A Massera-type Theorem on relative-periodic solutions for a second-order model of rectilinear locomotion

TL;DR

This paper proves that in a second-order model of rectilinear crawling, the existence of a bounded solution guarantees a global periodic attractor, with conditions on friction forces influencing this outcome.

Contribution

It establishes a Massera-type theorem for a second-order locomotion model, linking bounded solutions to the existence of a global attractor, and analyzes friction conditions affecting boundedness.

Findings

Bounded solutions imply a global periodic attractor.

Sufficient friction conditions ensure bounded solutions.

Generalized friction can prevent bounded solutions.

Abstract

We study the existence of a global periodic attractor for the reduced dynamics of a discrete toy model for rectilinear crawling locomotion, corresponding to a limit cycle in the shape and velocity variables. The body of the crawler consists of a chain of point masses, joined by active elastic links and subject to smooth friction forces, so that the dynamics is described by a system of second order differential equations. Our main result is of Massera-type, namely we show that the existence of a bounded solution implies the existence of the global periodic attractor for the reduced dynamics. In establishing this result, a contractive property of the dynamics of our model plays a central role. We then prove sufficient conditions on the friction forces for the existence of a bounded solution, and therefore of the attractor. We also provide an example showing that, if we consider more general friction forces, such as smooth approximations of dry friction, bounded solutions may not exist.