The Perron method: Jaap Eldering Utrecht University
Transcription
The Perron method: Jaap Eldering Utrecht University
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