Continuation and bifurcations of periodic orbits and symbolic dynamics in the Swift-Hohenberg equation

TL;DR

This paper investigates steady states of the Swift-Hohenberg equation, proving bifurcation of periodic solutions and demonstrating positive topological entropy, using computer-assisted rigorous bounds on Poincaré maps.

Contribution

It provides a rigorous computer-assisted proof of bifurcations and complex dynamics in the Swift-Hohenberg equation, including the creation of periodic orbits and entropy estimates.

Findings

Two smooth branches of periodic solutions are created at a saddle-node bifurcation.

The system exhibits positive topological entropy for a range of parameters.

The proof employs rigorous bounds on Poincaré maps using computer assistance.

Abstract

Steady states of the Swift--Hohenberg equation are studied. For the associated four--dimensional ODE we prove that on the energy level $E=0$ two smooth branches of even periodic solutions are created through the saddle-node bifurcation. We also show that these orbits satisfy certain geometric properties, which implies that the system has positive topological entropy for an explicit and wide range of parameter values of the system. The proof is computer-assisted and it uses rigorous computation of bounds on certain Poincar\'e map and its higher order derivatives.