When Shil'nikov Meets Hopf in Excitable Systems

Abstract

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.

Unfolding a Tangent Equilibrium-to-Periodic Heteroclinic Cycle

“…EP1t-points have also been seen in a three-dimensional model motivated by studies of semiconductor lasers with optical reinjection [20]. In this case, folds of a single curve of homoclinic orbits again accumulate on a pair of EP1t-points, in a manner similar to that seen in the nine-dimensional calcium model in [4]. The theoretical study of EP1t-points presented here is motivated by our desire to explain the observed folding and accumulation of homoclinic bifurcation curves in these and other systems.…”

“…and the values of all constants used in the numerical computations using these equations are given in the Appendix. As described in [4], a branch of E-homoclinic bifurcations is seen to fold many times (in a 'homoclinic snake') for s ≈ 6.7 and p varying between about 1.88 and 2.63. Figure 8.6 shows various parts of the bifurcation diagram and some representative phase portraits for this phenomenon.…”

“…For instance, Champneys et al [4] study ways in which branches of isolated travelling pulses interact with smallamplitude waves born in a Hopf bifurcation, in the context of excitable systems. By passing to a moving frame of reference, they reduce this question to the study of various codimension-two mechanisms for the termination of curves of homoclinic bifurcations as a Hopf bifurcation is approached.…”

“…Numerics suggest there exist folds of periodic orbits accumulating on the E-homoclinic orbits and folds accumulating on a P -homoclinic tangency, with the two families of folds being connected via cusps. Since the numerical evidence is not as clear here as for the previous two examples and for reasons of brevity we do not discuss these equations further here but refer the reader to the discussion in [4].…”

“…This paper has (partially) unfolded the dynamics occurring near an EP1t-point, i.e., near a codimension-two heteroclinic cycle connecting a hyperbolic equilibrium and a hyperbolic periodic orbit in the case that the connection from the equilibrium to the periodic orbit is of codimension one and the connection from the periodic orbit to the equilibrium occurs at a tangency of the unstable manifold of the periodic orbit and the stable manifold of the equilibrium. We were initially motivated to perform this study as part of a wider project which aimed to categorise some of the ways in which branches of homoclinic orbits of equilibria can terminate near Hopf bifurcations in excitable systems [4]; EP1t-points turn out to be relevant in this context.…”