Bodo Werner

Computational methods for bifurcation problems with symmetries—with special attention to steady state and Hopf bifurcation points

We show how group theoretical methods can be employed to utilize the symmetry of a bifurcation problem in numerical computations. We extend the approach by Werner (1988) by presenting methods for the detection of bifurcation points and the computation of (multiple) Hopf points. The essential numerical point is the utilization of certain reduced instead of full systems involving appropriate subgroups of the underlying symmetry group Γ. The group theoretical tool is an a priori knowledge of the interaction of certain subgroups Σ0 and Σ of Γ at (possibly multiple) steady state or Hopf bifurcation points (minimal Σ0-Σ-breaking bifurcation). We introduce a bifurcation graph which shows graphically this a priori information — its edges represent possible symmetry breaking bifurcations. Our analysis follows the lines of Golubitsky, Stewart and Schaeffer (1988) but it is aimed to numerical applications. We have chosen a 4-box-Brusselator model in order to explain our notions and ideas and to discuss the numerical procedure.

Regular Systems for Bifurcation Points with Underlying Symmetries

For the computation of bifurcation points the regularity of determining systems is a desirable property. We will present regular systems for simple and double bifurcation points of one-parameter dependent problems g(x,λ)=o which satisfy certain syitmetry-invariance conditions. Besides of the numerical relevance we will shcw that regular systems can be a rather powerful tool to obtain analytical bifurcation results.

Computation of Hopf bifurcation with bordered matrices

Hopf bifurcation for dynamical systems

Computation of Hopf Branches Bifurcating from Takens-Bogdanov Points for Problems with Symmetries

\dot x = f(x,\lambda )

Eine Erweiterung des Maximumprinzips für den Laplaceschen Operator

is characterized by rank deficiency two of

The dependence of critical parameter bounds on the monotonicity of a Newton sequence

A^2 + \omega ^2 I

Monotonie und Finite Elemente bei Elliptischen Differentialgleichungen

, where

Microscopic Car-Following Models Revisited: From Road Works to Fundamental Diagrams

A = f_x (x,\lambda ) \in \mathbb{R}^{n \times n}

Test Functions for Bifurcation Points and Hopf Points in Problems with Symmetries

is the Jacobian at the Hopf point and

Numerical computation of degenerate Hopf bifurcation points

\pm i\omega