Hopf Bifurcation in Differential Algebraic Equations and Applications to Circuit Simulation

Abstract

For the analysis of Hopf bifurcation in a dynamical system, i.e. of an autonomous differential equation

$$ \dot x = f(x,\lambda ) $$

where λ is the bifurcation parameter, we can use the fundamental theorem of Hopf [3]. For this we solve an eigenvalue problem for the Jacobi-matrix D ₁ f(x ₀,λ) at an equilibrium point x ₀. This theorem is very useful for application and theory as well. The existence of nontrivial periodic solutions can be shown and nonlinear oscillations can be computed.

In this paper we generalize the Theorem of Hopf for an implicit autonomous differential equation, in particular for a differential algebraic equation. Here we solve a generalized eigenvalue problem instead of an eigenvalue problem.

The important application of the simulation of electrical networks is covered by this approach. In this paper an utilization of the generalized theorem is presented and applications to circuit simulations are tested.