Hopf Bifurcation Of System Research Articles

Computing Hopf Bifurcations I

This paper addresses the problems of detecting Hopf bifurcations in systems of ordinary differential equations and following curves of Hopf points in two-parameter families of vector fields. The established approach to this problem relies upon augmenting the equilibrium condition so that a Hopf bifurcation occurs at an isolated, regular point of the extended system. We propose two new methods of this type based on classical algebraic results regarding the roots of polynomial equations and properties of Kronecker products for matrices. In addition to their utility as augmented systems for use with standard Newton-type continuation methods, they are also particularly well adapted for solution by computer algebra techniques for vector fields of small or moderate dimension.

Hopf-bifurcation in systems with spherical symmetry. I: Invariant tori

A Hopf-bifurcation scenario with symmetries is studied. Here, apart from the well known branches of periodic solutions, other bifurcation phenomena have to occur as it is shown in the second part of the paper using topological arguments. In this rst part of the paper we prove analytically that invariant tori with quasiperiodic motion bifurcate. The main methods used are orbit space reduction and singular perturbation theory.

Nongeneric Hopf Bifurcations in Functional-differential Equations

An algorithm is presented to prove the existence and determine stability of Hopf bifurcations in systems of functional differential equations. It is then applied to equations with “small” higher order terms and a study of bifurcations simultaneous to critical linear parts. The paper concludes with a thorough treatment of scalar integro-differential equations.

Combinatorial approaches to Hopf bifurcations in systems of interacting elements

We describe combinatorial approaches to the question of whether families of real matrices admit pairs of nonreal eigenvalues passing through the imaginary axis. When the matrices arise as Jacobian matrices in the study of dynamical systems, these conditions provide necessary conditions for Hopf bifurcations to occur in parameterised families of such systems. The techniques depend on the spectral properties of additive compound matrices: in particular, we associate with a product of matrices a signed, labelled digraph termed a DSR^[2] graph, which encodes information about the second additive compound of this product. A condition on the cycle structure of this digraph is shown to rule out the possibility of nonreal eigenvalues with positive real part. The techniques developed are applied to systems of interacting elements termed “interaction networks”, of which networks of chemical reactions are a special case.