Delayed feedback control of chaos: bifurcation analysis

Area of Science:

• Nonlinear dynamics
• Chaos theory
• Control theory

Background:

• Deterministic chaos is a complex phenomenon observed in nonlinear systems.
• Pyragas' delayed feedback control offers a method to influence chaotic behavior.
• The Rössler system is a well-studied model exhibiting chaotic dynamics.

Purpose of the Study:

• To investigate the impact of Pyragas' delayed feedback control on chaos in the Rössler system.
• To map the bifurcation diagram in the parameter space of time delay and feedback strength.
• To understand the emergence of multistability and new dynamical regimes.

Main Methods:

• Numerical simulation of the Rössler system with delayed feedback control.
• Construction of bifurcation diagrams in the (tau, K) parameter plane.
• Analysis of phase space to identify coexisting attractors and dynamical behaviors.

Main Results:

• A general bifurcation diagram was revealed, explaining previously observed phenomena.
• The diagram exhibits a multileaf structure, indicating multistability.
• Increased time delay (tau) leads to a greater number of coexisting attractors.
• New dynamical regimes, including tori and chaotic attractors, were induced by the feedback.
• The influence of delayed feedback parameters on limit cycle periods was estimated.

Conclusions:

• Delayed feedback control significantly alters the dynamics of the Rössler system, inducing complex behaviors.
• The multileaf bifurcation structure highlights the potential for extensive multistability.
• This control strategy can generate novel attractors and influence the periodicity of limit cycles.