Abstract. This paper considers a hierarchy of mathematical models of excitable media in one spatial dimension, specifically the FitzHugh-Nagumo equation and several models of the dynamics of intracellular calcium. A common feature of the models is that they support solitary traveling pulse solutions which lie on a characteristic C-shaped curve of wave speed versus parameter. This C lies to the left of a Ushaped locus of Hopf bifurcations that corresponds to the onset of small-amplitude linear waves. The central question addressed is how the Hopf and solitary wave (homoclinic orbit in a moving frame) bifurcation curves interact in these "CU systems." A variety of possible codimension-two mechanisms is reviewed through which such Hopf and homoclinic bifurcation curves can interact. These include Shil'nikov-Hopf bifurcations and the local birth of homoclinic chaos from a saddle-node/Hopf (Gavrilov-Guckenheimer) point. Alternatively, there may be barriers in phase space that prevent the homoclinic curve from reaching the Hopf bifurcation. For example, the homoclinic orbit may bump into another equilibrium at a so-called T-point, or it may terminate by forming a heteroclinic cycle with a periodic orbit. This paper presents the results of detailed numerical continuation results on different CU systems, thereby illustrating various mechanisms by which Hopf and homoclinic curves interact in CU systems. Owing to a separation of time scales in these systems, considerable care has to be taken with the numerics in order to reveal the true nature of the bifurcation curves observed.

Abstract. The dynamics occurring near a heteroclinic cycle between a hyperbolic equilibrium and a hyperbolic periodic orbit is analyzed. The case of interest is when the equilibrium has a onedimensional unstable manifold and a two-dimensional stable manifold while the stable and unstable manifolds of the periodic orbit are both two-dimensional. A codimension-two heteroclinic cycle occurs when there are two codimension-one heteroclinic connections, with the connection from the periodic orbit to the equilibrium corresponding to a tangency between the two relevant manifolds. The results are restricted to R 3 , the lowest possible dimension in which such a heteroclinic cycle can occur, but are expected to be applicable to systems of higher dimension as well.A geometric analysis is used to partially unfold the dynamics near such a heteroclinic cycle by constructing a leading-order expression for the Poincaré map in a full neighbourhood of the cycle in both phase and parameter space. Curves of orbits homoclinic to the equilibrium are located in a generic parameter plane, as are curves of homoclinic tangencies to the periodic orbit. Moreover, it is shown how curves of folds of periodic orbits, which have different asymptotics near the homoclinic bifurcation of the equilibrium and the homoclinic bifurcation of the periodic orbit, are glued together near the codimension-two point.A simple global assumption is made about the existence of a pair of codimension-two heteroclinic cycles corresponding to a first and last tangency of the stable manifold of the equilibrium and the unstable manifold of the periodic orbit. Under this assumption, it is shown how the locus of homoclinic orbits to the equilibrium should oscillate in the parameter space, a phenomenon known as homoclinic snaking. Finally, we present several numerical examples of systems that arise in applications, which corroborate and illustrate our theory.Key words: global bifurcation, Shil'nikov analysis, heteroclinic cycle, homoclinic orbit, homoclinic tangency, snaking.AMS subject classifications: 34C23, 34C37, 37C29, 37G20.1. Introduction. We are interested in the dynamics near a heteroclinic cycle connecting a hyperbolic equilibrium solution and a hyperbolic periodic orbit, referred to here as an EP-cycle. Specifically, we consider dynamics in R 3 , and assume that the equilibrium (denoted E) has a one-dimensional unstable manifold W u (E), while the periodic orbit (denoted P ) has a two-dimensional unstable manifold W u (P ). In this case, a heteroclinic connection from E to P generically will be of codimension one, while a connection from P to E generically will be of codimension zero. We denote by EP1-cycle an EP-cycle in R 3 where the connections from E to P and from P to E are both of the appropriate generic type. Such a heteroclinic cycle as a whole is then a codimension-one object, i.e., it occurs at isolated points in a generic one-parameter family of vector fields. In a two-parameter setting, EP1-cycles will occur on onedimensional curves in the two-par...

Resonant homoclinic flip bifurcations are codimension-three phenomena that act as organizing centres for codimension-two inclination flip, orbit flip and eigenvalue-resonance bifurcations of homoclinic orbits to a real saddle. In a recent paper by Homburg and Krauskopf unfoldings for several cases of resonant homoclinic flip bifurcations were proposed as bifurcation diagrams on a sphere around the central singularity.This paper presents a comprehensive numerical investigation into these unfoldings in a specific three-dimensional vector field, which was constructed by Sandstede to explicitly contain inclination flip and orbit flip bifurcations. For both orbit and inclination flips, different cases can be classified according to the eigenvalues of the saddle point. All possible cases are treated including complicated ones involving homoclinic-doubling cascades and chaos. In each case, by choosing a sufficiently small sphere around the codimension-three point in parameter space, the conjectured unfoldings are largely confirmed. However, for larger spheres interesting new codimension-three bifurcations occur, leading to a more complicated bifurcation structure. The results suggest an important trade-off between finding bifurcation curves numerically and introducing new bifurcations by enlarging the sphere too much.

The saddle-node Hopf bifurcation (SNH) is a generic codimension-two bifurcation of equilibria of vector fields in dimension at least three. It has been identified as an organizing centre in numerous vector field models arising in applications. We consider here the case that there is a global reinjection mechanism, because the centre manifold of the zero eigenvalue returns to a neighbourhood of the equilibrium. Such a SNH bifurcation with global reinjection occurs naturally in applications, most notably in models of semiconductor lasers.We construct a three-dimensional model vector field that allows us to study the possible dynamics near a SNH bifurcation with global reinjection. This model follows on from our earlier results on a planar (averaged) vector field model, and it allows us to find periodic and homoclinic orbits with global excursions out of and back into a neighbourhood of the SNH point. Specifically, we use numerical continuation techniques to find a two-parameter bifurcation diagram for a well-known and complicated case of a SNH bifurcation that involves the break-up of an invariant sphere. As a particular feature we find a concrete example of a phenomenon that was studied theoretically by Rademacher: a curve of homoclinic orbits that accumulates on a segment in parameter space while the homoclinic orbit itself approaches a saddle-periodic orbit.

We introduce a simple two-dimensional model that extends the Poincaré oscillator so that the attracting limit cycle undergoes a saddle node bifurcation on an invariant circle (SNIC) for certain parameter values. Arbitrarily close to this bifurcation, the phase-resetting curve (PRC) continuously depends on parameters, where its shape can be not only primarily positive or primarily negative but also nearly sinusoidal. This example system shows that one must be careful inferring anything about the bifurcation structure of the oscillator from the shape of its PRC.