M1 + M0 ≥ 1, . . . ,
(−1)k Mk = 1.
k=0
The zero-th order Morse inequality just states that M0 ≥ 1, i.e., a non-negative, coercive,
smooth functional enjoying the Palais-Smale property always has at least one (global) minima.
The first Morse inequality ensures that M1 − M0 ≥ −1, or M1 ≥ M0 − 1.
Corollary 30. Let the functional E be as in Corollary 29.
• If M and M0 are the number of critical points of E that are not local minima, and
those that are local minima, respectively. Then M0 ≤ M + 1.
• If M and M 0 are the number of critical points of E with a positive value, and those
with a vanishing value, respectively, then M 0 ≤ M + 1.
Proof. The first part comes directly from the first Morse inequality. For the second part notice
that M 0 ≤ M0 , and that critical points with a positive index cannot have a vanishing value
for E, because, as indicated above, global minima with E = 0 have necessarily zero index.
Therefore
∞
X
M=
Mk + (M0 − M 0 ) ≥ M1 ≥ M0 − 1 ≥ M 0 − 1.
k=1
Corollary 30 is sometimes called the generalized mountain-pass lemma, as it can be intuitively
stated by saying that the number of (mountain) passes is at least the number of valleys minus
one.
The generalization of these basic results to infinite dimensional Hilbert manifolds is treated
in places like [9].
18. Appendix II. Variational problems in dimension one.
The results in this section are quite standard, and can be found in many sources. See [15],
[22], [28].
Theorem 31. Suppose the density F (u, ξ, η) : R3 → R is such that:
(i) it is continuous, and convex in η for each fixed pair (u, ξ);
HILBERT’S 16TH
33
(ii) it is coercive in the sense C(|η|2 − 1) ≤ F (u, ξ, η) for some constant C > 0, and every
(u, ξ, η).
Then the variational problem
Z 1
2
F (u(s), u0 (s), u00 (s)) ds
Minimize in u(s) ∈ H (0, 1) : I(u) =
0
under the constraint u(0) = 0, u(1) = 1, admits an optimal solution. If the convexity, is
improved to strict convexity for all such pairs (u, ξ), the optimal solution is unique.
Proposition 32. Assume the density G(U, ξ, η) : RN ×R×R → R is smooth, strictly convex in
η for all pairs (U, ξ), and coercive in the sense of the preceding statement. For each U : R → RN
in H 2 (0, 1), consider the integrand F (u, ξ, η) = G(U(u), ξ, η) : R3 → R. Then the unique
minimizer whose existence is guaranteed in the previous theorem depends smoothly on U.
Proposition 33. Let F (u, ξ) : R2 → R comply with the convexity, and coercivity conditions as
in the previous statements. Suppose it is also smooth. Then the optimal function that minimizes
the integrand F (u(t), u0 (t)) under suitable end-point conditions on [0, 1], must satisfy
F (u(t), u0 (t)) − u0 (t)Fξ (u(t), u0 (t)) = constant in (0, 1).
This is sometimes called the second form of the Euler-Lagrange equation.
18.1. The derivative of a functional.
Lemma 34. Let H be a Hilbert space, and E : H → R, a smooth functional. Suppose we
have a way to compute the directional derivative hE 0 (u), vi for a certain u ∈ H, and for every
v ∈ H. Then the derivative v = E 0 (u) is the unique minimizer of the problem
1
kvk2 − hE 0 (u), vi.
2
18.2. The direct method of the Calculus of Variations for vector fields problems.
Minimize in v ∈ H :
Theorem 35. Let F (t, u, z, Z) : R×RN ×RN ×RN → R enjoy the two fundamental properties:
(i) Coercivity: there is an exponent p > 1, so that C(|Z|p − 1) ≤ F (t, u, z, Z) for all
(t, u, z, Z) ∈ R × RN × RN × RN .
(ii) Convexity: F is continuous, and strictly convex in Z for every triplet (t, u, z) ∈ R ×
R N × RN .
Let A be a weakly-closed subset of W 1,p ((0, 1); RN ). Then the variational problem
Z 1
Minimize in u ∈ A :
F (t, u(t), u0 (t), u00 (t)) dt
0
admits a unique minimizer in A. If, in addition, F is smooth, the unique minimizer u ∈ A will
be a solution of the system
d2
d
FZ (t, u, u0 , u00 ) − Fz (t, u, u0 , u00 ) + Fu (t, u, u0 , u00 ) = 0 in (0, 1).
dt2
dt
19. Appendix III. Some calculations.
We compute explicitly here the second derivative of the functional
Z 1
1 ⊥
E0 (u) =
(F (u) · u0 )2 dt,
2
0
for a path u which is a limit of critical paths of the perturbed functionals Eδ = E0 + δE, and
Z
1 1
E(u) =
|u00 |2 + |u0 |2 + |u|2 dt.
2 0
34
J. LLIBRE AND P. PEDREGAL
We in fact know (40) that u is such that
Div(u) ≡ 0,
(49)
(F⊥ (u) · u0 )2 = c.
In particular, due to the periodicity and the fact that that F⊥ (u) is continuous, we can conclude
that in half of the unit interval F⊥ (u)·u0 is a constant, and in the other half the value of F⊥ (u)·u0
is just the opposite. The first derivative is
d E0 (u + δU)
hE00 (u), Ui ≡
dδ δ=0
1
for U ∈ Hloc,per
([0, 1]; R2 ). It is found easily that
Z 1
0
hE0 (u), Ui =
(F⊥ (u) · U) ∇F⊥ (u)U · u0 + F⊥ (u) · U0 dt.
0
If we are interested in a more explicit form, we would put
F = (P, Q),
F⊥ = (−Q, P ),
u = (x, y),
U = (X, Y ),
and
hE00 (u), Ui
Z 1
d 1
=
(−Q(x + δX, y + δY )(x0 + δX 0 ) + P (x + δX, y + δY )(y 0 + δY 0 ))2 dt.
dδ δ=0 0 2
A careful computation yields
Z 1
(50) hE00 (u), Ui =
(−Qx0 + P y 0 ) [−(Qx X + Qy Y )x0 − QX 0 + (Px X + Py Y )y 0 + P Y 0 ] dt,
0
where P , Q, and its derivatives are evaluated at (x, y). For the second derivative, we further
put U = (X, Y ), and
d 00
(51)
hE0 (u), (U, U)i =
hE 0 (u + δU), Ui.
dδ δ=0 0
In performing this further derivative, we first notice that because (−Qx0 + P y 0 )2 is a constant,
according to (49) and hence the same factor without the square is piecewise-constant, we only
need to compute the derivative of the second factor in the integrand in (50). The derivative of
this second factor in the integrand for this second derivative in (50) is
h
0
−(Qx (x + δX, y + δY )X + Qy (x + δX, y + δY )Y )(x0 + δX 0 ) − Q(x + δX, y + δY )X +
i
0
(Px (x + δX, y + δY )X + Py (x + δX, y + δY )Y )(y 0 + δY 0 ) + P (x + δX, y + δY )Y .
Differentiation with respect to δ, and setting δ = 0 afterwards, leads to
−(Qxx (x, y)XX + Qxy (x, y)Y X + Qyx (x, y)XY + Qyy (x, y)Y Y )x0 −
0
0
(Qx (x, y)X + Qy (x, y)Y )X 0 − (Qx (x, y)XX + Qy (x, y)Y X +
(Pxx (x, y)XX + Pxy (x, y)Y X + Pyx (x, y)XY + Pyy (x, y)Y Y )y 0 +
i
0
0
(Px (x, y)X + Py (x, y)Y )Y 0 + (Px (x, y)XY + Py (x, y)Y Y .
Reorganizing terms, we can write
(Pxx Xy 0 + Pxy Y y 0 + Px Y 0 − Qxx Xx0 − Qxy Y x0 − Qx X 0 )X+
(Pyx Xy 0 + Pyy Y y 0 + Py Y 0 − Qyx Xx0 − Qyy Y x0 − Qy X 0 )Y −
0
0
(Qx X + Qy Y )X + (Px X + Py Y )Y .
HILBERT’S 16TH
35
The second derivative in (51) may be explicitly cast in the form
Z 1
(−Qx0 + P y 0 )×
hE000 (u), (U, U)i =
0
[(Pxx Xy 0 + Pxy Y y 0 + Px Y 0 − Qxx Xx0 − Qxy Y x0 − Qx X 0 )X+
(Pyx Xy 0 + Pyy Y y 0 + Py Y 0 − Qyx Xx0 − Qyy Y x0 − Qy X 0 )Y −
0
0
(Qx X + Qy Y )X + (Px X + Py Y )Y ] dt.
Proof of Claim 26. Suppose that u = (x, y) is a critical path for E0 , and the vector field
U = (X, Y ) is such that the vector field U = (X, Y ) is 1-periodic, the previous integral
vanishes. In particular, by taking U appropriately so as to compensate for the constant on
the second derivative coming from the second integral in (39), and the contribution on those
times where Qx0 + P y 0 changes sign, according to the observation made earlier concerning the
constancy of this expression, we can perform an integration by parts that does not involve that
first factor Qx0 + P y 0 (because it is constant), and find
Z 1
[(Pxx Xy 0 + Pxy Y y 0 + Px Y 0 − Qxx Xx0 − Qxy Y x0 − Qx X 0 )X+
0
(Pyx Xy 0 + Pyy Y y 0 + Py Y 0 − Qyx Xx0 − Qyy Y x0 − Qy X 0 )Y +
(Qx X + Qy Y )0 X − (Px X + Py Y )0 Y ] dt = 0,
for all such U. The arbitrariness of U leads to having
(Pxx Xy 0 + Pxy Y y 0 + Px Y 0 − Qxx Xx0 − Qxy Y x0 − Qx X 0 ) + (Qx X + Qy Y )0 ≡ 0,
(Pyx Xy 0 + Pyy Y y 0 + Py Y 0 − Qyx Xx0 − Qyy Y x0 − Qy X 0 ) − (Px X + Py Y )0 ≡ 0.
A careful calculation with the derivatives involved, and remembering that Div(u) ≡ 0 according
to (49), we can write this system in compact form
∇ Div(u) · U u0 ≡ 0.
Because u0 cannot vanish if c in (49) is not the zero constant, we can conclude that indeed
∇ Div(u) · U vanishes, and so U should be parallel to u0 .
20. Appendix IV. A direct treatment of singular systems
It is possible to deal with singular differential systems, i.e. those for which the curve Div = 0
has some singular points, directly without resorting to a perturbation argument as in Section
16. Though that perturbation argument is quite straightforward and rather clear, it hides how
the counting of critical paths takes place for those systems. It is our aim in this Appendix to
examine directly that class of non-generic differential systems.
In the case of a polynomial system, the curve Div = 0 is an algebraic curve of a certain finite
degree. As such, it may have a finite number of singular points. See Figure 4 for a specific
example with various singular points.
Theorem 3 is still valid. The difficulty arises when trying to count critical paths through
this theorem. In the non-singular case, connected components of the curve Div = 0 where
homeomorphic to an oval or to a line, and this fact simplified a lot the counting procedure.
However, in a singular situation, like the one in Figure 4, it is much more delicate. In fact,
if some of those singular points turn out to be contact points, and there are several branches
of the curve passing through them, the counting procedure can apparently yield many more
possibilities compared to a non-singular situation. In fact, we now know that it cannot be so,
but showing a direct argument is more involved, and more restrictive than the one in Section
16, because it requires to think in terms of a rather intriguing assumption, which we have called
the identification property.
36
J. LLIBRE AND P. PEDREGAL
Figure 4.
Definition 20.1.
(a) Let φ : R2 → R be a smooth function. We say that it complies with
the identification property if there is a positive integer p, so that whenever two curves
ui (t) : (−δ, δ), i = 1, 2, δ > 0, belong to C p , u1 (t) = u2 (t), for −δ < t ≤ 0 (or for
0 ≤ t < δ), u0i (0) does not vanish, and φ(ui ) ≡ 0, then they are identical u1 ≡ u2 .
(b) The differential system (6) is said to enjoy the identification property if its divergence
φ ≡ Div = Px + Qy does.
This property plays a crucial role in the counting of critical paths for the functional Eε in the
case of a singular system. It basically says that if we can restrict our analysis to smoother paths,
then feasible, critical paths through singular points cannot run through different branches. We
would like to make this rough idea rigorous.
We start by recalling our remarks at the end of Section 6, where we stressed how one could
perturb the error functional E0 by a small multiple of the square of some higher derivative
Z
1 1 p) 2
E(u) =
|u (t)| dt,
2 0
p
for p as high as we may need it to be. The class of paths would then be Hper
([0, 1]; R2 ). By
taking p directly related to the exponent p in the identification property, we can ensure that
our feasible paths will have the required smoothness.
We would then have to redo some of our previous results. Most of them are just simple generalizations of the p = 2 counterpart. But some others require further considerations. The first
class of statements include essentially those in Sections 8 (with an appropriate generalization
of Theorem 35), 9, 10, 11, and 12.
HILBERT’S 16TH
37
20.1. A refined result. Our basic Lemma 21 is probably the result that deserves a more clear
statement about the changes that should be introduced. For the sake of notation, given a
one-dimensional manifold in R2 determined by the equation φ = 0 for a smooth function φ, we
put T(x) for the tangent space at x, provided x is not a singular point of the manifold so that
∇φ(x) is not the null vector. If x is a singular point where this gradient vanishes, then we put
T(x) = {v ∈ R2 : vτ → v, xτ → x, vτ ∈ T(xτ ), |vτ | = 1},
the set of accumulation unit tangent vectors of that manifold at that singular point, i.e. the
“geometrical” tangent space.
Lemma 36. Let F (u, z, Z) : R2 × R2 × R2 → R be a smooth integrand, and p, an arbitrary
positive integer. Suppose that the functional
Z 1
F (u(t), u0 (t), up) (t)) dt
0
admits a critical point u : [0, 1] → R2 over the class of feasible paths
A = {v(t) ∈ H p ([0, 1]; R2 ) : vq) (0) = vq) (1), q = 0, 1, . . . , p − 1, φ(v(0)) = 0}
for a given smooth function φ : R2 → R. Assume that Fu (v, v0 , vp) ) and
Z t
0
p)
Fz (v, v , v ) −
Fu (v, v0 , vp) ) ds
0
1
2
belong to L ((0, 1); R ) for every feasible v ∈ A. Then:
(a) the vector field
(52)
dp−1
FZ (u, u0 , up) ) − Fz (u, u0 , up) )
dtp−1
is absolutely continuous in (0, 1), and
p−1
d
d
0
p)
0
p)
FZ (u, u , u ) − Fz (u, u , u ) + Fu (u, u0 , up) ) = 0 a.e. t in (0, 1).
dt dtp−1
(b) If T(u(0)) contains at least two independent directions, then the vector field in (52)
is continuous in all of [0, 1] (no jump at the end-points). On the contrary, if T(u(0))
contains just one direction, which coincides with ∇φ(u(0)) when the manifold φ = 0 is
regular at u(0), then the jump in the vector field in (52) at 0 must be parallel to this
single direction.
Proof. The proof of this new lemma follows along the line of the one for Lemma 21. The only
change involves the choice of v, and V
Z 1
tj v(t) dt = 0, j = 0, 1, 2, . . . , p − 1,
0
Z t
1
V(t) =
(t − s)p−1 v(s) ds,
(p − 1)! 0
so that
1
V (t) =
(p − k − 1)!
k)
t
Z
(t − s)p−k−1 v(s) ds,
k = 0, 1, . . . , p − 1,
0
with
V
p−1)
Z
(t) =
t
v(s) ds.
0
Some successive integrations by parts need to be taken into account with a bit of care.
Concerning the jump condition in (b) of the statement, we can have two different situations.
38
J. LLIBRE AND P. PEDREGAL
(i) Suppose the vector ∇φ(u(0)) is not the null vector. In this case, just as in the proof of
Lemma 21, we conclude that ∇φ(u(0)) must be parallel to the jump of the field in (52)
at t = 0.
(ii) If the vector ∇φ(u(0)) turns out to be zero so that the one-dimensional manifold determined by the equation φ = 0 is singular at u(0), we consider the set T(u(0)) of
accumulation unit tangent vectors of that manifold at that singular point. If T(u(0))
contains at least two independent vectors, then perturbations Uδ (0, 0) in (31) could
take at least those two independent values, and then the jump of the field (52) vanishes. However, if T(u(0)) contains just one direction, then all we can say is that the
jump of (52) is orthogonal to that single direction.
The condition given in (b), when u(0) is a singular point of the curve φ = 0 so that
∇φ(u(0)) = 0, can also be expressed through the hessian ∇2 φ(u(0)) of φ at the point u(0). In
particular, if that hessian is non-singular, so that u(0) is a non-degenerate critical point of φ,
then T(u(0)) indeed contains two independent directions. This is nothing but the well-known
Morse lemma about the behavior of a function near a non-degenerate critical point. However,
we have decided to write things through T(u(0)) because the condition of containing at least
two independent directions is more general than asking for the invertibility of the hessian.
The results in Sections 14, and 15 are again direct generalizations of the ones we already
have, or even they are applicable directly.
20.2. Counting critical paths for a singular system. We now come to the point of counting
limit cycles for a smooth, singular differential system enjoying the identification property.
We must distinguish two possible situations depending on the nature of the point where the
jump is taking place. The first corresponds to a singular point x for which there are at least
two independent tangent lines. If this is the situation, there can be no jump because the set
T(x) would span all of R2 , and, according to Lemma 36 above, these points cannot support a
discontinuity of the derivative for a critical path, and, hence, they cannot change branches at
these points.
But in order to treat the case of a singular point (single or multiple) where the set T(x) is a
single direction given by the field F, we are in need of assuming that F enjoys the identification
property. If we suppose that this property holds for our differential system (8), we can use
Lemma 36 for the precise value p + 1 where p is given in that property. Note that paths in
the space H p+1 ([0, 1]; R2 ) are C p , and so feasible paths enjoy continuity of derivatives up to
order p. When feasible paths pass through one such point they can follow at most one specific
branch of the many possibilities that might be given. This means that different branches cannot
interact with each other, and each branch can be treated, from our viewpoint, as if it were a
one-dimensional curve homeomorphic to a line, with the singular point counted twice or more
times, one for each branch. This argument implies that the number of possible critical paths
with image contained in the corresponding connected component of the curve Div = 0, can
still be controlled by the number of solutions of the system (2) because the combinatorial
optimization problem utilized in the proof of Theorem 2 is also valid even in the presence of
these singular points. The main concern is that each connected component may now have
various branches (recall Figure 4) in an uncontrolled way since, according to the identification
property, they act as if they were different connected components. Said differently, the issue is
if in the sum
N +2
+m−1
2
m may have a much higher value compared to the non-singular case. It cannot be so, because
there is no way to increase m without decreasing N . After all, singular points can only be
produced is we take away solutions of the system (2), in such a way that we may increase m at
HILBERT’S 16TH
39
the expense of decreasing N . The maximum is again attained when N is as large as possible,
but then there cannot be singular points at the curve Div = 0.
We finally very clearly state that:
Lemma 37. Every polynomial differential system complies with the identification property.
Proof. We refer to some basic material from [46]. In particular, in the proof of Lemma 2.3.1
(page 29), Puiseux’ series are used in the factorization of an algebraic curve D(x, y) = 0 near
a singular point. The factorization is of the form
D(x, y) = Πm
k=1 (y − ak (x))
where the ak (x)’s are Puiseux’ series. If one of these ak (x) turns out not to be analytic in the
usual sense, then one of our feasible smooth paths cannot go across the singular point through
such branch in any way. If ak (x) is analytic, then it can but only through that particular
branch. It cannot cross to a different aj (x) because the analyticity prevents from having all of
the derivatives equal for both branches.
21. Appendix V. More on Hilbert’s 16th problem
21.1. On the configuration of the limit cycles of the polynomial differential systems.
A topological configuration of limit cycles is a finite set C = {C1 , . . ., Cn } of disjoint simple
closed curves of the plane such that Ci ∩ Cj = ∅ for all i 6= j.
Given a topological configuration of limit cycles C = {C1 , . . . , Cn }, the curve Ci is primary
if there is no curve Cj of C contained into the bounded region limited by Ci .
0
}
Two topological configurations of limit cycles C = {C1 , . . . , Cn } and C 0 = {C10 , . . . , Cm
n
2
2
are (topologically) equivalent if there is a homeomorphism h : R → R such that h (∪i=1 Ci ) =
0
0
(∪m
i=1 Ci ). Of course, for equivalent configurations of limit cycles C and C , we have that n = m.
We say that a polynomial differential system (1) realizes the configuration C of limit cycles
if the set of all limit cycles of (1) is equivalent to C.
Llibre and Rodr´ıguez [34] proved the following result in 2004.
Theorem 38. Let C = {C1 , . . . , Cn } be a topological configuration of limit cycles, and let r be
its number of primary curves. Then the following statements hold.
(a) The configuration C is realizable by some polynomial differential system.
(b) The configuration C is realizable as algebraic limit cycles by a polynomial differential
system of degree ≤ 2(n + r) − 1. Moreover, such a polynomial differential system has a
first integral of Darboux type.
Statement (a) of Theorem 38 follows immediately from statement (b).
Statement (a) of Theorem 38 was solved, for the first time, by Schecter and Singer [41], and
Sverdlove [45], but they do not provide an explicit polynomial vector field satisfying the given
configuration of limit cycles, as it was provided in the proof of statement (b) of Theorem 38.
If f = f (x, y) is a polynomial we denote its partial derivatives with respect to the variables
x and y as fx and fy , respectively. Christopher [10] proved the following result in 2001.
Theorem 39. Let f = 0 be a non–singular algebraic curve of degree n, and D, a first degree
polynomial chosen so that the straight line D = 0 lies outside all bounded components of f = 0.
Choose the constants α and β so that αDx + βDy 6= 0, then the polynomial differential system
of degree n,
x˙ = αf − Dfy ,
y˙ = βf + Dfx ,
has all the bounded components of f = 0 as hyperbolic limit cycles. Furthermore, the differential
system has no other limit cycles.
40
J. LLIBRE AND P. PEDREGAL
Theorem 39 improves a similar result due to Winkel [47], but the polynomial differential
system obtained by Winkel has degree 2n − 1.
Given a topological configuration of k limit cycles we can consider an equivalent topological
configuration formed by circles. Consider then the algebraic curve f = 0 formed by the product
of all the circles. Applying Theorem 39 to the curve f = 0, we obtain a polynomial differential
system of degree n = 2k, which realizes the given topological configuration of k limit cycles with
algebraic limit cycles. A difference between the polynomial differential systems of Theorems
38 and 39, is that the first always has a first integral, and the second, in general, has no first
integrals.
In short, both Theorems 38 and 39 show that any topological configuration of limit cycles is
realizable with algebraic limit cycles for some polynomial differential system, and provide the
degree of such polynomial differential systems. But there are many questions which remains
open, as for instance: what are the possible topological configurations of limit cycles realizable for
the polynomial differential systems of a given degree? This question is definitely more difficult
than the question to provide a uniform upper bound for the maximum number of limit cycles
that the polynomial differential systems of a given degree can have.
21.2. The 16-th Hilbert problem restricted to algebraic limit cycles. Associated with
the polynomial differential system (1), there is the polynomial vector field
(53)
F = P (x, y)
∂
∂
+ Q(x, y) .
∂x
∂y
The algebraic curve f (x, y) = 0 of R2 is called an invariant algebraic curve of the polynomial
vector field F, or of the polynomial differential system (1), if for some polynomial K = K(x, y),
we have
∂f
∂f
Ff = P
+Q
= Kf.
∂x
∂y
The polynomial K is called the cofactor of the invariant algebraic curve f = 0.
Since on the points of an algebraic curve f = 0, the gradient ∇f = (fx , fy ) of the curve
is orthogonal to the vector field F (see (53)), this vector field is tangent to the curve f = 0.
Hence, the curve f = 0 is formed by orbits of the vector field F. This justifies the name of
invariant algebraic curve given to the algebraic curve f = 0 satisfying (53) for some polynomial
K: it is invariant under the flow defined by the vector field F.
The next well-known result tell us that we can restrict our attention to the irreducible
invariant algebraic curves, for a proof see for instance [31]. Here, as it is usual, R[x, y] denotes
the ring of all polynomials in the variables x and y, and coefficients in R.
Proposition 40. Let f ∈ R[x, y], and let f = f1m1 · · · frmr be its factorization in irreducible
factors over R[x, y]. Then for a polynomial vector field F, f = 0 is an invariant algebraic curve
with cofactor Kf if and only if fi = 0 is an invariant algebraic curve for each i = 1, . . . , r with
cofactor Kfi . Moreover Kf = m1 Kf1 + . . . + mr Kfr .
Consider the space Σ0 of all real polynomial vector fields (53) of degree n having real irreducible invariant algebraic curves.
An algebraic limit cycle is an oval of an algebraic curve which is a limit cycle of a polynomial
differential system (1).
A simpler version of the second part of the 16th Hilbert’s problem with respect to the number
of limit cycles is: Is there an uniform upper bound for the maximal number of algebraic limit
cycles of any polynomial vector field of Σ0 ? We cannot provide an answer to this question for
general real algebraic curves, but we give the answer for the following class of algebraic curves.
We say that a set fj = 0, for j = 1, . . . , k, of irreducible algebraic curves is generic if it
satisfies the following five conditions:
HILBERT’S 16TH
41
(i) There are no points at which fj = 0 and all its first derivatives vanish (i.e. fj = 0 is a
non–singular algebraic curve).
(ii) The highest order homogeneous terms of fj have no repeated factors.
(iii) If two curves intersect at a point in the affine plane, they are transversal at this point.
(iv) There are no more than two curves fj = 0 meeting at any point in the affine plane.
(v) There are no two curves having a common factor in the highest order homogeneous
terms.
The next result was proved by Llibre, Ram´ırez and Sadovskaia [32] in 2010.
Theorem 41. For a polynomial vector field F of degree n ≥ 2 having all its irreducible invariant
algebraic curves generic, the maximum number of algebraic limit cycles is at most 1+(n−1)(n−
2)/2 if n is even, and (n − 1)(n − 2)/2 if n is odd. Moreover these upper bounds are sharp.
For cubic polynomial vector fields having all their irreducible invariant algebraic curves
generic, Theorem 41 says that one is the maximum number of algebraic limit cycles, but there
are examples of cubic polynomial vector fields having two algebraic limit cycles. Such vector
fields have non–generic invariant algebraic curves. For example, the polynomial differential
system of degree 3
x˙ = 2y(10 + xy),
y˙ = 20x + y − 20x3 − 2x2 y + 4y 3 ,
has two algebraic limit cycles contained into the invariant algebraic curve 2x4 −4x2 +4y 2 +1 = 0,
see Proposition 19 of [35].
Until now, all polynomial vector fields having non–generic invariant algebraic curves and
more algebraic limit cycles than the upper bounds given in Theorem 41 for the generic case,
have odd degree, and at most one more limit cycle than the upper bound of Theorem 41. So,
in [32] we conjectured the following.
Conjecture. The maximum number of algebraic limit cycles that a polynomial differential
system of degree n can have is 1 + (n − 1)(n − 2)/2.
Note that the conjectures is true when n is even and we restrict the algebraic limit cycles to
generic invariant algebraic curves.
After [32], three other paper have appeared related to the algebraic limit cycles of polynomial
differential systems. One due to the same authors [33], and other two due to Xiang Zhang
[49, 50].
21.3. Limit cycles and the inverse integrating factor. Another useful result on the limit
cycles of a C 1 differential system in the plane is the following one due to Giacomini, Llibre and
Viano [21], see an easier proof in [34]. This result has been used in the proofs of Theorems 38
and 41. First we need a definition.
Let U be an open subset of R2 . A function V : U → R is an inverse integrating factor of a
1
C vector field F defined on U if V verifies the linear partial differential system
∂V
∂P
∂Q
∂V
+Q
=
+
V
P
∂x
∂y
∂x
∂y
in U .
Theorem 42. Let X be a C 1 vector field defined in the open subset U of R2 . Let V : U → R
be an inverse integrating factor of X. If γ is a limit cycle of X, then γ is contained in {(x, y) ∈
U : V (x, y) = 0}.
Acknowledgements
The first author is partially supported by a MINECO/FEDER grant MTM2008-03437 and
MTM2013-40998-P, an AGAUR grant number 2014SGR568, an ICREA Academia, the grants
FP7-PEOPLE-2012-IRSES 318999 and 316338, FEDER-UNAB-10-4E-378. The second author
42
J. LLIBRE AND P. PEDREGAL
is partially supported by MINECO/FEDER grants MTM2010-19739, and by MTM2013-47053P.
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1 Departament de Matema
` tiques, Universitat Auto
` noma de Barcelona, 08193 Bellaterra, Barcelona,
Catalonia, Spain
E-mail address: [email protected]
2
E.T.S. Ingenieros Industriales. Universidad de Castilla La Mancha. Campus de Ciudad Real,
Spain
E-mail address: [email protected]