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Mapping chaos: Bifurcation patterns and shrimp structures in the Ikeda map.

Diego F M Oliveira¹

¹School of Electrical Engineering and Computer Science, College of Engineering & Mines, University of North Dakota, Grand Forks, North Dakota 58202-8367, USA.

Chaos (Woodbury, N.Y.)

|December 9, 2024

Summary

This study reveals how dissipation affects the Ikeda map, showing shrimp-shaped structures that mark transitions to chaos. Lyapunov exponents identify stable and chaotic dynamics in nonlinear optical systems.

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Area of Science:

Nonlinear Dynamics
Optical Systems
Chaos Theory

Background:

The Ikeda map is a key model for understanding nonlinear phenomena in optical systems.
Investigating bifurcations and chaotic behavior is crucial for predicting system dynamics.

Purpose of the Study:

To examine the dynamical properties of the Ikeda map.
To analyze the influence of dissipation parameters on bifurcations and chaos.
To characterize the transition from regular to chaotic dynamics.

Main Methods:

Analysis of period-doubling bifurcations.
Investigation of parameter variations, specifically dissipation.
Utilizing Lyapunov exponents to differentiate between stable and chaotic regimes.

Main Results:

Discovery of intricate shrimp-shaped structures indicating transitions.
Detailed analysis of period-doubling bifurcations leading to chaos.
Clear distinction between stable and chaotic regions using Lyapunov exponents.

Conclusions:

Dissipation significantly influences the Ikeda map's dynamics, creating complex transitional structures.
The study deepens the understanding of nonlinear and chaotic dynamics in optical contexts.
Lyapunov exponents are effective tools for characterizing dynamical states.