Delay-induced homoclinic bifurcations in modified gradient bistable systems and their relevance to optimization

Area of Science:

• Dynamical Systems and Control Theory
• Nonlinear Dynamics
• Mathematical Physics

Background:

• Nonlinear dynamical systems with time delay are common in various applications.
• Analyzing these systems is challenging due to delay-induced effects dependent on nonlinearities.
• Time delays can significantly alter system behavior and stability.

Purpose of the Study:

• To explore a specific class of nonlinear systems with time delay derived from gradient systems.
• To investigate the delay-induced effects, particularly homoclinic bifurcations.
• To understand the role of time delay in optimizing multistable systems.

Main Methods:

• Consideration of gradient dynamical systems with a two-well potential function.
• Introduction of a delayed argument into the system's right-hand side function.
• Graphical interpretation and analysis of delay-induced phenomena like homoclinic bifurcations.

Main Results:

• Demonstration of qualitatively predictable delay-induced effects in simple systems.
• Observation of a chain of homoclinic bifurcations that eliminate local attractors.
• Identification of delay-induced reorganization of manifolds as the key phenomenon.

Conclusions:

• The study provides a foundation for understanding nonlinear multistable systems with time delay.
• Homoclinic bifurcations, induced by time delay, can allow phase trajectories to explore all local minima.
• The findings elucidate mechanisms for the role of time delay in optimization processes.