Dynamics of Periodically Perturbed Homoclinic Solutions

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1 Dynamics of Periodically Perturbed Homoclinic Solutions Qiudong Wang University of Arizona

2 Part I Structure of Homoclinic Tangles

3 Unperturbed equation Let (x,y) R 2 be the phase variables. dx dt = αx+f(x,y), dy = βy +g(x,y) dt where f,g are high order terms at (0,0). Dissipative saddle: 0 < β < α. Homoclinic solution: (0, 0) is with a homoclinic solution l = {l(t) = (a(t),b(t)) R 2, t R}.

4 Periodically perturbed equation where dx dt = αx+f(x,y)+µp(x,y,t), dy = βy +g(x,y)+µq(x,y,t) dt for some T > 0. P(x,y,t+T) = P(x,y,t), Q(x,y,t+T) = Q(x,y,t) Long history dated back to Poincaré Two Generic Scenarios: Scenario (a) Scenario (b) There is an established way of analysis

5 Established Way of Analysis Part I Theoretic Analysis Compute Melnikov function to verify that we are in scenario (a) Use Smale s construction to confirm the existence of complicated dynamics. Part II Numerical Evidence To find a few values of µ that offer numerical plots of chaos y(kt) x(kt) x(kt) kt x X(ω) ω

6 Long Existing Problems There is a substantial gap between the theory and the numerical plots Numerical plots are likely something with an attractive basin of positive Lebesgue measure But the attractive basin of a horseshoe is measure zero Dynamical structure of entire homoclinic tangles Horseshoe is only a participating part Other participating part? The way all participating parts fit together?

7 Our Results Standing Assumption: Melnikov function z = M(t) is a Morse function and its graph is transversal to the t-axis. Scenario (a): Case of intersections (With Ali Oksasgolu, JDE, 2011) We have periodic occurrence of three types of tangles as µ 0, (i) Transient Tangle: Λ contains nothing else but one horseshoe of infinitely many branches. (ii) Tangle Dominated by Sinks: Λ contains nothing else but some sinks and one horseshoe of infinitely many branches. (Stable dynamics) (iii) Tangle with Henon-like Attractors: Positive measure set of µ for which Λ admit strange attractors with SRB measures. (Chaos)

8 Scenario (b): Case of non-intersections (With W. Ott, CPAM, 2011) Dynamics depend on the magnitude of forcing frequency ω (i) For small ω: A Attractive Quasi-periodic tori (ii) As ω increases: Transition to chaos (iii) For large ω: Non-hyperbolic attractors: Horseshoes Newhouse tangency (and the associated sinks and Hénon-like attractors) Rank one attractors (Wang & Young)

9 Tool of Study: The separatrix map Looking into a different direction t = 0 The time T map t = T t New Return Map Separatrix map has been used before Used by many in Hamiltonian dynamics It was the main tool for Treschev to construct his theory on Arnold diffusion It was also used by Afraimovich and Shilnikov to prove the existence of horseshoes in scenario (b) assuming ω is large

10 Our contribution: (i) Through separatrix map we can apply many theory of non-uniformly hyperbolic maps SRB measures Newhouse tangency Hénon-like attractors (Mora & Viana) Rank one attractors (Wang & Young) (ii) We acquired a comprehensive understanding on the structure of homoclinic tangles (iii) Our theory matches results of numerical simulation in perfection

11 Part II Numerical Investigation

12 Equation of Study Step 1: We start with the Duffing equation d 2 q dt 2 q +q2 = 0. Step 2: Add non-linear dumping terms d 2 q dt 2 +(λ γq2 ) dq dt q +q2 = 0. For λ > 0 small, there exists γ λ so that it has a homoclinic loops to a dissipative saddle. Step 3: Add periodic perturbations d 2 q dt 2 +(λ γ λq 2 ) dq dt q +q2 = µsin2πt

13 A directly observable object is a dynamical object with an attractive basin of positive Lebesgue measure. Claims To Verify There are three major dynamical scenarios competing in parameter space of µ: (I) Homoclinic tangle contains no directly observable objects. (II) Sinks representing stable dynamics (III) Hénon-like attractors admitting SRB measures representing chaos As µ 0, Scenarios (I)-(III) would repeat periodically in an accelerated fashion as µ 0.

14 Numerical Procedure Numerical scheme Fourth-order Runge-Kutta routine Values of λ and γ λ = 0.5, γ λ = Initial phase position (q 0, p 0 ) = (0.01, 0.0). Parameters varied µ ranged from 10 3 to t 0 ranged in [0,1). Compute the solution for each fixed combination of µ and t 0.

15 Simulation Results (Physica D, 2010) Transient Tangle Let t 0 run over [0,1) with an increment of all 1,000 solutions leave the neighborhood of the homoclinic solution. Homoclinic tangle contains no object directly observable. Periodic Sink 0.5 (a) 0 p(kt) q(kt) 1.5 (b) 4000 (c) q(kt) Q(ω) k k

16 Hénon-Like Attractor 0.5 (a) 0 p(kt) q(kt) 1.5 (b) 4000 (c) q(kt) Q(ω) k k These are plots of a chaotic attractor close to a periodic solution. They represent a Hénon-like attractor associated to Newhouse tangency (Mora-Viana)

17 Periodicity of Dynamical Behavior Theoretical Multiplicity = e βt = µ Dynamics Actual Ratio Transient Hénon-like Periodic Sink Transient Hénon-like Periodic Sink Transient Hénon-like Periodic Sink Transient Hénon-like Periodic Sink Transient Hénon-like Periodic Sink Transient Hénon-like Periodic Sink

18 Summary (I) A Theory on Homoclinic Tangles We provided a comprehensive description on the overall dynamical structure of homoclinic tangles from a periodically perturbed homoclinic solution. (II) Theories on maps come together Horseshoes, Newhouse sinks and Hénon-like attractors all fall into their places as part of a larger picture. (III) Applications Our results can be applied to the analysis of given equations, such as Duffing equation. (IV) A systematic numerical investigation The behavior of numerical solutions are entirely predictable and are consistent with the predictions of our theory.

19 The separatrix map R t = 0 The time T map t = T t New Return Map It is only partially defined on the annulus. We rigorously derived a formula for R from the equation.

20 The structure of homoclinic tangle R is defined on vertical strips. The action of R on one strip Compressing in vertical direction Stretching in horizontal direction, to infinity in length towards both end Folded, and put back into A. The image moves horizontally in a constant speed with respect to a lnµ 1 as µ 0.

21 Happy Birthday, Ken I am forever grateful For all your taught to and did for me