Mixed dynamics of 2-dimensional reversible maps with a symmetric couple of quadratic homoclinic tangencies

Amadeu Delshams,J Tomás Lázaro,Marina Gonchenko,Sergey V Gonchenko,

doi:10.3934/dcds.2018196

Abstract

We study dynamics and bifurcations of 2-dimensional reversible maps having a symmetric saddle fixed point with an asymmetric pair of nontransversal homoclinic orbits (a symmetric nontransversal homoclinic figure-8). We consider one-parameter families of reversible maps unfolding the initial homoclinic tangency and prove the existence of infinitely many sequences (cascades) of bifurcations related to the birth of asymptotically stable, unstable and elliptic periodic orbits.

Highlights

In his seminal works in 1974 and 1979, Newhouse [27, 28] developed the concept of structural instability. This notion set up a cornerstone in Bifurcation Theory since it provided “large” domains in the space of dynamical systems where “chaotic non-hyperbolic” behaviour was robust
Newhouse proved [26, 28] that, near any 2-dimensional diffeomorphism with a homoclinic tangency there exist open regions consisting of diffeomorphisms exhibiting nontransversal intersections between stable and unstable manifolds of hyperbolic basic sets
In case (a) the diffeomorphism has a symmetric couple of saddle periodic points O1 and O2 = R(O1), as well as two heteroclinic orbits Γ12 ⊂ W u(O1) ∩ W s(O2) and Γ21 ⊂ W u(O2) ∩ W s(O1) such that R(Γ21) = Γ21, R(Γ12) = Γ12

Summary

Introduction

In his seminal works in 1974 and 1979, Newhouse [27, 28] developed the concept of structural instability. In case (a) the diffeomorphism has a symmetric couple of saddle periodic (fixed) points O1 and O2 = R(O1), as well as two heteroclinic orbits Γ12 ⊂ W u(O1) ∩ W s(O2) and Γ21 ⊂ W u(O2) ∩ W s(O1) such that R(Γ21) = Γ21, R(Γ12) = Γ12.

Objectives

Results

Conclusion