1. No category

The Perron method:
what, why, and how to apply it to NHIMs
Jaap Eldering
Utrecht University
23 February 2011
VU Dynamic Analysis Seminar
1
Talk outline
The Perron method
What is it: a basic explanation of this method for proving
existence of (un)stable manifolds of a hyperbolic
fixed point.
Why is it useful? Some of the (dis)advantages and an
example application to a hyperbolic periodic orbit.
How to apply it to Normally Hyperbolic Invariant
Manifolds (NHIMs), as a generalization of
hyperbolic fixed points.
I am specifically interested in noncompact NHIMs.
2
Talk outline
The Perron method
What is it: a basic explanation of this method for proving
existence of (un)stable manifolds of a hyperbolic
fixed point.
Why is it useful? Some of the (dis)advantages and an
example application to a hyperbolic periodic orbit.
How to apply it to Normally Hyperbolic Invariant
Manifolds (NHIMs), as a generalization of
hyperbolic fixed points.
I am specifically interested in noncompact NHIMs.
2
What: the Perron method
I
Alternative to Hadamard’s graph transform (1901).
I
Attributed to Oskar Perron (1929), although Perron
´
attributes the method to Emile
Cotton (1911).
I
Based on a variation of constants integral to construct a
contraction on a space of bounded solution curves.
3
What: the Perron method
E+
U
Wlo
Consider an ODE around a
fixed point v (0) = 0:
S
Wlo
E−
x˙ = v (x) = Dv (0) · x + r (x).
If the fixed point is hyperbolic,
then there is a linear splitting
Rn = E+ ⊕ E− ,
Dv (0) = A+ ⊕ A− ,
with eigenvalue bounds λ− < 0 < λ+ .
4
What: the Perron method
We are looking for the stable manifold of the nonlinear problem:
S
Wloc
= x ∈ Rn lim x(t) = 0
t→∞
These are precisely the points x whose forward orbit x(t) is
bounded. So together, all forward bounded solution curves x(t)
S by the graph x = h(x ) of the initial value
describe Wloc
+
−
components.
5
What: the Perron method
We are looking for the stable manifold of the nonlinear problem:
S
Wloc
= x ∈ Rn lim x(t) = 0
t→∞
These are precisely the points x whose forward orbit x(t) is
bounded. So together, all forward bounded solution curves x(t)
S by the graph x = h(x ) of the initial value
describe Wloc
+
−
components.
5
What: the Perron method
The ODE is equivalent to the variation of constants
integral equation
Zt
x(t) = e t Dv (0) · x(0) + e (t−τ) Dv (0) · r (x(τ)) dτ.
0
6
What: the Perron method
The ODE is equivalent to the variation of constants
integral equation
Zt
T : x(t) 7→ e t Dv (0) · x(0) + e (t−τ) Dv (0) · r (x(τ)) dτ.
0
We view this as a map T , which has precisely solution
curves x as fixed points: T (x) = x.
6
What: the Perron method
We split coordinates x = (x+ , x− ), change the initial time t0 in
the unstable part and let t0 → ∞
Zt
t A+
x+ (t) 7→ e
· x+ (0)
+ e (t−τ) A+ · r+ ((x+ , x− )(τ)) dτ,
0
Zt
+ e (t−τ) A− · r− ((x+ , x− )(τ)) dτ.
x− (t) 7→ e t A− · x− (0)
0
We consider this rewritten map T for bounded curves
x ∈ B(R; Rn ) only.
7
What: the Perron method
We split coordinates x = (x+ , x− ), change the initial time t0 in
the unstable part and let t0 → ∞
Zt
(t−t0 ) A+
x+ (t) 7→ e
· x+ (t0 ) + e (t−τ) A+ · r+ ((x+ , x− )(τ)) dτ,
t
Z t0
+ e (t−τ) A− · r− ((x+ , x− )(τ)) dτ.
x− (t) 7→ e t A− · x− (0)
0
We consider this rewritten map T for bounded curves
x ∈ B(R; Rn ) only.
7
What: the Perron method
We split coordinates x = (x+ , x− ), change the initial time t0 in
the unstable part and let t0 → ∞
Zt
(t−t0 ) A+
x+ (t) 7→ e
· x+ (t0 ) + e (t−τ) A+ · r+ ((x+ , x− )(τ)) dτ,
t
Z t0
+ e (t−τ) A− · r− ((x+ , x− )(τ)) dτ.
x− (t) 7→ e t A− · x− (0)
0
We consider this rewritten map T for bounded curves
x ∈ B(R; Rn ) only.
7
What: the Perron method
We split coordinates x = (x+ , x− ), change the initial time t0 in
the unstable part and let t0 → ∞
Z∞
x+ (t) 7→
...
− e (t−τ) A+ · r+ ((x+ , x− )(τ)) dτ,
t
Zt
+ e (t−τ) A− · r− ((x+ , x− )(τ)) dτ.
x− (t) 7→ e t A− · x− (0)
0
We consider this rewritten map T for bounded curves
x ∈ B(R; Rn ) only.
7
What: the Perron method
Now T is a contraction (straightforward estimates, using
e t A± 6 C e t λ± for t ≷ 0), so there is a unique bounded x(t)
solution curve for a given parameter x− (0). This determines the
corresponding x+ (0) so we have a map x+ = h(x− )!
By a direct implicit function argument, h is as smooth as T so
S
v ∈ C k =⇒ Wloc
= Graph(h) ∈ C k .
8
What: the Perron method
Now T is a contraction (straightforward estimates, using
e t A± 6 C e t λ± for t ≷ 0), so there is a unique bounded x(t)
solution curve for a given parameter x− (0). This determines the
corresponding x+ (0) so we have a map x+ = h(x− )!
By a direct implicit function argument, h is as smooth as T so
S
v ∈ C k =⇒ Wloc
= Graph(h) ∈ C k .
8
Why: (dis)advantages
I
It trivially generalizes to non-autonomous systems.
I
For hyperbolic fixed points, smoothness of T
U , W S ) is easily proven.
(and thus Wloc
loc
I
By construction gives dynamical information:
growth estimates of solution curves.
However: it doesn’t easily generalize to normally hyperbolic
invariant manifolds (NHIMs) because solution curves are global
objects that cannot be confined to coordinate charts.
9
Why: (dis)advantages
I
It trivially generalizes to non-autonomous systems.
I
For hyperbolic fixed points, smoothness of T
U , W S ) is easily proven.
(and thus Wloc
loc
I
By construction gives dynamical information:
growth estimates of solution curves.
However: it doesn’t easily generalize to normally hyperbolic
invariant manifolds (NHIMs) because solution curves are global
objects that cannot be confined to coordinate charts.
9
Why: (dis)advantages
I
It trivially generalizes to non-autonomous systems.
I
For hyperbolic fixed points, smoothness of T
U , W S ) is easily proven.
(and thus Wloc
loc
I
By construction gives dynamical information:
growth estimates of solution curves.
However: it doesn’t easily generalize to normally hyperbolic
invariant manifolds (NHIMs) because solution curves are global
objects that cannot be confined to coordinate charts.
9
Why: (dis)advantages
I
It trivially generalizes to non-autonomous systems.
I
For hyperbolic fixed points, smoothness of T
U , W S ) is easily proven.
(and thus Wloc
loc
I
By construction gives dynamical information:
growth estimates of solution curves.
However: it doesn’t easily generalize to normally hyperbolic
invariant manifolds (NHIMs) because solution curves are global
objects that cannot be confined to coordinate charts.
9
Why: (dis)advantages
An example: finding
the stable manifold to
a periodic orbit γ(t).
γ(t)
E+
U
Wlo
S
Wlo
E−
Change of coordinates:
y = x − γ(t) turns this into
a non-autonomous problem.
Handle center direction along
˙
γ(t)
by shifting hyperbolic spectral separation to λ0 < 0.
S as a smooth manifold
Apply Perron method to obtain Wloc
U by time-reversal).
(and Wloc
10
Why: (dis)advantages
An example: finding
the stable manifold to
a periodic orbit γ(t).
γ(t)
E+
U
Wlo
S
Wlo
E−
Change of coordinates:
y = x − γ(t) turns this into
a non-autonomous problem.
Handle center direction along
˙
γ(t)
by shifting hyperbolic spectral separation to λ0 < 0.
S as a smooth manifold
Apply Perron method to obtain Wloc
U by time-reversal).
(and Wloc
10
Why: (dis)advantages
An example: finding
the stable manifold to
a periodic orbit γ(t).
γ(t)
E+
U
Wlo
S
Wlo
E−
Change of coordinates:
y = x − γ(t) turns this into
a non-autonomous problem.
Handle center direction along
˙
γ(t)
by shifting hyperbolic spectral separation to λ0 < 0.
S as a smooth manifold
Apply Perron method to obtain Wloc
U by time-reversal).
(and Wloc
10
Why: (dis)advantages
An example: finding
the stable manifold to
a periodic orbit γ(t).
γ(t)
E+
U
Wlo
S
Wlo
E−
Change of coordinates:
y = x − γ(t) turns this into
a non-autonomous problem.
Handle center direction along
˙
γ(t)
by shifting hyperbolic spectral separation to λ0 < 0.
S as a smooth manifold
Apply Perron method to obtain Wloc
U by time-reversal).
(and Wloc
10
How: apply to NHIMs
A normally hyperbolic invariant manifold M:
M
11
How: apply to NHIMs
A submanifold M ⊂ Q is a NHIM if:
invariant
splitting
hyperbolic
12
How: apply to NHIMs
A submanifold M ⊂ Q is a NHIM if:
invariant ∀ t ∈ R : Φt (M) = M
splitting
hyperbolic
12
How: apply to NHIMs
A submanifold M ⊂ Q is a NHIM if:
invariant ∀ t ∈ R : Φt (M) = M
splitting TM Q = TM ⊕ N+
⊕ N− ,
DΦt = DΦt0 ⊕ DΦt+ ⊕ DΦt−
hyperbolic
12
How: apply to NHIMs
A submanifold M ⊂ Q is a NHIM if:
invariant ∀ t ∈ R : Φt (M) = M
splitting TM Q = TM ⊕ N+
⊕ N− ,
DΦt = DΦt0 ⊕ DΦt+ ⊕ DΦt−
hyperbolic ∀ t > 0 : DΦt− 6 C e ρ− t , ρ− < 0
∀ t 6 0 : DΦt+ 6 C e ρ+ t , ρ+ > 0
∀ t ∈ R : DΦt0 6 C e ρ0 |t| , ρ0 < min(−ρ− , ρ+ ).
12
How: apply to NHIMs
A submanifold M ⊂ Q is a NHIM if:
invariant ∀ t ∈ R : Φt (M) = M
splitting TM Q = TM ⊕ N+
⊕ N− ,
DΦt = DΦt0 ⊕ DΦt+ ⊕ DΦt−
hyperbolic ∀ t > 0 : DΦt− 6 C e ρ− t , ρ− < 0
∀ t 6 0 : DΦt+ 6 C e ρ+ t , ρ+ > 0
∀ t ∈ R : DΦt0 6 C e ρ0 |t| , ρ0 < min(−ρ− , ρ+ ).
Special case: a hyperbolic fixed point M = {0}, ρ0 = 0.
12
How: apply to NHIMs
Two results to generalize.
Existence of (un)stable manifolds:
I
Use same technique as for periodic orbit.
I
Yields a fibration of isochronous (un)stable manifolds.
Persistence under perturbations:
I
Simple for a fixed point: use implicit function theorem.
I
Nontrivial for a NHIM and smoothness is lost.
13
How: apply to NHIMs
Two results to generalize.
Existence of (un)stable manifolds:
I
Use same technique as for periodic orbit.
I
Yields a fibration of isochronous (un)stable manifolds.
Persistence under perturbations:
I
Simple for a fixed point: use implicit function theorem.
I
Nontrivial for a NHIM and smoothness is lost.
13
How: apply to NHIMs
Two results to generalize.
Existence of (un)stable manifolds:
I
Use same technique as for periodic orbit.
I
Yields a fibration of isochronous (un)stable manifolds.
Persistence under perturbations:
I
Simple for a fixed point: use implicit function theorem.
I
Nontrivial for a NHIM and smoothness is lost.
13
How: apply to NHIMs
The standard Perron operator only works for linearization in a
neighborhood around a hyperbolic fixed point. Here, no control
on global dynamics on M, i.e. solutions cannot be localized.
Instead consider a tubular neighborhood M × Y and curves
x(t), y (t) with y bounded. Only linearize the Y component:
v (x, y ) = vx (x, y ) ⊕ (A(x) · y + r (x, y )).
7→ x 0 (t) = ΦM (t, t0 , x0 , y ),
Z∞
0
Ty : x(t), y (t) 7→ y (t) = . . . −
ΨY (t, τ, x) · r (x(τ), y (τ)) dτ.
Tx : y (t), x0
t
Now T = Ty ◦ Tx is a contraction leading to the unique
˜
persisting manifold M.
14
How: apply to NHIMs
The standard Perron operator only works for linearization in a
neighborhood around a hyperbolic fixed point. Here, no control
on global dynamics on M, i.e. solutions cannot be localized.
Instead consider a tubular neighborhood M × Y and curves
x(t), y (t) with y bounded. Only linearize the Y component:
v (x, y ) = vx (x, y ) ⊕ (A(x) · y + r (x, y )).
7→ x 0 (t) = ΦM (t, t0 , x0 , y ),
Z∞
0
Ty : x(t), y (t) 7→ y (t) = . . . −
ΨY (t, τ, x) · r (x(τ), y (τ)) dτ.
Tx : y (t), x0
t
Now T = Ty ◦ Tx is a contraction leading to the unique
˜
persisting manifold M.
14
How: apply to NHIMs
The standard Perron operator only works for linearization in a
neighborhood around a hyperbolic fixed point. Here, no control
on global dynamics on M, i.e. solutions cannot be localized.
Instead consider a tubular neighborhood M × Y and curves
x(t), y (t) with y bounded. Only linearize the Y component:
v (x, y ) = vx (x, y ) ⊕ (A(x) · y + r (x, y )).
7→ x 0 (t) = ΦM (t, t0 , x0 , y ),
Z∞
0
Ty : x(t), y (t) 7→ y (t) = . . . −
ΨY (t, τ, x) · r (x(τ), y (τ)) dτ.
Tx : y (t), x0
t
Now T = Ty ◦ Tx is a contraction leading to the unique
˜
persisting manifold M.
14