Desynchronization of chaos in coupled logistic maps

Area of Science:

• Nonlinear Dynamics and Chaos Theory
• Complex Systems Analysis
• Statistical Mechanics

Background:

• Identical chaotic oscillators can achieve synchronization, restricting motion to lower-dimensional invariant manifolds.
• Riddling occurs when embedded orbits in synchronized states become transversely unstable, despite the state remaining attractive on average.

Purpose of the Study:

• To investigate the transition mechanisms to riddling in coupled chaotic systems.
• To differentiate between soft and hard transitions to riddling based on bifurcation types.
• To characterize the resulting basin structures and dynamics.

Main Methods:

• Analysis of a system of two coupled logistic maps.
• Identification and tracking of asynchronous cycles and their invariant manifolds.
• Examination of bifurcations, including supercritical, subcritical, contact, and boundary crises.
• Construction of a phase diagram to map transition regimes.

Main Results:

• The transition to riddling is classified as soft (supercritical bifurcation) or hard (subcritical bifurcation).
• Supercritical bifurcations lead to locally riddled basins via mixed absorbing areas.
• Subcritical bifurcations can result in globally riddled basins of attraction.
• Three distinct scenarios for riddling onset and basin transformation were identified, involving contact bifurcations and boundary crises.

Conclusions:

• The nature of the bifurcation dictates the global or local extent of basin riddling.
• Mixed absorbing areas and asynchronous chaotic states play crucial roles in basin structure evolution.
• A comprehensive phase diagram illustrates the parameter-dependent transitions in coupled chaotic systems.