1. No category

Physica 13D (1984) 82-89
North-Holland, Amsterdam
E X T E N D E D C H A O S A N D D I S A P P E A R A N C E OF KAM T R A J E C T O R I E S
David B E N S I M O N and Leo P. K A D A N O F F
The James Franck Institute, University of Chicago, 5640 S. Ellis, Chicago, IL 60637, USA
R e c e i v e d 26 O c t o b e r 1983
In two-dimensional area-preserving maps, KAM trajectories serve as natural boundaries between stochastic regions. As
they disappear, points may leak out from one region to the next. This escape rate is defined and related to the action-generating
function. Analytic and numerical results are presented to describe the behavior of the escape rate near the disappearance of a
KAM trajectory.
I. Introduction
The study of area-preserving maps in two dimensions is motivated by the fact that these maps
exhibit the non-trivial dynamics of the simplest
class of conservative systems (e.g., Hamiltonian
systems with two degrees of freedom). In these
mapping problems a sequence of points X / = (5, 0/)
j = 0,1, 2 . . . . is generated by the mapping function
T via Xj + 1 = T(Xj). We assume that the mapping
is continuous and invertible so that we can study
the behavior of the entire sequence Xj ( j = 0,
+ 1, + 2 . . . . ). Depending upon the initial point,
X 0, these sequences m a y show several different
behaviors including:
Cyclic. The sequence repeats with period q: Xj+q
=Xj.
Invariant (KAM) curve. The sequence never
repeats but the points X/ fill out a curve F. That
is, for each x on F, we can find a subsequence Xjo
of Xj such that
These different orbits are exhibited in fig. 1.
Notice that if a K A M curve F encloses a region
R r containing a fixed point X* of T (i.e., T(X*)
= X*), then
Vx~Rr, n~Z:
In words: chaos cannot move across K A M curves.
If the mapping T depends upon some parameter
k, as for example, the standard m a p Tk,
(r,O)~(r',O')=
1.0 L
'
J
'
I
'
0.6 ~--
,
I
'
J
'
~ _
+
o
o 2~-
+
o
~ , .
,..,~"
R "
"¢~~..:
-0.2 ;..~,-"c~'~'~"~"
n
It is also believed that for some starting points,
{Xj} shows an area-filling chaotic behavior,
namely:
Chaos. There exists a region R c of finite area
such that if x lies anywhere in R c there is a
subsequence XA which approaches x.
r-~-~-sin2~r0, r'+0
(1.1)
lim Xj = x .
r/...~ O0
T " ( x ) ~ R r.
....
-
~
-0.4
-I.0
0
,
I
0.2
,
I
0.4
,
I
0.6
,
I
0.8
,
1.0
THETA
Fig. 1. Three typical orbits of an area preserving map: the
standard map at k = k c. a) Periodic orbit; b) K A M trajectory;
c) bounded chaotic orbit.
0167-2789/84/$03.00 © Elsevier Science Publishers B.V.
(North-Holland Physics Publishing Division)
D. Bensimon and LP. Kadanoff/ Extended chaos and disappearance of KAM trajectories
then as k is varied and reaches some critical initial
value k c, a K A M curve may disappear, i.e., become discontinuous, Cantor set like. In that case a
barrier to the extension of chaos vanishes and the
chaotic region may thus suddenly grow in size. It
is the purpose of this paper to understand what
may happen in this sudden growth. Many of the
results of the present paper were obtained in parallel by MacKay, Meiss and Percival with whom we
had fruitful exchanges [1].
2. The escape rate from a chaotic region
Notice that a KAM curve bounds an invariant
region: If R r contains a fixed point and if F is a
K A M curve, the function Tk maps R r into itself:
Tk( Rr)= Rr .
being a K A M curve. In parallel, we can define an
escape rate for any region as
E(Tk, R) =
L(Tk,R)
~2(R)
(2.5)
This escape rate serves as a bound on the rate at
which points may leak out of R. Notice that if we
apply q steps of Tk we have
L(Tq, R)<-qL(Tk,R),
(2.6)
so that the number of steps Q required to reduce
the area in TkQ(R)nR to a proportion O of the
original area I2(R) is bounded by
1-O
Q> e(rk,R)"
(2.7)
(2.1)
Once the K A M curve has disappeared there exists
no invariant region corresponding to R r. If we
look for such a region Rv, bounded by a y which
is a closed curve but not a KAM curve, then
Tk(Rv) 4: R v.
(2.2)
If Rv contains a fixed point then the region Tg(Rv)
will partially overlap with Rv,
(2.3)
Here ~2(Rv) is just the area of region Rv. As k
approaches the critical value and R v approaches
Rr, for F being the critical KAM curve, then the
region of overlap will get closer and closer to the
entire region R r and the left-hand inequality in
eq. (2.3) will get closer and closer to an equality.
In this limit we can say that the area escaping
from Rr,
L(T , R,) =
83
n E - R,)
(2.4)
(where E - Rr is the complement of Rv), goes to
zero.
Eq. (2.4) represents the leakiness of the curve "t
under the mapping Tk. This leakiness is zero for -f
Hence to get the best possible bounds on the
permitted motion we should wish to minimize the
escape rate (2.5). For this reason we consider, in
this paper, the construction of the minimum escape
rate from a given region [2].
Imagine that we wish to look for the escape
from some region R 1 into another disjoint region
R 2. Then consider all regions Rv containing R 1
( R r ~ R1) and not overlapping R 2 ( R v c E - R2).
Under these constraints we define t h e m i n i m u m
escape rate from a neighborhood of R 1 into a
neighborhood of R 2 to be
e( Tk, R1, R2) = i n f [ E ( T k , R , ) ] ,
(2.8)
with the minimum (inf) being taken over all Rv
which satisfy the constraints.
Since we are interested in the case in which
L(Tk, R) may be very small, we also calculate the
slightly simpler minimum escaping area
l(T k, R1, R 2) = inf[ L ( T k, R,)].
(2.9)
We expect that the Rv's which give the minimum
in (2.8) and (2.9) are essentially identical.
Given these definitions, we set out to estimate e
and l. To begin, we approximate a curve which is
"almost" the K A M curve by constructing one
84
D. Bensimon and L.P. Kadanoff/ Extended chaos and disappearance of KA M trajectories
which goes through all the elements of cycles
which approximate the KAM curve. This construction is meaningful even after the KAM curve
has disappeared. We note that this curve gives an
extremal of l: moving away from the unstable
cycle elements increases / while moving away from
the stable cycle elements decreases 1 (see appendix
A). This extremum does give a bound on l, which
can be estimated with the aid of a scaling theory
for near critical K A M curves. Then we look at
obtaining a better bound by considering curves
which pass only through elements of an unstable
cycle.
1.2
'
I
'
i
I
i
[
i
.... r+lTl rlTIT] I]ITI]-I
1.0
0.8
0.6
R
0.4
0.2
0
-0.2
I
0.2
0
0.4
I
I
0.6
i
I
0.8
,
1.0
THETA
Fig. 2. The escaped area (dashed region) for a trivial test
region: the square (0 _<r < 1; 0 _<8 < 1).
3. Bounding the escape rate by cycles
For specificity, we concern ourselves with the
mapping (1.1) defined on a cylinder, i.e., with
0 = O + 1, and consider the K A M trajectory with
winding number W = (~/5 + 1)/2, which disappears at k = k c = 0.97163540 . . . . This curve is
shown in fig. 1.
The initial regions (R 1 and R2) are chosen to lie
between two K A M curves by taking Rx to be a
narrow strip around r = 1/2, while R 2 includes
similar strips about r = + 3/2. We consider the
test region Rv to be the area bounded by the
curves Y0 and ~/1, where ~'0 is defined parametrically by writing
Yo:
O=O(t),
r=r(t),
0_<t<l,
with
0(0) = 0(1) + 1 ,
O's in [0,1]. Then the bounding curves are (see
fig. 2)
Y0:
r(t) =0,
O(t) = t;
o(t)=t,
while the images of these curves are
k
2~r sin2rrt,
k
O'(t) = t - ~ sinZlrt,
r'(t) =
T(yo):
and T(3q) is the same curve displaced one step
upward. Since the area enclosed by Y0 and 7t is
one, the escape rate and the escaping area are
identical and given by
E(Tk,Rv)=
O=O(t),
r=r(t)+
(3.2)
r(0) = r(1),
and "tl is defined as the same curve displaced one
step upward:
Yt"
(3.1)
r ( t ) = 1,
l.
A trivial estimate of the escape rate will be
obtained if we choose Rv to include all r ' s and
L(Tk,Rr)
- - ~Ikl
.
(3.3)
Notice that for k = k c this trivial estimate gives
roughly 10% of the area escaping upon one iteration.
It is instructive to obtain the result (3.3) in
another way. Notice that the curves Y0 and its
image T(Y0) pass through the stable fixed point at
(r, 0) -- (0,0) and the unstable fixed point at (r, 0)
D. Bensimon and LP. Kadanoff / Extended chaos and disappearance of KAM trajectories
= (0, ½). The escaped area is
o.,ok
½L(Tk,Rr)=Sol/2r(t)~dt
0"f/
Lv
e1/2
, dO'(t)
-Jo r ' ( t ) ~ d t .
(3.4)
Remark that the motion is generated by an action
principle in which
r'(t)=
r(t)=
a A(O,O') o-or)
} '
aO'
0 ' = 0 (t)
O---~A(O,O')o=o(t)
\
<°' t
0.501
\
,
0
-
/1
'\, x-o'x-,~-'x / ,dI
I
0.4
0.8
THETA
1
'
I
'
I
'
I
0.71
,
0.70
O'=O'(t)
with
0.69
A(0,0') = -½(0-0') 2--(2,r) 2 cos2rr0.
Thus,
'
o.p
0.72
(3.5)
85
(3.6)
0.68
0.2
0.4
0.3
0.5
THETA
L(Tk, Rr) is given by
Fig. 3. a) A curve "t (full line) and its image y ' = Tk(-t )
(dashed line), passing through all stable (0) and unstable (x)
elements of the p / q = 5 / 8 cycle; b) segment of the above
curves lying in (0.68 < r < 0.72; 0.2 < 0 _< 0.55). The escaped
area is the dashed region.
L(Tk,RT)=2fol/2(dO
0
dt 00
dO O) A(O,O')dt.
dt' 00"
+----
Since the integrand is a total derivative, we find
L(Tk, Rv)=2(Au-As).
(3.7)
Here A u =A(½, ½) is the action for the unstable
cycle while A s = A(0,0) is the action for the stable
one. Notice that in the derivation of eq. (3.7) we
have used no properties of the paths 70 and 71
except the facts that they cross their image paths
"/~ = T(70) and "/~= T(71) at only the cycle points.
Hence, the results (3.7) and (3.3) hold for all paths
which pass through the fixed points and have no
other crossing points. This result is thus extremal
(indeed constant!) under variations of the paths.
This result may be immediately generalized to
higher order cycles. Let "/o pass through the 2q
dements of a stable and unstable cycle of length q
as in fig. 3. Let the image path cross the original
path only at these 2q points. The stable cycle has
0-values 87 and the unstable one 0-values 0p. Then
the escaped area is the union of the shaded regions
shown. The net result is a form for L which is
L(Tg'Rv)= 2E f 0ui' rdO- fo;ur'dO'.
JvO;
m
By exactly the same calculation as before, we can
reduce the result to
L(Tk, Rv) = 2(Aqu -
a qs),
(3.8)
where A q and A q are the total actions for the
stable and unstable cycles of length q, namely
q-l{
Aq= E -½(#j"-Oj~-i
j=o
)2
}
---c°s2*r~
(2Ir) 2
(a = u,s).
~
(3.9)
D. Bensimon and L.P. Kadanoff/ Extended chaos and disappearanceof KA M trajectories
86
This result is, once again, path independent so
long as the original path considered, To passes
through the cycle elements and crosses its image
path at these points only. This approach yields a
sequence of estimates for L(T, R) and E(T, R),
by choosing a succession of cycles converging to
the K A M curve (or Cantor set for k > kc). For the
Golden mean (W = (v~- + 1)/2) these cycles are of
period [3] q. = F., where F. is the n th Fibonacci
number, and have Oj+q, = ~ + p , , with p, = F n _ 1.
For k < k c, the leakage rate Lq.(Tk, R) goes to
zero faster than exponentially [4]. For k = k c,
zq"(Tk, R) "') 0 algebraically in q~ [5, 6, 7]. In fact
from the scaling and renormalization group analysis [5, 6, 7] of the action for the Golden mean
K A M curve near its disappearance, we expect that
Lq"(Tk, R ) = qZ(~°+Y°)L*~
(q, l k -
QI~),
(3,10)
with: d o = x o + Y0 = 3.049960... ; I, = 0.987463...
and where L ~ ( ~ ) is a scaling function which
applies respectively when k > k¢ or k < k¢. Its
exponential behavior (~ ~ oo) is
k < kc:
L* ( ~ ) - e - ~
k>kc:
L * ( ~ ) - ~ do.
(T a constant),
(3.11)
Namely, in the supercritical case we expect
lim zq"(Tk,
qn"-) oO
R) =
I~ -
kcl "d°-
(3.12)
However, the region k > k¢ is not directly accessible to this analysis since the "stable" cycles tend
to bifurcate and increase in number as k passes
above k c. Hence, we seek a formulation based
only upon unstable cycles.
There is another reason for looking to the unstable cycles. If we move our curve "t slightly from
the unstable cycle elements, L(T k, Rv) increases
at a rate proportional to the separation squared
(appendix A). But a corresponding motion away
from the stable cycle elements decreases L (Tk , R v ).
Since we are looking for a minimum of the escape
rate we should seek to get away from the stable
t
l
1/
f
I
/
I
I
I
I
I
.C?"
I
/
/
I
I
I
~
I
o.~
I
I
x.,.
I \1-'
Fig. 4. The escaped area (dashed region) for a curve V passing
through unstable cycle elements only. A and C are neighboring
points of the unstable cycle; B and B' are homoclinic points.
elements, and use a curve which passes through
unstable elements only.
4. Numerical results using unstable cycles only
The curve T passing through all the elements of
an unstable cycle is generated in the following way
(see fig. 4). Choose two neighboring points A and
C of an unstable cycle with winding number W =
p / q ~ F,_I/F.. The curve y between these two
points is composed of the unstable manifold of A
and the stable manifold of C which intersect at the
homoclinic point B*. Iterating that segment (ABC)
of ~,, q - 1 times generates a continuous curve T
passing through all the elements of the unstable
cycle. Thus the curve
v'= Tk(V)
is identical with T except for the segment ABC.
Due to the contraction along the stable manifold
of C, homoclinic point B is mapped to homoclinic
point B'. Since the mapping is area preserving T'
must then intersect T in 2m + 1 points (for the
standard map at the Golden mean: m = 0). The
escaped area, S, is the dashed area in fig. 4 between T and T'. The numerical results- concerning
* The unstable (stable) manifolds at two neighboring points
of the unstable cycle are determined by solving for the eigenvalues of the tangent map and applying the mapping on points
along the relevant eigenvectors forward with Tk (or backward
with rk- 1).
87
D. Bensimon and L.P. Kadanoff/ Extended chaos and disappearance of KA M trajectories
the dependence of S on k - validate the theoretical
predictions:
(a) k < k c - W e observed the escaped area to
scale as:
S - q - d ° L * (ql k -- k c r )
0
.
0
taking d o = 3.049960 . . . . we could estimate ~/=
6.00 and ~,---0.99. T h e observed value of v is in
agreement with the R.G. value: v = 0.987463 . . . .
T h e scaling function L * ( x ) is shown in fig. 5.
(b) k = k c - We observed a p o w e r law decay of
the leakage rate, S, as Q increases (fig. 6)
s - q-do.
Jlltiilll/
0
0.08
0.02
i
'
]
'
I
0.06
'
'
0.08
' ~
I
0.06
x o +Y0 = do = 3.050 ___
0.02
0 i~
X1()46
5
.
L'
.
.
1
.
.
'~\
4
.
--.,,,\
3
~
2 --
0
"%~
,
.
,
I
0.2
I ~ 1 ~ I
0.1
0.2
0.3
I
,
I ~ I0.4
0.5
k
0.12
~
.9o
+
.92
o
.94
-
x
955
-
•
.965
-
.96
*
~-#xo.
,
.
I
Symbol
"*~x.~
"~-
0
0.04
0.04
T h e observed p o w e r is:
0.002.
•_Av
~
0.03
°° l
_ q-do exp [ - yql k - kcl" ] ,
-
6
-
0.10
-
0.08
-
'
I
'
I
'
I
0.06
0.04
,
0.4
x-qlk-kcl
0.6
0.02
0
0
~
Fig. 5. The scaling function for k< kc: L*(x). Notice the
two branches which are generated by n even (odd) Fibonacci
cycles (q, = F,,).
0.1
0.2
0.3
Fig. 7. The escaped area (dashed region), S for a curve passing through unstable cycle elements only at k= 1.5 (> kc).
a) p / q = 3/5, S = 1.6352 × 10-3; b) p / q = 5/8, S = 1.6374
X 10-3; C) p / q = 8/13, S = 1.6356 × 10 -3.
- 4
-10
I
0.4
I
I
0.8
I
]
~
1.2
I
1.6
M
I
2.0
I
I
2.4
loglo(q)
Fig. 6. The escaped area, S versus q ( = F,) at k : kc.
(c) k > k c - T h e results in that regime are particularly interesting since we expect the escaped area
to converge to a constant non-zero value at large
q. There exists, however, several numerical difficulties in converging to the correct unstable cycle for
supercritical values of k. T o o v e r c o m e these difficulties we used an algorithm which consisted in
predicting the initial point [ro(k), # 0 ( k ) ] - on the
relevant s y m m e t r y l i n e - b a s e d on the values of
( r o, 8o) for previous smaller values of k. An itera-
88
D. Bensimon and L.P. Kadanoff/ Extended chaos and disappearance of K A M trajectories
tive Newton method was then used to improve the
determination of the initial point to an accuracy
< 10-lo. To check for the correctness of the convergence, we verified that the order of the unstable
cycle elements was the same as for the k = 0 case.
A typical example of the form the escaped area
exhibits as higher cycles are used is shown in fig. 7
(at k = 1.5). Note the convergence of the leakage
rate to four significant digits. In these numerical
studies we could not study the behavior of the
escaped area for arbitrarily large cycles, since the
eigenvalue of the tangent map for large cycles is
extremely high (for example: k = 2.28 X 105 for
= 55/89 at k = 1.1). Such a high eigenvalue
means that after three iterations, with double precision arithmetic, one has lost all information about
the initial point.
The scaling function L~ (x) is shown in fig. 8.
Its asymptotic behavior is given by eq. (3.11) with
a critical exponent
Acknowledgements
d o -- 3.026 + 0.036.
The dashed area in fig. 9a is the escaped area in
the vicinity of 0. Now shift a segment of 3' a
distance e below 3', and consider the image 3" of
p/q
Note however the different form of the scaling
function
and
for n-even (odd)
Fibonacci cycles (q, = F,). We do not understand
that result in the framework of the renormalization
group for 2-D area-preserving maps [5, 6, 7].
(L*(x)
I
I
0-
'
'
Symbol
L*(x))
'
I
k
x
t..I..i
--4
,
#
'
'
I .99
i.O0
I
11.05
--
Appendix A
We will prove that a curve passing through
stable or unstable cycle dements extremizes the
escaped area.
Consider fig. 9a: In the vicinity of the q-cycle
point 0 one can use the tangent map to relate the
curve 3' to its image 3" (after q iterations),
(0/
IY
fr'
/ ~
I ' /
/
+o 1.98
.985
-2
'
We would like to acknowledge helpful discussions and correspondence on this work with R.S.
MacKay, I. Percival and J.D. Meiss.
This work is supported in parts by Grants NSF
D M R M R L 79-24007 and NSF 80-20609.
Y'
Y
(b/
..
/
/ I I
YES//
#
ISi
F
I,
-6
-8
-~
-2
log
I
0
~
,
,
.. r,
d-IIIIIIIA
~
X
C(I,-,)
I
2
[x--q(k-k )v]
Fig. 8. The scaling function for k > k c : L * ( x ) . Notice the
two branches which are generated by n even (odd) Fibonacci
cycles ( q. = Fn).
Fig. 9. a) The escaped area for a curve y passing through cycle
element O; b) the escaped area for the same curve "r moved
away slightly (by e) in the neighborhood of cycle element 0.
D. Bensimon and LP. Kadanoff/ Extended chaos and disappearance of KAM trajectories
that new line, which by now does not pass through
0 (see fig. 9b). The escaped area, S', is the dashed
one in fig. 9b. Clearly,
S ' = S -- SOAC,D, + SABCD.
XA-- M22
, -
Inserting (A.4) and (A.5) into (A.2):
/iS = S ' - S = (Mll + M22 - 2) e2"
21M2al
y = (M21x -- e)//M11,
with the lines y = 0 and y = - e :
/M2x,
(1-mxx)
XB=
M21 e,
- M22 and
(A.2)
Points A and B are determined by finding the
intersection of the line C'F' (the image of CF):
x^ =
89
(A.3)
(A.6)
(We have introduced the absolute value of M21,
since in fig. 9 M21 is implicitly assumed to be
positive.) Therefore, unstable cycles ( T r M > 2)
minimize the escaped area, and stable cycles
(ITr M I< 2) maximize it.
References
Then
SABCD=e( 1
XA+XB)2
=el1 _ ( 2 - M l l
(A.4)
-To evaluate SOAC,o, we use the fact that the mapping is area conserving and thus
So^c'D' = So^'co.
A' is the pre-image of A, thus: YA= 0 = M21" X^,
[1] R.S. MacKay, J.D. Meiss and I.C. Percival, Physica 13D
(1984) 55 (previous paper).
[2] E. Wigner, J. Chem. Phys. 5 (1937) 720 and LC. Keck, Adv.
Chem. Phys. 13 (1967) 85 consider analogous minimizations
of flow rates in phase space.
[3] J.M. Greene, J. Math. Phys. 20 (1979) 1183.
[4] J.N. Mather, preprint, Princeton (1982).
[5] S.J. Shenker and L.P. Kadanoff, J. Stat. Phys. 27 (1982)
631.
[6] L.P. Kadanoff, Phys. Rev. Lett. 47 (1981) 1641.
[7] R.S. MacKay, Proc. Conf. "Order in Chaos", Los Alamos
(1982), published in Physica 7D (1983) 283. See also thesis
(Princeton). The relation between the critical exponents
(v, do) used here and those found by MacKay (& a,B) are:
v = (logwS)-l; do=logw(afl) with W=(vr5 + 1)/2.