Structured scale dependence in the Lyapunov exponent of a Boolean chaotic map

Area of Science:

• * Chaos theory
• * Boolean networks
• * Nonlinear dynamics

Background:

• * Experimental chaotic maps exhibit complex dynamics.
• * Scale-dependent Lyapunov exponents reveal system behavior across different scales.
• * Discontinuities in maps can introduce unique dynamical features.

Purpose of the Study:

• * To investigate the origin of structures in the scale-dependent Lyapunov exponent of an experimental chaotic map.
• * To understand how discontinuities in the map influence chaotic behavior.
• * To elucidate the role of discrete logic elements in shaping system dynamics.

Main Methods:

• * Realization of chaos in an autonomous Boolean network using asynchronous logic gates.
• * Construction of a map operator with pulse-width stretching and folding.
• * Feedback of the operator's output to its input for continuous map iteration.
• * Development of a simple model to analyze the scale-dependent Lyapunov exponent.

Main Results:

• * Observed structures in the scale-dependent Lyapunov exponent attributed to map discontinuities.
• * Demonstrated that chaos arises from an autonomous Boolean network with a specific map operator.
• * Identified discrete logic elements within the map operator's stretching function as the cause of structured scale-dependence.

Conclusions:

• * Discontinuities in experimental chaotic maps lead to structured scale-dependent Lyapunov exponents.
• * The autonomous Boolean network effectively generates chaos through pulse-width manipulation.
• * Discrete logic elements are crucial in determining the scale-dependent dynamics of chaotic systems.