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Unfolding homoclinic connections formed by corner intersections in piecewise-smooth maps
1Institute of Fundamental Sciences, Massey University, Palmerston North, New Zealand.
Chaos (Woodbury, N.Y.)
|August 1, 2016
Summary
This study introduces homoclinic corners in piecewise-smooth dynamical systems. These bifurcations occur when stable and unstable manifolds intersect at non-smooth points, leading to unstable periodic solutions.
Area of Science:
- Dynamical Systems
- Bifurcation Theory
- Piecewise-Smooth Systems
Background:
- Manifolds of invariant sets in piecewise-smooth maps are piecewise-smooth.
- Parameter variations can cause intersections between stable and unstable manifolds at non-differentiable points.
- This phenomenon is termed a homoclinic corner, analogous to homoclinic tangency in smooth systems.
Purpose of the Study:
- To analyze generic homoclinic corners for saddle fixed points in planar piecewise-smooth continuous maps.
- To understand the unfolding of these bifurcations.
- To investigate the behavior of nearby periodic solutions.
Main Methods:
- Analysis of stable and unstable manifolds of invariant sets.
- Study of bifurcations in piecewise-smooth continuous maps.
- Investigation of saddle fixed points in planar systems.
Main Results:
- A sequence of border-collision bifurcations can limit to a homoclinic corner.
- All periodic solutions in the vicinity of a homoclinic corner are unstable.
- Homoclinic corners represent a codimension-one bifurcation in piecewise-smooth systems.
Conclusions:
- Homoclinic corners are a significant feature in the dynamics of piecewise-smooth systems.
- Understanding these bifurcations is crucial for predicting system behavior.
- The presence of homoclinic corners implies instability of nearby periodic orbits.
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