Dynamics in hybrid complex systems of switches and oscillators

TL;DR

This paper investigates the complex dynamics of hybrid systems combining oscillators and switches, revealing multiple stable states and transitions driven by feedback, with implications for understanding diverse real-world systems.

Contribution

It introduces a theoretical framework for analyzing large hybrid systems of oscillators and switches, demonstrating coexistence of multiple stable states and transition mechanisms.

Findings

Coexistence of incoherent oscillators with switches off and synchronized oscillators with switches on.

Existence of periodic switching states with oscillators synchronized and switches alternating states.

Transitions between states can be deterministic or due to finite-size fluctuations.

Abstract

While considerable progress has been made in the analysis of large systems containing a single type of coupled dynamical component (e.g., coupled oscillators or coupled switches), systems containing diverse components (e.g., both oscillators and switches) have received much less attention. We analyze large, hybrid systems of interconnected Kuramoto oscillators and Hopfield switches with positive feedback. In this system, oscillator synchronization promotes switches to turn on. In turn, when switches turn on they enhance the synchrony of the oscillators to which they are coupled. Depending on the choice of parameters, we find theoretically coexisting stable solutions with either (i) incoherent oscillators and all switches permanently off, (ii) synchronized oscillators and all switches permanently on, or (iii) synchronized oscillators and switches that periodically alternate between the on and off states. Numerical experiments confirm these predictions. We discuss how transitions between these steady state solutions can be onset deterministically through dynamic bifurcations or spontaneously due to finite-size fluctuations.