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

Paolo Gidoni, Alessandro Margheri

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.

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

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.

Paper Structure

This paper contains 9 sections, 10 theorems, 77 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A model of rectilinear crawling locomotion
  3. Relative-periodic solutions in rectilinear locomotion
  4. A Massera-type Theorem
  5. Existence of a limit relative-periodic behaviour
  6. Time-dependent viscous friction
  7. Proof of Lemma \ref{['lemma:stiff_dissip']}
  8. Acknowledgements.
  9. Proof of Theorem \ref{['th:glob_existence']}

Key Result

Theorem 2.1

Assume that hyp:A1hyp:A2, hyp:A3, hyp:A5 and hold. Fixed any $t_0\in\mathbb{R}$, the solutions of system eqcrawl are defined on $[t_0,+\infty).$

Figures (3)

  • Figure 1: The toy model of rectilinear crawling locomotion
  • Figure 2: Plots of the shape and velocity of the barycenter of the solution of System \ref{['eq:example2']} in Example \ref{['ex:resonance']} with parameters $k=2$, $A=3$, $F_1(t,v)=F_2(t,v)=\frac{1}{4}\arctan(v)(5-\arctan(v)$ and initial conditions $(x_1(0),\dot{x}_1(0), x_2(0),\dot{x}_2(0))=(0,10,15,0)$.
  • Figure 3: Plots of the three solutions of System \ref{['eq:example']} of Example \ref{['ex:linear']} with the following initial conditions: $(x_1(0),\dot{x}_1(0), x_2(0),\dot{x}_2(0))=(\frac{1}{6},-7,-\frac{1}{6}, 2)$ for the orange, solid line, $(x_1(0),\dot{x}_1(0), x_2(0),\dot{x}_2(0))=(\frac{5}{2},2,-\frac{5}{2}, 3)$ for the violet, dashdotted line, $(x_1(0),\dot{x}_1(0), x_2(0),\dot{x}_2(0))=(\frac{3}{2},10,-\frac{3}{2}, 1)$ for the green, dashed line.

Theorems & Definitions (21)

  • Theorem 2.1
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Theorem 4.1
  • Corollary 4.2
  • Theorem 4.3
  • proof
  • Corollary 4.4
  • ...and 11 more