Bifurcations without parameters: some ODE and PDE examples
| dc.creator | Fiedler, Bernold | |
| dc.creator | Liebscher, Stefan | |
| dc.date | 2003-04-28 | |
| dc.date.accessioned | 2026-07-07T04:57:31Z | |
| dc.date.available | 2026-07-07T04:57:31Z | |
| dc.description | Standard bifurcation theory is concerned with families of vector fields $dx/dt = f(x,λ)$, $x \in \R^n$, involving one or several constant real parameters $λ$. Viewed as a differential equation for the pair $(x,λ)$, we observe a foliation of the total phase space by constant $λ$. Frequently, the presence of a trivial stationary solution $x=0$ is also imposed: $0 = f(0,λ)$. Bifurcation without parameters, in contrast, discards the foliation by a constant parameter $λ$. Instead, we consider systems $dx/dt = f(x,y), dy/dt = g(x,y)$. Standard bifurcation theory then corresponds to the special case $y=λ, g=0$. 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. | |
| dc.identifier | https://arxiv.org/abs/math/0304453 | |
| dc.identifier | http://arxiv.org/abs/math/0304453 | |
| dc.identifier | Proceedings of the ICM, Beijing 2002, vol. 3, 305--316 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/67285 | |
| dc.subject | Dynamical Systems | |
| dc.subject | 34C23, 34C29, 34C37 | |
| dc.title | Bifurcations without parameters: some ODE and PDE examples | |
| dc.type | text |