Bifurcations without parameters: some ODE and PDE examples

TL;DR

This paper explores bifurcation phenomena in dynamical systems without varying parameters, analyzing cases where normal hyperbolicity fails due to eigenvalues being zero or purely imaginary, revealing complex dynamics beyond classical bifurcation theory.

Contribution

It introduces and analyzes bifurcations without parameters, extending classical bifurcation theory to systems lacking explicit parameter dependence, with detailed eigenvalue-based classifications.

Findings

Identifies bifurcation scenarios without parameters involving zero and imaginary eigenvalues.

Provides mathematical analysis of bifurcations like Hopf and Takens-Bogdanov without parameters.

Highlights complex dynamics arising from non-hyperbolic trivial solutions in parameter-free systems.

Abstract

Standard bifurcation theory is concerned with families of vector fields $dx/dt = f(x,\lambda)$, $x \in \R^n$, involving one or several constant real parameters $\lambda$. Viewed as a differential equation for the pair $(x,\lambda)$, we observe a foliation of the total phase space by constant $\lambda$. Frequently, the presence of a trivial stationary solution $x=0$ is also imposed: $0 = f(0,\lambda)$. Bifurcation without parameters, in contrast, discards the foliation by a constant parameter $\lambda$. Instead, we consider systems $dx/dt = f(x,y), dy/dt = g(x,y)$. Standard bifurcation theory then corresponds to the special case $y=\lambda, g=0$. To preserve only the trivial solution $x=0$, instead, we only require $0 = f(0,y) = g(0,y)$ for all $y$. A rich dynamic phenomenology arises, when normal hyperbolicity of the trivial stationary manifold $x=0$ fails, due to zero or purely imaginary eigenvalues of the Jacobian $f_x(0,y)$. Specifically, we address the cases of failure of normal hyperbolicity due to a simple eigenvalue zero, a simple purely imaginary pair (Hopf bifurcation without parameters), a double eigenvalue zero (Takens-Bogdanov bifurcation without parameters), and due to a double eigenvalue zero with additional time reversal symmetries. The results are joint work with Andrei Afendikov, James C. Alexander, and Stefan Liebscher.