Canard cascading in networks with adaptive mean-field coupling

TL;DR

This paper investigates canard cascading in adaptive networks, revealing it as a robust, scalable phenomenon linked to heteroclinic orbits, demonstrated through coupled semiconductor lasers, and organizing unstable states into an attractive limit cycle.

Contribution

It uncovers the dynamical mechanisms behind canard cascading in adaptive networks, introducing the concept of heteroclinic canard orbits that organize unstable states into a stable cycle.

Findings

Canard cascading is a robust, adaptive network effect.

Multiple saddle slow manifolds are linked by heteroclinic orbits.

The CC cycle is attractive despite unstable quasi-stationary states.

Abstract

Canard cascading (CC) is observed in dynamical networks with global adaptive coupling. It is a fast-slow phenomenon characterized by a recurrent sequence of fast transitions between distinct and slowly evolving quasi-stationary states. In this letter, we uncover the dynamical mechanisms behind CC, using an illustrative example of globally and adaptively coupled semiconductor lasers, where CC represents sequential switching on and off the lasers. Firstly, we show that CC is a robust and truly adaptive network effect that is scalable with network size and does not occur without adaptation. Secondly, we uncover multiple saddle slow manifolds (unstable quasi-stationary states) linked by heteroclinic orbits (fast transitions) in the phase space of the system. This allows us to identify CC with a novel heteroclinic canard orbit that organises different unstable quasi-stationary states into an intricate fast-slow limit cycle. Although individual quasi-stationary states are unstable (saddles), the CC cycle as a whole is attractive and robust to parameter changes.