Destabilization Mechanisms of Periodic Pulse Patterns Near a Homoclinic Limit

Abstract

It has been observed in the Gierer-Meinhardt equations that destabilization mechanisms are rather complex when spatially periodic pulse patterns approach a homoclinic limit. While decreasing the wave number k, the character of destabilization alternates between two kinds of Hopf instabilities. In the first kind, a conjugated pair of so-called 1eigenvalues crosses the imaginary axis exciting perturbations that are in phase with the periodic solution. In the second kind, a pair of −1-eigenvalues crosses the imaginary axis exciting anti-phase perturbations. In (parameter, wave number)space, the curves H ±1 corresponding to ±1-Hopf instabilities intersect infinitely often as they oscillate about each other, while both converge to the Hopf destabilization point of the homoclinic limit on the line k = 0, i.e. they perform a Hopf dance. In an appropriate singular limit, the curves H ±1 cover the boundary of the region of stable pulse solutions-the so-called Busse balloon. The Busse-balloon boundary is non-smooth at intersections of H +1 and H −1 due to an associated higher order phenomenon: the belly dance. The analysis of these phenomena in the Gierer-Meinhardt system relies crucially on specific characteristics of the equations. In this paper, we employ recently developed spectral methods to show that both the Hopf and belly dance are persistent mechanisms that occur in a general class of singularly perturbed reaction-diffusion systems beyond the 'slowly linear' Gierer-Meinhardt equations. Moreover, we establish an explicit sign criterion to determine whether the homoclinic limit is the last or the first 'periodic' solution to destabilize. We illustrate our results by explicit calculations in a slowly nonlinear model system.

The dynamics of disappearing pulses in a singularly perturbed reaction–diffusion system with parameters that vary in time and space

“…there typically are two (quasi-steady) eigenvalues that may cause the destabilization. The associated two critical eigenfunctions are also (almost) periodic, either with the same period of the underlying pattern, or with twice that period -which is in agreement with analytical insights in the destabilization mechanisms of 'perfect' spatially periodic patterns [6,15,16] (see also the two conjectures in section 4.1.2). These critical eigenfunctions are plotted in Figure 4 for a stationary regular 2-pulse pattern for h(x) ≡ 0 and a fixed near its bifurcation value -i.e.…”

“…At present, it is not clear how we get from the simple, one-pulse eigenfunctions to these more involved (periodic) eigenfunctions as patterns evolve towards regularity. These two types of destabilisations are intertwined in an involved way, which is explained by the appearance of 'Hopf dances' [16,15]. We refrain from going in the details here.…”

“…Consequently the homoclinic pulse, the solitary V-pulse, is furthest away from any other pulses and has the lowest u 0 j -value. It should therefore be the most stable configuration, as stated by Ni [29,16,15].…”

Pattern formation in reaction-diffusion systems — an explicit approach

“…The saddle node curve represents the onset of the pulse self-replication mechanism the Gray-Scott model is famous for [22,51,67] and the intersection of the Hopf curve is the first of countably many intersections -and thus co-dimension 2 points -accumulating on the 'homoclinic tip' at the extreme left hand side of the Busse balloon at/near k = 0, i.e. at/near the homoclinic limit where T → ∞, with T = the period of the (stable) spatially periodic patterns, see [23,24].…”

“…However, in [60] it was claimed that 'the homoclinic pattern is the last to become unstable' in the generalized Gierer-Meinhardt model [36]. This conjecture has been shown to hold in a large class of 2-component reaction-diffusion models [23,24]. Therefore, the Busse balloon can indeed be considered as crucial -but certainly not yet well-studied -bridging concept between the classical Turing patterns near onset and localized far from equilibrium patterns.…”