Intermittent Two-Point Dynamics at the Transition to Chaos for Random Circle Endomorphisms

Area of Science:

• Dynamical systems theory
• Statistical mechanics
• Nonlinear dynamics

Background:

• Understanding transitions between order and chaos is crucial in complex systems.
• Circle endomorphisms are fundamental models for studying chaotic dynamics.
• Lyapunov exponents characterize the sensitivity to initial conditions, distinguishing regular from chaotic behavior.

Purpose of the Study:

• To characterize the transition from synchronization to chaos in random circle endomorphisms.
• To establish the existence of intermittent two-point dynamics.
• To prove the existence of infinite stationary measures for systems with zero Lyapunov exponent.

Main Methods:

• Analysis of random circle endomorphisms.
• Investigation of dynamical properties at zero Lyapunov exponent.
• Mathematical formulation of intermittent dynamics and stationary measures.

Main Results:

• Established the existence of intermittent two-point dynamics.
• Proved the existence of infinite stationary measures.
• Provided a dynamical characterization of the transition from synchronization (negative Lyapunov exponent) to chaos (positive Lyapunov exponent).

Conclusions:

• The zero Lyapunov exponent regime exhibits complex dynamics, bridging synchronized and chaotic states.
• Intermittent dynamics and infinite stationary measures are key features of this transition.
• These findings offer new insights into the onset of chaos in random dynamical systems.