Aubry Mather Theory from a Topological Viewpoint

Aubry Mather Theory from a Topological Viewpoint III. Applications to Hamiltonian instability Marian Gidea,2 Northeastern Illinois University, Chicago 2 Institute for Advanced Study, Princeton WORKSHOP ON INTERACTIONS BETWEEN DYNAMICAL SYSTEMS AND PARTIAL DIFFERENTIAL EQUATIONS (JISD202) May 28 - June, 202 Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 / 25

Outline Hamiltonian instability 2 3 References Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 2 / 25

Hamiltonian instability Hamiltonian systems Symplectic manifold: (M, ω) M = (2n) dimensional smooth manifold ω =closed non-degenerate differential 2-form Hamiltonian function (total energy): H : M R smooth function Hamiltonian vector field: X H defined by i XH ω = dh Hamiltonian flow: φ : R M M given by dφt dt L XH H = 0 and (φ t ) (ω) = ω = X H φ t Darboux coordinates: local coordinates (q, p) on M, with q = (q,..., q n ), p = (p,..., p n ), s.t. ω = dq dp = n i= dq i dp i In Darboux coordinates: X H = H p { q = H p Hamilton s equations: ṗ = q H q p H q Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 3 / 25

Examples Hamiltonian instability Set of particles in a conservative force field F = V : H(p, q) = n i= 2m i p 2 i + V (q) { dqi dt = m i p i = H p i dp i dt d = m 2 q i i dt 2 Pendulum: H(p, q) = { dq dt = m p dp dt = sin(q) Rotator: H(I, θ) = 2 I 2 { dθ dt = I di dt = 0 = F (q) = d dq i V (q) 2m p2 + ( cos(q)) = H q i N-body problem: H(p, q) = n i= 2m i p i 2 i<j Gm i m j q i q j Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 4 / 25

Hamiltonian instability Integrable Hamiltonians Poisson bracket of two functions: {f, g} = ω(x f, X g ) Integral of motion f : M R if f constat along solutions {f, H} = 0 H is always an integral of motion H is (Liouville) integrable f = H, f 2,..., f n integrals of motion, independent, and in involution, i.e., {f i, f j } = 0 Liouville-Arnold Theorem: Assume H is integrable and the level set f (c) is compact and connected, where f = (f,..., f n ) and c is a regular value. Then there is a neighborhood U of f (c) and action-angle coordinates (I, φ) = (I,..., I n, φ,..., φ n ) on U such that: U is foliated by invariant tori T I T n T I = {I = const.} φ t TI = {φ φ + tω} where ω = I / φ Example: the N-body problem is integrable for n = 2 and non-integrable for n 3 Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 5 / 25

Hamiltonian instability Nearly integrable hamiltonian systems Fundamental problem of dynamics (Poincaré) long term behavior of nearly integrable Hamiltonian systems: H ε (I, φ) = H 0 (I ) + εh (I, φ), (I, φ) R n T n Stability: K compact C(ε) s.t. I ε (t) I 0 < C(ε) for any (I 0, φ 0 ) K, with C(ε) 0 as ε 0 Equations of motion: İ = H ε / φ = ε H / φ φ = H ε / I = H 0 / I + ε H / I For ε = 0, the system is integrable permanent stability For ε 0 small KAM Theorem: most tori survive the perturbation provided H ε is sufficiently smooth and 2 H 0 / I 2 is invertible Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 6 / 25

Hamiltonian instability Stability vs. instability KAM theorem: if H ε has 2-degrees of freedom, then the phase space is separated by KAM tori into regions where I (t) I (0) < C ε for all t KAM tori do not separate the energy manifold for system with more than 2 degrees of freedom Nekhoroshev stability: in higher dimensions, exponentially long-term stability; however unstable orbits are possible Applications: stability particle accelerators, plasma confinement devices instability gravity assisted space missions Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 7 / 25

Example Hamiltonian instability Instability in the coupled standard map I n+ = I n + K sin(θ n ) + ε sin(θ n + φ n ) θ n+ = θ n + I n+ J n+ = J n +K sin(φ n )+ε sin(θ n +φ n ) φ n+ = φ n + J n+ Choice of parameters: K = 0.85, K = 0.5, ε = 0 2 (I 0, θ 0 ) boundary of a BZI (J 0, φ 0 ) hyperbolic fixed point n = 0 3 and n = 0 6 3 2 0 - -2-3 -3-2 - 0 2 3 Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 8 / 25

Hamiltonian instability Example Instability in the coupled standard map In+ = In + K sin(θn ) + ε sin(θn + φn ) θn+ = θn + In+ Jn+ = Jn +K 0 sin(φn )+ε sin(θn +φn ) φn+ = φn + Jn+ Choice of parameters: K = 0.85, K 0 = 0.5, ε = 0 2 (I0, θ0 ) boundary of a BZI (J0, φ0 ) hyperbolic fixed point n = 03 and n = 06 Marian Gidea (IAS) Aubry Mather theory 3 2 0 - -2-3 -3-2 - 0 May 28 - June, 202 2 3 8 / 25

Hamiltonian instability Arnold s mechanism of instability Arnold s example [Arnold (964)]: H µ,ε (p, q, I, φ, t) = H 0 + µh µ + εh ε = I 2 2 + p2 2 +µ(cos q )+ε(cos q ) (sin φ+cos t) Add an extra variable 3-degrees of freedom autonomous Hamiltonian For ε = 0: 4-dim s normally hyperbolic manifold its stable and unstable manifolds coincide 3-dim s intersection with energy manifold this is filled out with 2-dim s tori they possess stable and unstable manifolds the stable and unstable manifolds coincide The perturbation is carefully chosen (non-generic) it does not destroy any invariant tori Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 9 / 25

Hamiltonian instability Arnold s mechanism of instability For ε 0 small (0 < ε µ ) perturbation does not affect the tori stable and unstable manifolds of nearby tori intersect transversally transition tori obstruction argument orbits for which action I changes O() over time Arnold s mechanism of instability: transition chains of tori invariant tori of irrational rotation vectors linked via transverse heteroclinic orbits trajectories shadowing transition chains Arnold diffusion conjecture: generic perturbations of integrable systems trajectories that visit a large fraction of the phase space; also symbolic dynamics Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 0 / 25

Mechanical system: H ε (p, q, I, φ, t) = ± ( 2 p2 + V (q) ) + h 0 (I ) + εh(p, q, I, φ, t; ε), (p, q, I, φ, t) R T R T T [Holmes, Marsden,982] Claim: Under some assumptions, that are verifiable and generic, there exists ε 0 such that for each ε 0 in [ ε 0, ε 0 ], the Hamiltonian system has orbits x(t) s.t., for some T : q + - p Ι φ I (x(t )) I (x(0)) = O(). : KAM theorem gaps of size O( ε) are created between KAM tori; the splitting of the stable and unstable manifolds is only O(ε) h t Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 / 25

Mechanical system: H ε (p, q, I, φ, t) = ± ( 2 p2 + V (q) ) + h 0 (I ) + εh(p, q, I, φ, t; ε), (p, q, I, φ, t) R T R T T [Holmes, Marsden,982] Claim: Under some assumptions, that are verifiable and generic, there exists ε 0 such that for each ε 0 in [ ε 0, ε 0 ], the Hamiltonian system has orbits x(t) s.t., for some T : I (x(t )) I (x(0)) = O(). : KAM theorem gaps of size O( ε) are created between KAM tori; the splitting of the stable and unstable manifolds is only O(ε) Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 / 25

Normal hyperbolicity H ε (p, q, I, φ, A, t) = ±( 2 p2 + V (q)) + h 0 (I ) + A + εh(p, q, I, φ, t; ε) autonomous Fix energy, F ε time- map Λ 0 = {p = 0, q = 0, I, φ} - normally hyperbolic invariant manifold (NHIM) for F 0 F 0 restricted to Λ 0 exact symplectic twist map pendulum rotator X For ε 0 small, Λ ε NHIM for F ε F ε restricted to Λ ε exact symplectic twist map The unperturbed system Λ M is a NHIM for F : M M: TM = T Λ E u E s C > 0, 0 < λ < µ < v Ex s DF x k(v) Cλk v, k 0 v Ex u DF x k(v) Cλ k v, k 0 v T x Λ DFx k(v) Cµ k v, k Z Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 2 / 25

Scattering map Details [Delshams, de la Llave, Seara,2008] Non-degeneracy condition on V W u (Λ ε ), W s (Λ ε ) intersect transversally along a homoclinic manifold Γ ε (fixed) at an angle O(ε) Wave maps: Ω ± : Γ ε Λ ε Ω ± (x) = x ± {x} = W s,u (x ± ) Γ ε Restrict Γ ε Ω ± diffeomorphisms Scattering map: S ε : D ε D + ε given by S ε = Ω + (Ω ) If S ε (x ) = x + then I (x + ) I (x ) = O(ε) Two dynamics: Inner dynamics: twist map F ε on Λ ε Outer dynamics: scattering map S ε on Λ ε Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 3 / 25

Averaging and KAM methods Inner dynamics: averaging normal form +O(ε 2 ) far from resonances: O(ε 3/2 )-spaced primary KAM tori resonances of order j: gaps O(ε j/2 ) (j =, 2) within resonant regions: O(ε 3/2 )-spaced primary KAM tori, secondary KAM tori, invariant manifolds of hyperbolic periodic points Outer dynamics: S ε (T ) T with T, T at O(ε) Resonances determine BZI s the boundaries of the BZI s are Lipschitz tori Transition chains of primary KAM tori and boundary tori interspersed with BZI s Outcome: existence of diffusing orbits through a different mechanism from [Delshams, de la Llave, Seara,2006a]; similar to [Cheng,Yan,2004] Dε Λε S Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 4 / 25

Topological mechanism for diffusion (i) Λ 2-dimensional NHIM, F area preserving monotone twist map on Λ (ii) {T i } i Z Λ primary, invariant, Lipschitz tori in Λ, satisfying certain technical conditions (iii) (T i ) i=ik +,...,i k+ forms a transition chain W u (T i ) W s (T i+ ) for all i (iv) The region in Λ between T ik and T ik + is a BZI (v) Inside each BZI between T ik and T ik+ there exist {Σ ω k, Σ ω k 2,..., Σ ω k mk } Aubry-Mather (A-M) sets GAP Theorem [M.G.,Robinson,2007,2009,200,202] (ɛ i ) i Z, {ns k } s=,...,mk, z M such that the orbit of z goes ε i -close to each T i, the orbit of z follows each Σ ω k s for ns k iterates. Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 5 / 25

Outline of the proof of the main theorem Construct correctly aligned windows {R i, R i } i Z in Λ along the transition chains of tori and across the BZI s window topological rectangle a window is correctly aligned with another window the image of the first window crosses the second window all the way through Correct alignment under different mappings R i is correctly aligned with R i+ under S R i+ is correctly aligned with R i+ under F K i+ for some K i+ large enough Lambda Lemma {R i, R i } can be thickened to m-dimensional windows {W i, W i } that are correctly aligned under some powers of F Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 6 / 25

Two results Proposition. Given Z a BZI bounded by T and T 2 (not topologically transitive). For any z T, z 2 T 2, U 0, V 0 neighborhoods of z, z 2, there exists an orbit that goes from U 0 to V 0. Proposition 2. Given Z a BZI bounded by T, T 2, and {Σ ωs } i=,...,m vertically ordered. For any z T, z 2 T 2, U 0 neighborhood of z, V 0 neighborhood of z 2, {n s } s=,...,m an increasing sequence in N, there exists an orbit F n (z) that starts in an U 0, ends in V 0, and shadows each Σ ωs for n s iterates s.t. 3 2 0 - -2-3 -3-2 - 0 2 3 π x (F j (w s )) < π x (F j (ζ)) < π x (F j (w 2 s )) for n s -consecutive j s, and some w s, w 2 s Σ ωs. Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 7 / 25

Proof of Proposition choose Σ ρ < Σ ρ < Σ ρ A-M sets W ε (p ) neighborhood of p Σ ρ cl[ j=0 F j (W ε (p ))] T [Kaloshin,2003] U m U m... U 0 s.t. F jm (U m ) intersects W ε (p ) + (h m, 0) F jm (U m ) crosses the gaps [aρ m, bρ m ] of Σ ρ and [aρ m, b m ρ of Σ ] ρ [aρ m, bρ m ] and [aρ m, b m ρ shifted apart ] F j (U ) F jm (U m ) forms an arch over some part of Σ ρ a neighborhood U of a point in Σ ρ Similarly an arch over a part of some A-M set Σ ρ2 near T 2 a neighborhood V of a point in Σ ρ2 Mather connecting property: orbit from U to V orbit from U 0 to V 0 Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 8 / 25

Proof of Proposition choose Σ ρ < Σ ρ < Σ ρ A-M sets W ε (p ) neighborhood of p Σ ρ cl[ j=0 F j (W ε (p ))] T [Kaloshin,2003] U m U m... U 0 s.t. F jm (U m ) intersects W ε (p ) + (h m, 0) F jm (U m ) crosses the gaps [aρ m, bρ m ] of Σ ρ and [aρ m, b m ρ of Σ ] ρ [aρ m, bρ m ] and [aρ m, b m ρ shifted apart ] Birkhoff Zone of Instability Σ ρ'' f j0 (U ) V ε (p ) Σ ρ' p f jm (U ) copy of V ε (p ) F j (U ) F jm (U m ) forms an arch over some part of Σ ρ a neighborhood U of a point in Σ ρ V T f jm (ζ f j0 ) (ζ ) Σ ρ Similarly an arch over a part of some A-M set Σ ρ2 near T 2 a neighborhood V of a point in Σ ρ2 Mather connecting property: orbit from U to V orbit from U 0 to V 0 Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 8 / 25

Proof of Proposition 2 Theorem (Hall): Given {Σ ωs } s Z. For every {n s } s=,...,m N, {F j (ζ)} that follows each Σ ωs for n s iterates, i.e., π x (F j (ws )) < π x (F j (ζ)) < π x (F j (ws 2 )) for n s -consecutive j s, and some ws, ws 2 Σ ωs. Recursive argument: start with adjacent points w, w 2 in Σ ω construct windows R F j (w )F j (w 2), j =, n, with R F j (w ),F j (w 2) correctly aligned with R F j+ (w ),F j+ (w 2) under F monotone orbits p 0 near y = 0 that gets near y =, and p near y = that gets near y = 0 and stays there for m iterates then R F n +k +m (w ),F n +k +m (w 2) correctly aligned with R F k +m (p0),f k +m (p ), of width > hence R F n +k +m (w ),F n +k +m (w 2) correctly aligned with R w 2,w2 2 3 2 0 - -2-3 -3-2 - 0 2 3 Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 9 / 25

Proof of Proposition 2 l 0 > 0, R z 0,z0 2 such that F l 0 (U 0 ) cl(r z 0,z0 2 ) has a component that is a positive diagonal D 0 in R z 0,z0 2 j 0 > 0, R w,w 2 with w, w 2 Σ ω such that F j 0 (D 0 ) cl(r w,w 2 ) has a component D that is a positive diagonal in R w,w 2 upper and lower edges of D contained in F j 0 (bd(u 0 )) Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 20 / 25

Proof of Proposition 2 [Hall,989] R R 2... R m negative diagonals of R w,w 2 s.t. F js +ns (R s ) is a positive diagonal in R w s,ws 2, where [w s, ws 2 ] is a gap in Σ ωs R m intersects F j 0 (bd(u 0 )) Similar argument about T 2 There exists an orbit that goes from bd(u 0 ) to bd(v 0 ) and shadows each Σ ωs Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 2 / 25

Outline of the proof of the main theorem Crossing over isolated invariant tori and resonances Non-generic Divide the region into BZI s separated by isolated invariant tori or separatrices Cross each BZI using Proposition, Proposition 2, and the Birkhoff-Smale s Homoclinic Orbit mechanism S Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 22 / 25

Outline of the proof of the main theorem Crossing over isolated invariant tori and resonances Non-generic Divide the region into BZI s separated by isolated invariant tori or separatrices Cross each BZI using Proposition, Proposition 2, and the Birkhoff-Smale s Homoclinic Orbit mechanism S S Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 22 / 25

Outline of the proof of the main theorem Linking windows along transition chains with windows across BZI Starting from two successive BZI s, we construct windows forward along one transition chain and we construct windows backward along the other transition chain until they meet By the torus T j where they meet we squeeze a new window R 2j+ s.t. R 2j+ correctly aligned with R 2j+ under identity R 2j+ correctly aligned with R 2j+2 under F K j with K j sufficiently large R' 2j+ R 2j+ S R 2j S R 2j+2 R 2j+3 T j+ T j T j- Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 23 / 25

Outline of the proof of the main theorem Linking windows along transition chains with windows across BZI Starting from two successive BZI s, we construct windows forward along one transition chain and we construct windows backward along the other transition chain until they meet By the torus T j where they meet we squeeze a new window R 2j+ s.t. R 2j+ correctly aligned with R 2j+ under identity R 2j+ correctly aligned with R 2j+2 under F K j with K j sufficiently large S R 2j S R 2j+2 R 2j+3 T j+ T j T j- Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 23 / 25

Outline of the proof of the main theorem Linking windows along transition chains with windows across BZI Starting from two successive BZI s, we construct windows forward along one transition chain and we construct windows backward along the other transition chain until they meet By the torus T j where they meet we squeeze a new window R 2j+ s.t. R 2j+ correctly aligned with R 2j+ under identity R 2j+ correctly aligned with R 2j+2 under F K j with K j sufficiently large S R 2j F K'j (R' 2j+ ) S R 2j+2 R 2j+3 T j+ T j T j- Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 23 / 25

Outline of the proof of the main theorem Existence of shadowing orbit We obtain a sequence of windows in Λ... R 2j, R 2j+, R 2j+2, R 2j+3,... with R 2j correctly aligned with R 2j+ under S, R 2j+ correctly aligned with R 2j+2 under F K j, etc... The Shadowing lemma for normally hyperbolic manifold an orbit in M which visits the ε j -neighborhoods of the tori (T j ) j=jk +,...,j k and crosses the BZI s between T jk and T jk +, for all j, k. Orbits found similar to those through variational methods [Xia,998],[Cheng and Yan,2004] but no minimizers may take a long time to squeeze through Aubry-Mather sets T jk- T jk Σ jk Σ jk+ T jk+ T jk+2 Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 24 / 25

[] A. Delshams, R. de la Llave and T. M. Seara. A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model. Mem. Amer. Math. Soc. 79 (844) (2006), viii+4. [2] A. Delshams, R. de la Llave and T. M. Seara. Geometric properties of the scattering map of a normally hyperbolic invariant manifold. Adv. Math. 27(3) (2008), 096-53. [3] A. Delshams, M. Gidea, R. de la Llave and T. M. Seara. Geometric approaches to the problem of instability in Hamiltonian systems. An informal presentation. Hamiltonian Dynamical Systems and Applications. Springer, Dordrecht, 2008, pp. 285-336. [4] M. Gidea and R. de la Llave. Topological methods in the instability problem of Hamiltonian systems. Discrete Contin. Dyn. Syst. 4(2) (2006), 295-328. [5] M. Gidea and C. Robinson. Diffusion along transition chains of invariant tori and Aubry-Mather sets. Ergod. Th. & Dynam. Sys. (202), to appear. Marian Gidea (IAS) Aubry Mather theory May 28 - June, 202 25 / 25 References References