# Structurally stable families of periodic solutions in sweeping processes of networks of elastoplastic springs

## TL;DR

This paper investigates the stability of periodic solutions in elastoplastic spring networks, showing that while certain attractors can be destroyed by perturbations of the constraint, they remain stable under changes to the system's physical parameters.

## Contribution

The paper provides a simple example demonstrating the structural stability of periodic attractors against parameter perturbations, addressing an open problem in the dynamics of elastoplastic systems.

## Key findings

- Periodic attractors resist perturbations of physical parameters.
- Small perturbations of the moving constraint can destroy periodic attractors.
- The stability of periodic solutions depends on the type of perturbation.

## Abstract

Networks of elastoplastic springs (elastoplastic systems) have been linked to differential equations with polyhedral constraints in the pioneering paper by Moreau (1974). Periodic loading of an elastoplastic system, therefore, corresponds to a periodic motion of the polyhedral constraint. According to Krejci (1996), every solution of a sweeping process with a periodically moving constraint asymptotically converges to a periodic orbit. Understanding whether such an asymptotic periodic orbit is unique or there can be an entire family of asymptotic periodic orbits (that form a periodic attractor) has been an open problem since then. Since suitable small perturbation of a polyhedral constraint seems to be always capable to destroy a potential family of periodic orbits, it is expected that none of potential periodic attractor is structurally stable. In the present paper we give a simple example to prove that even though the periodic attractor (of non-stationary periodic solutions) can be destroyed by little perturbation of the moving constraint, the periodic attractor resists perturbations of the physical parameters of the mechanical model (i.e. the parameters of the network of elastoplastic springs).