Global stability of synchronous and out-of-phase oscillations in central pattern generators

Abstract

Coupled arrays of Andronov-Hopf oscillators are investigated. These arrays can be diffusively or repulsively coupled, and can serve as central pattern generator models in animal locomotion and robotics. It is shown that repulsive coupling generates out-of-phase oscillations, while diffusive coupling generates synchronous oscillations. Specifically, symmetric solutions and their corresponding amplitudes are derived, and contraction analysis is used to prove global stability and convergence of oscillations to either symmetric out-of-phase or synchronous states, depending on the coupling constant. Next, the two mechanisms are used jointly by coupling multiple arrays. The resulting dynamics is analyzed, in a model inspired by the CPGmotorneuron network that controls the heartbeat of a medicinal leech.

Rhythm motion control in bio‐inspired fishtail based on central pattern generator

“…It follows from (20), (23) and (25) which implies that A −1 is bounded and 0 ∈ ϱðA Þ, the resolvent set of A . We use Sobolev's embedding theorem to conclude that A −1 is a compact operator on H. □ Remark 1 From the above proof, we know that for λ > 0 small enough, we have R λI − ðA − MIÞ ð Þ ¼ H where RðL Þ represents the range of the operator L .…”

“…Therefore, there exists at least one closed orbit in this area. We imply that the closed trajectory of the cubic autonomous system is an isolated closed trajectory [22,23], called the asymptotically stable limit cycle.…”