Amplitude death in oscillators coupled by a one-way ring time-delay connection

Area of Science:

• Nonlinear dynamics
• Complex systems
• Control theory

Background:

• Amplitude death is a stabilization phenomenon in coupled nonlinear oscillators.
• Diffusive coupling typically prevents amplitude death in identical oscillators.
• Time-delay coupling offers a potential mechanism for inducing amplitude death.

Purpose of the Study:

• To investigate amplitude death in a one-way ring time-delay coupled system of N identical nonlinear oscillators.
• To analyze the conditions under which amplitude death can be induced.
• To develop a systematic graphical method for stability analysis.

Main Methods:

• Analytical confirmation of amplitude death induction via diffusive time-delay coupling.
• Proposal of a one-way ring time-delay coupled system (N=1: delayed-feedback control; N=2: time-delay coupled oscillators).
• Stability analysis based on the eigenvalues of the Jacobi matrix at fixed points.
• Development and illustration of a graphical stability testing procedure.

Main Results:

• Amplitude death is shown to never occur at a steady state when the Jacobi matrix has an odd number of real positive eigenvalues.
• A simple, systematic graphical procedure for stability testing is presented.
• The procedure is successfully illustrated using coupled Rössler and Lorenz oscillators.

Conclusions:

• Time-delay coupling is crucial for achieving amplitude death in identical nonlinear oscillators.
• The proposed one-way ring system provides a framework for studying amplitude death.
• The developed graphical method offers an accessible tool for stability analysis in such systems.