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Rapid Repetition Rate Fluctuation Measurement of Soliton Crystals in a Microresonator

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Temporal dissipative solitons in the Morris-Lecar model with time-delayed feedback.

M Stöhr¹, M Wolfrum¹

¹Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117 Berlin, Germany.

Chaos (Woodbury, N.Y.)

|March 1, 2023

Summary

We investigated temporal dissipative solitons in excitable systems with time-delayed feedback. A homoclinic orbit flip destabilizes single solitons, leading to stable pulse packages, revealing complex dynamics in excitable models.

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Area of Science:

Nonlinear Dynamics
Computational Neuroscience
Mathematical Biology

Background:

Excitable systems, like the Morris-Lecar model, exhibit complex dynamics crucial for understanding biological processes.
Time-delayed feedback introduces intricate behaviors, including the formation and stability of temporal dissipative solitons.
Understanding soliton dynamics is key to modeling phenomena ranging from neural signaling to pattern formation.

Purpose of the Study:

To analyze the dynamics and bifurcations of temporal dissipative solitons in an excitable system (Morris-Lecar model) subjected to time-delayed feedback.
To elucidate the role of homoclinic and heteroclinic bifurcations in shaping soliton behavior and stability.
To explain the emergence of stable pulse packages from destabilized multi-pulse solutions.

Main Methods:

Utilized the Morris-Lecar model as a prototypical excitable system.
Employed concepts from classical homoclinic bifurcation theory, treating large-delay soliton solutions as homoclinic solutions.
Analyzed delay-differential equations and their relationship to advanced argument equations.

Main Results:

Demonstrated that a homoclinic orbit flip of a single-pulse soliton destabilizes equidistant multi-pulse solutions.
Showed the emergence of stable pulse packages due to this destabilization.
Identified that this transition is triggered by a heteroclinic orbit flip in the feedback-free system, linked to the model's excitability.

Conclusions:

Time-delayed feedback in excitable systems can lead to complex soliton dynamics and the formation of stable pulse packages.
Homoclinic and heteroclinic bifurcations are critical mechanisms governing these transitions.
The study provides a theoretical framework for understanding pulse behavior in excitable media with delayed feedback.