# Grazing-sliding bifurcations creating infinitely many attractors

## TL;DR

This paper investigates how grazing-sliding bifurcations in piecewise-smooth systems can lead to the creation of infinitely many stable periodic orbits, revealing complex bifurcation structures.

## Contribution

It demonstrates that at a grazing-sliding bifurcation, a single stable periodic orbit can bifurcate into infinitely many stable periodic orbits, extending previous results.

## Key findings

- Infinitely many stable periodic solutions can emerge at grazing-sliding bifurcations.
- Numerical continuation reveals subsequent bifurcations destroying these orbits.
- Theoretical results are supported by numerical experiments on abstract ODE systems.

## Abstract

As the parameters of a piecewise-smooth system of ODEs are varied, a periodic orbit undergoes a bifurcation when it collides with a surface where the system is discontinuous. Under certain conditions this is a grazing-sliding bifurcation. Near grazing-sliding bifurcations structurally stable dynamics are captured by piecewise-linear continuous maps. Recently it was shown that maps of this class can have infinitely many asymptotically stable periodic solutions of a simple type. Here this result is used to show that at a grazing-sliding bifurcation an asymptotically stable periodic orbit can bifurcate into infinitely many asymptotically stable periodic orbits. For an abstract ODE system the periodic orbits are continued numerically revealing subsequent bifurcations at which they are destroyed.