Controlling complex rhythms: A hierarchical approach to limit cycle switching

TL;DR

This paper introduces a hierarchical control method for reliably switching between multiple limit cycles in nonlinear systems, with applications in neuro-engineering and synthetic biology.

Contribution

It presents a novel hierarchical, stepwise periodic modulation approach to control multirhythmicity in complex dynamical systems.

Findings

Hierarchical control enables reliable limit cycle switching.

Method preserves system properties during transitions.

Applicable to neuro-engineering and synthetic biology.

Abstract

Limit cycles are self-sustained, closed trajectories in phase space representing (un)-stable, periodic behavior in nonlinear dynamical systems. They underpin diverse natural phenomena, from neuronal firing patterns to engineering oscillations. The presence of multiple concentric limit cycles reflects distinct behavioral symmetries within a system. In this work, we investigate the hierarchical dynamical transitions from one limit cycle to another, driven by oscillatory excitation while preserving other system properties. We demonstrate that controlling multirhythmicity through hierarchical, stepwise periodic modulation enables reliable switching between rhythmic states. This hierarchical control framework is crucial for applications in neuro-engineering and synthetic biology, where precise, robust modulation of complex rhythmic behaviors enhances system functionality and adaptability.