Dynamical behaviors of a system modeling wave bifurcations

We rigorously show that a class of systems of partial differential equations (PDEs) modeling wave bifurcations supports stationary equivariant bifurcation dynamics through deriving its full dynamics on the center manifold(s). This class of systems is related to the theory of hyberbolic conservation laws and supplies a new class of PDE examples for stationary $O(2)$-bifurcation. A direct consequence of our result is that the oscillations of the dynamics are 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 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 obtain 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.

Keywords

spectrum, resolvent, equivariant bifurcation, center manifold, symmetry, implicit function theorem

2010 Mathematics Subject Classification

34Bxx, 34Gxx, 35Pxx, 37Gxx

Full Text (PDF format)

Received 9 March 2018

Accepted 20 June 2018

Published 7 March 2019