Computer assisted proof of branches of stationary and periodic solution, and Hopf bifurcations, for dissipative PDEs

TL;DR

This paper presents a computer-assisted method to rigorously prove the existence of stationary and periodic solutions, including Hopf bifurcations, in dissipative PDEs exemplified by the Brussellator system.

Contribution

It introduces a novel computational approach for verifying solution branches and bifurcations in dissipative PDEs, with a focus on stationary and periodic solutions.

Findings

Successfully proved existence of solution branches for the Brussellator system.

Validated the occurrence of Hopf bifurcations leading to periodic solutions.

Demonstrated the effectiveness of the method through rigorous computer-assisted proofs.

Abstract

We discuss an approach to the computer assisted proof of the existence of branches of stationary and periodic solutions for dissipative PDEs, using the Brussellator system with diffusion and Dirichlet boundary conditions as an example, We also consider the case where the branch of periodic solutions emanates from a branch of stationary solutions through a Hopf bifurcation.