Bifurcations without parameters: some ODE and PDE examples
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.
- Publication:
-
arXiv Mathematics e-prints
- Pub Date:
- April 2003
- DOI:
- 10.48550/arXiv.math/0304453
- arXiv:
- arXiv:math/0304453
- Bibcode:
- 2003math......4453F
- Keywords:
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- Dynamical Systems;
- 34C23;
- 34C29;
- 34C37
- E-Print:
- Proceedings of the ICM, Beijing 2002, vol. 3, 305--316