Dynamical behavior of a system modeling wave bifurcations with higher order viscosity

TL;DR

This paper rigorously analyzes the bifurcation dynamics of a PDE system modeling wave bifurcations with higher order viscosity, revealing that oscillations are not due to rotation waves despite Euclidean symmetry.

Contribution

It derives the full center manifold dynamics for a class of PDEs with multiple bifurcation parameters and addresses unique mathematical challenges posed by the zero matrix linear operator.

Findings

Oscillations are not caused by rotation waves.

The system supports stationary equivariant bifurcation dynamics.

The analysis handles complex bifurcation parameter interactions.

Abstract

We rigorously show that a class of systems of partial differential equations modeling wave bifurcations supports stationary equivariant bifurcation dynamics through deriving its full dynamics on the center manifold(s). A direct consequence of our result is that the oscillations of the dynamics are \textit{not} due to rotation waves though the system exhibits Euclidean symmetries. The main difficulties of carrying out the program are: 1) the system under study contains multi bifurcation parameters and we do not know \textit{a priori} how they come into play in the bifurcation dynamics. 2) the representation of the linear operator on the center space is a $2\times 2$ zero matrix, which makes the characteristic condition in the well-known normal form theorem trivial. We overcome the first difficulty by using projection method. We managed to overcome the second subtle difficulty by using a conjugate pair coordinate for the center space and applying duality and projection arguments. Due to the specific complex pair parametrization, we could naturally get a form of the center manifold reduction function, which makes the study of the current dynamics on the center manifold possible. The symmetry of the system plays an essential role in excluding the possibility of bifurcating rotation waves.