Coherent structures near the boundary between excitable and oscillatory media

Abstract

We investigate reaction-diffusion systems near parameter values that mark the transition from an excitable to an oscillatory medium. We analyze existence and stability of traveling waves near a steep pulse that arises as the limit of excitation pulses when parameters cross into the oscillatory regime. Traveling waves near this limiting profile are obtained by analyzing a codimension-two homoclinic saddle-node/orbit-flip bifurcation. The main result shows that there are precisely two generic scenarios for such a transition, distinguished by the sign of an interaction coefficient between pulses. In both scenarios, we find stable fast fronts, unstable slow fronts, stable excitation pulses, and trigger and phase waves. Both trigger and phase waves are stable for repulsive interaction and both are unstable for attractive interaction.

Spatial Wavenumber Selection in Recurrent Precipitation

“…There can actually be multiple stable spreading mechanisms as illustrated in [2], where there are both a stable pulled front and a stable pushed front, defined in Section 3.1, below. In the simplest explicit example in [2], the two invasion mechanisms are distinguished by a topological winding number of the solution profile in the regime between leading edge and wake. All this said, we will not attempt to provide a complete picture of the invasion process, here, but focus on two aspects: in this section, we exhibit a regime where pushed fronts exist and create homogeneous states in their wake, and in the next section, we give a counting arguments for the existence of coherent pulled fronts.…”

“…For this, one needs to control the effects of nonlinearity and, in particular, discuss the possible appearance of pushed fronts [35]. There can actually be multiple stable spreading mechanisms as illustrated in [2], where there are both a stable pulled front and a stable pushed front, defined in Section 3.1, below. In the simplest explicit example in [2], the two invasion mechanisms are distinguished by a topological winding number of the solution profile in the regime between leading edge and wake.…”

Strain and defects in oblique stripe growth

“…For c ≫ 1, we find at leading order, after a a reduction to a slow manifold, cu y = 1 + κ sin(u) − k x , which possesses heteroclinic orbits for k z = 1 − κ, connecting the saddle-node equilibria u = π/2 mod 2π. These heteroclinics between saddle-node equilibria are robust up to a heteroclinic codimension-two bifurcation [11,4]. The associated phase-portraits in the u − u x -plane are shown in Figure 5.3 and can be easily confirmed using elementary phase-plane analysis and monotonicity in c.…”

“…4 shows the surface k x (k y , c x ) from different angles, exhibiting the singularities that occur in the compactification at the boundaries c x , k y ∈ {0, ∞}.…”

Spiral waves: linear and nonlinear theory

“…Note that one typically thinks of excitable media as organized around excitation pulses and their periodic concatenation, so-called trigger waves, whereas periodic media are organized around spatially homogeneous oscillations and their spatial modulation, so-called phase waves. At the transition from oscillatory to excitable media, excitation pulses terminate in a saddle-node bifurcation [13,21], while homogeneous oscillations end in homoclinic or Hopf bifurcations. Phase waves can however be continued to trigger waves [13,20], and we show in Figure 9 that spiral waves emitting those phase and trigger waves, respectively, are connected in parameter space.…”