Isabel S. Labouriau

Degenerate Hopf bifurcation formulas and Hilbert's 16th problem

This paper presents explicit formulas for the solution of degenerate Hopf bifurcation problems for general systems of differential equations of dimension

Bifurcation of the Hodgkin and Huxley equations: A new twist

n \geqq 2

Dynamics near a heteroclinic network

, with smooth vector fields. The main new result is the general solution of the problem for a weak focus of order 3. For bifurcation problems with a distinguished parameter, we present the formulas for the defining conditions of all cases with codimension

Degenerate Hopf Bifurcation and Nerve Impulse

\leqq 3

Switching near a network of rotating nodes

. The formulas have been applied to Hilbert’s 16th problem, yielding a new proof of Bautin’s theorem, and correcting an error in Bautin’s formula for the third focal value. The approach used is the Lyapunov–Schmidt method. A review of five other approaches is given, along with literature references and comparisons to the present work.

Degenerate Hopf bifurcation and nerve impulse, part II

The Hodgkin and Huxley equations model action potentials in squid giant axons. Variants of these equations are used in most models for electrical activity of excitable membranes. Computational tools based upon the theory of nonlinear dynamical systems are used here to illustrate how the dynamical behavior of the Hodgkin Huxley model changes as functions of two of the system parameters.

Chaotic double cycling

We study the dynamical behaviour of a smooth vector field on a three-manifold near a heteroclinic network. Under some generic assumptions on the network, we prove that every path on the network is followed by a neighbouring trajectory of the vector field—there is switching on the network. We also show that near the network there is an infinite number of hyperbolic suspended horseshoes. This leads to the existence of a horseshoe of suspended horseshoes with the shape of the network.Our results are motivated by an example constructed by Field (1996 Lectures on Bifurcations, Dynamics, and Symmetry (Pitman Research Notes in Mathematics Series vol 356) (Harlow: Longman)), where we have observed, numerically, the existence of such a network.

SIMPLE VECTOR FIELDS WITH COMPLEX BEHAVIOR

It has been established by other authors that the clamped Hodgkin and Huxley equations for the nerve impulse have two branches of periodic solutions arising through Hopf bifurcation. In this paper these solution branches are shown to join, using singularity theory methods developed by M. Golubitsky and W. Langford (J. Differential Equations, 41 (1981), pp. 375–415). The equations are perturbed by varying parameters like temperature and average membrane permeability to certain ions. A hidden organizing centre for the equations is obtained, and its unfolding provides a topological description of the periodic orbits that bifurcate from the equilibrium solution.

Spiralling dynamics near heteroclinic networks

We study the dynamics of a Z 2 ⊕ Z 2-equivariant vector field in the neighbourhood of a heteroclinic network with a periodic trajectory and symmetric equilibria. We assume that around each equilibrium the linearization of the vector field has non-real eigenvalues. Trajectories starting near each node of the network turn around in space either following the periodic trajectory or due to the complex eigenvalues near the equilibria. Thus, in a network with rotating nodes, the rotations combine with transverse intersections of two-dimensional invariant manifolds to create switching near the network; close to the network, there are trajectories that visit neighbourhoods of the saddles following all the heteroclinic connections of the network in any given order. Our results are motivated by an example where switching was observed numerically by forced symmetry breaking of an asymptotically stable network with O(2) symmetry.

Spatial hidden symmetries in pattern formation

The bifurcation from equilibrium of periodic solutions of the Hodgkin and Huxley equations for the nerve impulse is studied. In earlier work singularity theory techniques were used to establish that these equations have a branch of periodic solutions undergoing two Hopf bifurcations, and the equations were conjectured to be equivalent to a member of a one-parameter family of generalized Hopf bifurcation problems. Here the invariants for equivalence to this family and the value of the modal parameter are computed (see [W. W. Farr et al., “Degenerate Hopf bifurcation formulas and Hilbert’s 16th problem,” SIAM J. Math. Anal., 20 (1989), pp. 13–30]). The value of this parameter determines the type of bifurcation, and in this way it is decided which of the proposed bifurcation diagrams are actually to be found. Thus a topological description of periodic orbits of the Hodgkin and Huxley equations near the equilibrium solution is obtained. In this way, a periodic solution branch is found that does not arise thro...