The following pictures illustrate some elements of the nonlinear dynamics near equilibrium manifolds. Interesting dynamics arise at points where normal hyperbolicity of this manifold is violated. Due to the analogies of our results and methods with bifurcation theory, we call the emerging theory bifurcation without parameters. In particular, analogues to Andronov-Hopf points and Bogdanov-Takens points are displayed.

Elliptic Hopf point

Hopf bifurcation (without parameters) from a line of equilibria, three-dimensional picture of stable and unstable manifolds in the elliptic case.

Hyperbolic Hopf point

Hopf bifurcation (without parameters) from a line of equilibria, three-dimensional picture of stable and unstable manifolds and trajectories in the hyperbolic case.

Takens-Bogdanov point with elliptic Hopf

Takens-Bogdanov bifurcation (without parameters) from a plane of equilibria, picture of stable and unstable manifolds of the Poincare map in case of a bifurcating elliptic Hopf point.

Takens-Bogdanov point with hyperbolic Hopf

Takens-Bogdanov bifurcation (without parameters) from a plane of equilibria, picture of stable and unstable manifolds of the Poincare map in case of a bifurcating hyperbolic Hopf point.