A minimal hyperbolic system for unstable shock waves

TL;DR

This paper analyzes a minimal hyperbolic system that exhibits complex nonlinear and chaotic behaviors, including bifurcations and stability changes in traveling-wave solutions, serving as both a model for chaos and a test problem for numerical methods.

Contribution

It introduces a minimal hyperbolic system with chaotic solutions and provides detailed numerical analysis of bifurcations and stability properties of its traveling waves.

Findings

Traveling waves undergo bifurcations as parameters vary

The system exhibits chaotic solutions

Numerical stability analysis confirms complex behaviors

Abstract

We present a computational analysis of a 2$\times$2 hyperbolic system of balance laws whose solutions exhibit complex nonlinear behavior. Traveling-wave solutions of the system are shown to undergo a series of bifurcations as a parameter in the model is varied. Linear and nonlinear stability properties of the traveling waves are computed numerically using accurate shock-fitting methods. The model may be considered as a minimal hyperbolic system with chaotic solutions and can also serve as a stringent numerical test problem for systems of hyperbolic balance laws.