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Cycles homoclinic to chaotic sets; robustness and resonance
1Institut Non Lineaire de Nice, 1361 Route des Lucioles, 06560 Valbonne, France.
Chaos (Woodbury, N.Y.)
|June 1, 1997
Summary
This study explores "cycling chaos" in dynamical systems, where stable homoclinic cycles lead to intermittent chaotic bursts. The timing and stability of these chaotic transients are characterized by Lyapunov exponents.
Area of Science:
- Dynamical Systems
- Chaos Theory
- Nonlinear Dynamics
Background:
- Invariant subspaces can host robust homoclinic cycles leading to chaotic sets.
- Stable homoclinic cycles exhibit quiescent chaotic behavior punctuated by transient bursts.
- The time between bursts lengthens as trajectories approach the cycle.
Purpose of the Study:
- Characterize homoclinic cycles and their stability using normal Lyapunov exponents.
- Investigate the role of the 'footprint' (Lyapunov exponent spectrum) in cycle stability for skew-product systems.
- Analyze the creation of chaotic attractors in parametrically forced systems.
Main Methods:
- Analysis of normal Lyapunov exponents to characterize cycle stability.
- Examination of systems with skew-product structure.
- Numerical simulations of a homoclinic cycle forced by a Rossler attractor.
Main Results:
- Identified persistent states that are attracting but not Lyapunov stable.
- Observed approximately periodic states.
- Demonstrated the crucial dependence of cycle stability on the 'footprint' in skew-product systems.
- Observed creation of nearby chaotic attractors at resonance of transverse Lyapunov exponents.
Conclusions:
- Normal Lyapunov exponents effectively characterize homoclinic cycles and their stability.
- The 'footprint' is critical for understanding asymptotic stability and attractivity in chaotically forced systems.
- Resonance phenomena can lead to the emergence of new chaotic attractors.
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