Periodic Solutions to Dissipative Hyperbolic Systems. II: Hopf Bifurcation for Semilinear Problems

TL;DR

This paper establishes conditions for Hopf bifurcation in semilinear hyperbolic systems, providing a framework for existence, uniqueness, and smooth dependence of time-periodic solutions near zero stationary solutions.

Contribution

It introduces a novel set of conditions and a bifurcation formula for hyperbolic PDEs, extending bifurcation theory beyond classical parabolic and ODE cases.

Findings

Derived a bifurcation direction formula.

Proved existence and uniqueness of bifurcating solutions.

Highlighted the importance of dissipativity in hyperbolic PDEs.

Abstract

We consider boundary value problems for semilinear hyperbolic systems of the type $$ \partial_tu_j + a_j(x,\la)\partial_xu_j + b_j(x,\la,u) = 0, \; x\in(0,1), \;j=1,\dots,n $$ with smooth coefficient functions $a_j$ and $b_j$ such that $b_j(x,\la,0) = 0$ for all $x \in [0,1]$, $\la \in \R$, and $j=1,\ldots,n$. We state conditions for Hopf bifurcation, i.e., for existence, local uniqueness (up to phase shifts), smoothness and smooth dependence on $\la$ of time-periodic solutions bifurcating from the zero stationary solution. Furthermore, we derive a formula which determines the bifurcation direction. The proof is done by means of a Liapunov-Schmidt reduction procedure. For this purpose, Fredholm properties of the linearized system and implicit function theorem techniques are used. There are at least two distinguishing features of Hopf bifurcation theorems for hyperbolic PDEs in comparison with those for parabolic PDEs or for ODEs: First, the question if a non-degenerate time-periodic solution depends smoothly on the system parameters is much more delicate. And second, a sufficient amount of dissipativity is needed in the system, and a priori it is not clear how to verify this in terms of the data of the PDEs and of the boundary conditions.