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

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.