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Heteroclinic primary intersections and codimension one Melnikov method for volume-preserving maps.

Hector E. Lomeli¹, James D. Meiss

¹Department of Mathematics, Instituto Tecnologico Autonomo de Mexico, Mexico, DF 01000.

Chaos (Woodbury, N.Y.)

|June 5, 2003

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Researchers explored the topology of intersecting stable and unstable manifolds in 3D dynamical systems. They found the primary intersection is a submanifold, with bifurcations observed numerically.

Area of Science:

Dynamical Systems and Chaos Theory
Differential Geometry
Topology

Background:

Volume-preserving diffeomorphisms in 3D space are fundamental to understanding complex dynamical systems.
The behavior of hyperbolic fixed points and their associated stable and unstable manifolds dictates system dynamics.
Intersections of these manifolds are crucial for revealing chaotic behavior and topological structures.

Purpose of the Study:

To elucidate the topology of intersecting stable and unstable manifolds for families of volume-preserving diffeomorphisms in R(3).
To understand how these topological features change with system parameters.
To investigate the nature of the primary intersection between these manifolds.

Main Methods:

Studied families of volume-preserving diffeomorphisms with hyperbolic fixed points.

Related Experiment Videos

Employed a codimension-one Melnikov function for perturbative computation of manifold intersections.

Conducted numerical experiments to observe bifurcations and topological changes.

Main Results:

The primary intersection of stable and unstable manifolds is generically a neat submanifold within a fundamental domain.
Perturbative calculations using the Melnikov function provide insights into intersection properties.
Numerical simulations revealed various bifurcations in the homotopy class of the primary intersections.

Conclusions:

The study clarifies the topological structure of manifold intersections in a specific class of 3D dynamical systems.
Parameter-dependent bifurcations indicate complex dynamics and potential for chaos.
The findings contribute to the understanding of global dynamics and topological invariants in Hamiltonian systems.