Continuation and collapse of homoclinic tangles
This work advances the understanding of homoclinic tangles in dynamical systems, providing a theoretical and numerical framework for analyzing bifurcations in systems like the Hénon map.
The paper studies the bifurcation structure of homoclinic tangles in parameterized systems, showing that the maximal invariant set near a homoclinic tangency can be characterized by bifurcation equations indexed by symbolic sequences. For the Hénon family, they numerically continue multi-humped orbits and explain the homoclinic network using graph-theoretical arguments.
By a classical theorem transversal homoclinic points of maps lead to shift dynamics on a maximal invariant set, also referred to as a homoclinic tangle. In this paper we study the fate of homoclinic tangles in parameterized systems from the viewpoint of numerical continuation and bifurcation theory. The bifurcation result shows that the maximal invariant set near a homoclinic tangency, where two homoclinic tangles collide, can be characterized by a system of bifurcation equations that is indexed by a symbolic sequence. For the Hénon family we investigate in detail the bifurcation structure of multi-humped orbits originating from several tangencies. The homoclinic network found by numerical continuation is explained by combining our bifurcation result with graph-theoretical arguments.