Ascertain existence of an SPO–UPO pair for N>3 in the neuromechanical N-link model

Ascertain whether, for the N-link neuromechanical undulatory locomotion model (an inextensible elastic rod with resistive-force-theory coupling and nearest-neighbour phase oscillators with local exteroceptive torque-load feedback), there always exists an unstable periodic orbit paired with the observed single global stable periodic orbit when N>3 across the parameter ranges considered.

Background

The paper analyzes periodic orbits in a neuromechanical model consisting of an elastic N-link body in a dissipative medium, with internal actuation governed by a chain of coupled oscillators modulated by local torque-load feedback. For N=3 (Purcell’s swimmer), both stable and unstable periodic orbits are identified, with the unstable orbit being a saddle.

For N>3, numerical evidence shows a single global stable periodic orbit over large parameter ranges. However, due to numerical difficulties in detecting unstable periodic orbits in high-dimensional state spaces, the authors cannot conclusively determine whether a corresponding unstable periodic orbit always coexists with the stable one.