A general mechanism of diffusion in Hamiltonian Systems DYNAMICAL - PowerPoint PPT Presentation

A general mechanism of diffusion in Hamiltonian Systems DYNAMICAL SYSTEMS: FROM GEOMETRY TO MECHANICS Marian Gidea 1 Rafael de la Llave 2 and Tere Seara 3 1 Yeshiva University, New York 2 Georgia Institute of Technology, Atlanta 3 Universitat Politecnica de Catalunya, Barcelona U. Roma Tor Vergata, February 5–8, 2019 U. Roma Tor Vergata, February 5–8, 2019 Tere M-Seara (UPC) A general mechanism for instability in Hamiltonian systems / 34

Goals of the talk • The problem of Arnold diffusion consists in studying in which Hamiltonian systems the effects of perturbations can accumulate over time to produce effects much larger than the size of the perturbations. Specially in integrable systems. U. Roma Tor Vergata, February 5–8, 2019 Tere M-Seara (UPC) A general mechanism for instability in Hamiltonian systems / 34

Goals of the talk • We will describe a recent mechanism based on the presence of Normally Hyperbolic Invariant Manifolds with stable and unstable manifolds which intersect. • The mechanism is rather robust. • It does not need that the perturbations are Hamiltonian (applies to small dissipation problems or for space craft maneuvers that involve burns). • Can be applied to concrete problems • Enjoys remarkable genericity properties since it does not require non-generic assumptions (for instance convexity). U. Roma Tor Vergata, February 5–8, 2019 Tere M-Seara (UPC) A general mechanism for instability in Hamiltonian systems / 34

Outline 1 Background 2 Shadowing lemmas for NHIM’s 3 Perturbative results 4 A general diffusion result 5 Application: Diffusion in a priori unstable systems U. Roma Tor Vergata, February 5–8, 2019 Tere M-Seara (UPC) A general mechanism for instability in Hamiltonian systems / 34

Background Normal hyperbolicity Normally hyperbolic invariant manifold (NHIM) : f : M → M , C r -smooth, r ≥ r 0 , m = dim M . f (Λ) ⊂ Λ, n c = dim Λ. TM = T Λ ⊕ E u ⊕ E s n s = dim E s , n u = dim E u . m = n c + n s + n u ∃ C > 0, 0 < λ < µ − 1 < 1, s.t. ∀ x ∈ Λ v ∈ E s x ⇔ � Df k x ( v ) � ≤ C λ k � v � , ∀ k ≥ 0 v ∈ E u x ⇔ � Df k x ( v ) � ≤ C λ − k � v � , ∀ k ≤ 0 v ∈ T x Λ ⇔ � Df k x ( v ) � ≤ C µ | k | � v � , ∀ k ∈ Z In this case W u , s (Λ) = � x ∈ Λ W u , s ( x ) U. Roma Tor Vergata, February 5–8, 2019 Tere M-Seara (UPC) A general mechanism for instability in Hamiltonian systems / 34

Background Scattering map: homoclinic channel Assume that f has a Normally Hyperbolic Invariant Manifold (NHIM) Λ Assume W u (Λ) intersects transversally W s (Λ) along a homoclinic manifold Γ satisfying certain extra transversality conditions (Γ is transverse to the foliation). We call Γ an homoclinic channel. U. Roma Tor Vergata, February 5–8, 2019 Tere M-Seara (UPC) A general mechanism for instability in Hamiltonian systems / 34

Background Scattering map Definition Wave maps: Ω ± : Γ → Λ, Ω ± ( x ) = x ± ⇔ x ∈ W s , u ( x ± ) ∩ Γ Restrict Γ so that Ω ± diffeomorphisms Scattering map: s : Ω − (Γ) → Ω + (Γ) given by s = Ω + ◦ (Ω − ) − 1 Properties s is symplectic, if M , Λ , f are symplectic [Delshams,de la Llave,Seara,2008] s ( x − ) = x + � d ( f − m ( x ) , f − m ( x − )) → 0, d ( f n ( x ) , f n ( x + )) → 0, as m , n → ∞ U. Roma Tor Vergata, February 5–8, 2019 Tere M-Seara (UPC) A general mechanism for instability in Hamiltonian systems / 34

Shadowing lemmas for NHIM’s A general Shadowing Lemma for NHIM’s Theorem 1 [Gidea, de la Llave, S.] Given f : M → M , is a C r -map, r ≥ r 0 , Λ ⊆ M NHIM, Γ ⊆ M homoclinic channel. s = s Γ : Ω − (Γ) → Ω + (Γ) is the scattering map associated to Γ. Assume that Λ and Γ are compact. Then, for every δ > 0 there exists m ∗ ∈ N and a family of functions i : N 2 i +1 → N , i ≥ 0, such that, for every pseudo-orbit { y i } i ≥ 0 in Λ of the form n ∗ y i +1 = f m i ◦ s ◦ f n i ( y i ) , for all i ≥ 0, with m i ≥ m ∗ and n i ≥ n ∗ i ( n 0 , . . . , n i − 1 , n i , m 0 , . . . , m i − 1 ), there exists an orbit { z i } i ≥ 0 of f in M such that, for all i ≥ 0, z i +1 = f m i + n i ( z i ) , and d ( z i , y i ) < δ. n ∗ and m ∗ i also depend on the angle between ( W u , W s ) along Γ Related result: Gelfreich, Turaev Arnold Diffusion in a priori chaotic symplectic maps, Commun. Math. Phys., 2017, talk of A. Clarke U. Roma Tor Vergata, February 5–8, 2019 Tere M-Seara (UPC) A general mechanism for instability in Hamiltonian systems / 34

Shadowing lemmas for NHIM’s A general Shadowing Lemma for NHIM’s: Proof The result is true if we use several scattering maps to build the pseudo-orbit: y i +1 = f m i ◦ s α i ◦ f n i ( y i ) We have two proofs, one uses the topological method of correctly aligned windows. The one we present here uses the obstruction argument. We build a nested sequence of closed balls B i +1 ⊂ B i ⊂ B δ ( y 0 ) ( y 0 is the first point of the pseudo-orbit), such that: if z 0 ∈ B k = � 0 ≤ i ≤ k B i , z 0 ∈ B δ ( y 0 ) z i +1 = f m i + n i ( z i ) ∈ B δ ( y i +1 ) for i = 0 , 1 . . . , k , for any k ∈ N . Moreover, taking z 0 ∈ B ∞ = � i ≥ 0 B i � = ∅ , one has that: z i +1 ∈ B δ ( y i +1 ) for any i ∈ N . The argument will be done by induction. At every step of the process we will have several choices which give us different orbits U. Roma Tor Vergata, February 5–8, 2019 Tere M-Seara (UPC) A general mechanism for instability in Hamiltonian systems / 34

Shadowing lemmas for NHIM’s Choice of m ∗ We will take δ > 0 and consider V Λ and V Γ contained in neighborhoods of size δ of the compact manifolds Λ and Γ. We define m ∗ = m ∗ ( δ ) such that: given any point p ∈ Γ, for any m ≥ m ∗ , one has that f ± m ( p ) ∈ V Λ . Moreover, this property also holds for points in W u , s (Λ) ∩ V Γ when iterating them backwards or forward respectively. We will give an extra condition to m ∗ . U. Roma Tor Vergata, February 5–8, 2019 Tere M-Seara (UPC) A general mechanism for instability in Hamiltonian systems / 34

Shadowing lemmas for NHIM’s General step • Assume we have p ∈ Γ and let p − , p + ∈ Λ, such that s ( p − ) = p + • x ∈ W s ( f − k ( p − )), B = B ρ ( x ), ρ > 0 small enough B ⊂ B δ ( f − k ( p − )) ⊂ V Λ , • W s ( p + ) intersects transversally W u (Λ) at the homoclinic point p • Lambda Lemma: there exists k ∗ > 0 such that: if k > k ∗ , there exists a point ¯ x ∈ W s ( p + ) ∩ V Γ such that f − k (¯ x ) ∈ B . x such that f − k (¯ x ) ∈ f − k ( V ) ⊂ B . • By continuity, ∃ V ⊂ V Γ centered at ¯ U. Roma Tor Vergata, February 5–8, 2019 Tere M-Seara (UPC) A general mechanism for instability in Hamiltonian systems / 34

Shadowing lemmas for NHIM’s The value of k ∗ depends on ρ (and δ ) and also on the angle of intersection of the stable and unstable manifolds of Λ along Γ. x and its neighborhood V depend on the k > k ∗ we choose. The point ¯ U. Roma Tor Vergata, February 5–8, 2019 Tere M-Seara (UPC) A general mechanism for instability in Hamiltonian systems / 34

Shadowing lemmas for NHIM’s Inductive construction • We construct the shadowing orbit { z i } once the pseudo-orbit { y i } is given. • Remember y i +1 = f m i ( s ( f n i ( y i ))), then z i +1 = f m i ( f n i ( z i )). i , and m ∗ do not depend of the given • The required values of n ∗ pseudo-orbit, but only on the numbers n i , m j . U. Roma Tor Vergata, February 5–8, 2019 Tere M-Seara (UPC) A general mechanism for instability in Hamiltonian systems / 34

Shadowing lemmas for NHIM’s Inductive construction. First step Fisrt step: p − = f n 0 ( y 0 ), p + = s ( f n 0 ( y 0 )) Choose x 0 ∈ W s ( y 0 ) and B 0 = B ρ 0 ( x 0 ) of radius ρ 0 > 0: x 0 ∈ B 0 ∩ W s ( y 0 ) � = ∅ . B 0 ⊂ B δ ( y 0 ) ⊂ V Λ , There exists m ∗ = k ∗ ( ρ 0 , δ ) such that, taking k = n 0 > n ∗ 0 = m ∗ , x 0 ∈ W s ( s ( f n 0 ( y 0 ))) ∩ V Γ and a a ball V 0 ⊂ V Γ : ∃ ¯ such that f − n 0 (¯ x 0 ) ∈ f − n 0 ( V 0 ) ⊂ B 0 ⊂ B δ ( y 0 ) ⊂ V Λ . U. Roma Tor Vergata, February 5–8, 2019 Tere M-Seara (UPC) A general mechanism for instability in Hamiltonian systems / 34

Shadowing lemmas for NHIM’s Inductive construction. Intermediate step The value of ρ 0 and therefore the value of m ∗ will be fixed from now on. Remember: y 1 = f m 0 ( s ( f n 0 ( y 0 )). x 0 ∈ W s ( s ( f n 0 ( y 0 ))) ¯ Therefore f m 0 (¯ x 0 ) ∈ W s ( f m 0 ( s ( f n 0 ( y 0 ))) = W s ( y 1 ). U. Roma Tor Vergata, February 5–8, 2019 Tere M-Seara (UPC) A general mechanism for instability in Hamiltonian systems / 34

Shadowing lemmas for NHIM’s Inductive construction. Intermediate step We know that, if m 0 > m ∗ , 1 f m 0 (¯ x 0 ) ∈ W s ( f m 0 ( s ( f n 0 ( y 0 ))) = W s ( y 1 ) ∈ V Λ . By continuity there exists a ball U 1 centered at f m 0 (¯ x 0 ) such that: 2 x 0 ) ∈ U 1 f − m 0 ( U 1 ) ⊂ V 0 ⊂ V Γ . U 1 ⊂ B δ ( y 1 ) ⊂ V Λ , f m 0 (¯ U. Roma Tor Vergata, February 5–8, 2019 Tere M-Seara (UPC) A general mechanism for instability in Hamiltonian systems / 34