Spectra and Stability of Spatially Periodic Pulse Patterns: Evans Function Factorization via Riccati Transformation

Rijk, Björn de; Doelman, Arjen; Rademacher, Jens D. M.

doi:10.1137/15m1007264

Cited by 28 publications

(40 citation statements)

References 39 publications

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“…The relative time scales of the variation of a(t) versus the intrinsic rate of change of the N-pulse pattern as it evolves over M N is a decisive ingredient that shapes this competition. A more subtle, but at least as important, ingredient is the -at present not understood -(slow) dynamics of the quasi-steady eigenfunctions as they evolve from the irregular setting of being localized around one pulse location to the global Floquet-type eigenfunctions -see [7] and the references therein -associated with regular spatially periodic patterns. Figure 24 -Sketch of the classic 'mass on incline' problem (a).…”

Section: Discussion and Outlookmentioning

confidence: 99%

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

Bastiaansen

Doelman

2019

Physica D: Nonlinear Phenomena

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We consider the evolution of multi-pulse patterns in an extended Klausmeier equation -as generalisation of the wellstudied Gray-Scott system, a prototypical singularly perturbed reaction-diffusion equation -with parameters that change in time and/or space. As a first step we formally show -under certain conditions on the parameters -that the full PDE dynamics of a N-pulse configuration can be reduced to a N-dimensional dynamical system that describes the dynamics on a N-dimensional manifold M N . Next, we determine the local stability of M N via the quasi-steady spectrum associated to evolving N-pulse patterns. This analysis provides explicit information on the boundary ∂M N of M N . Following the dynamics on M N , a N-pulse pattern may move through ∂M N and 'fall off' M N . A direct nonlinear extrapolation of our linear analysis predicts the subsequent fast PDE dynamics as the pattern 'jumps' to another invariant manifold M M , and specifically predicts the number N − M of pulses that disappear during this jump. Combining the asymptotic analysis with numerical simulations of the dynamics on the various invariant manifolds yields a hybrid asymptotic-numerical method that describes the full pulse interaction process that starts with a N-pulse pattern and typically ends in the trivial homogeneous state without pulses. We extensively test this method against PDE simulations and deduce a number of general conjectures on the nature of pulse interactions with disappearing pulses. We especially consider the differences between the evolution of (sufficiently) irregular and regular patterns. In the former case, the disappearing process is gradual: irregular patterns loose their pulses one by one, jumping from manifold M k to M k−1 (k = N, . . . , 1). In contrast, regular, spatially periodic, patterns undergo catastrophic transitions in which either half or all pulses disappear (on a fast time scale) as the patterns pass through ∂M N . However, making a precise distinction between these two drastically different processes is quite subtle, since irregular N-pulse patterns that do not cross ∂M N typically evolve towards regularity. sometimes also called the generalised Klausmeier-Gray-Scott system [34,40]. This model is a generalization of the original ecological model by Klausmeier on the interplay between vegetation and water in semi-arid regions [23] -which was proposed to describe the appearance of vegetation patterns as crucial intermediate step in the desertification process that begins with a homogeneously vegetated terrain and ends with the non-vegetated bare soil state: the desert -see [8,28,32] and the references therein for observations of these patterns and their relevance for the desertification process. In (1.1), U(x, t) represents (the concentration of) water and V(x, t) vegetation; for simplicity -and as in [34,35,36,38, 40] -we consider the system in a 1-dimensional unbounded domain, i.e. x ∈ R; parameter a models the rainfall and m the mortality of the vegetation. Since the diffusion of water occurs on a mu...

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Section: Discussion and Outlookmentioning

confidence: 99%

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

Bastiaansen

Doelman

2019

Physica D: Nonlinear Phenomena

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“…In this paper, we will determine the fine-structure of the Busse balloon near a homoclinic tip of Hopf type as sketched in Figure 2. We will do so by employing the recently developed spectral methods of [3,4,5,8] for the stability of spatially periodic and homoclinic (pulse) patterns to a general class of singularly perturbed two-component reactiondiffusion systems 1) or, in the 'fast' spatial scale x = ε −1x ,…”

Section: Busse Balloonmentioning

confidence: 99%

Destabilization Mechanisms of Periodic Pulse Patterns Near a Homoclinic Limit

Doelman¹,

Rademacher²,

Rijk³

et al. 2018

SIAM J. Appl. Dyn. Syst.

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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.

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“…Here, the situation is a bit more subtle since γ p (ξ; σ) is not a subset of W s (M ε ) ∩ W u (M ε ) (but exponentially close to it). We refrain from going into the details and refer to [15] for a (much more) general existence result and to [21] for a deeper exploration into the extremely rich structure of periodic (and aperiodic) symmetric solutions to singularly perturbed 2-component reaction-diffusion systems.…”

Section: The Construction Of Symmetric Homoclinic Pulses In Singularlmentioning

confidence: 99%

“…4(c) and Lemma 4.5. This can be done using the Floquet theory based concept of γ-eigenvalues [35] -see [15] (also for a much more general setting than considered here). [42,43].…”

Section: Stability Analysis In Singularly Perturbed Systemsmentioning

confidence: 99%

Pattern formation in reaction-diffusion systems — an explicit approach

Doelman¹

2019

Complexity Science

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Pattern formation is the sub-area of complexity science in which the dynamics of nonlinear spatial processes is studied. Reaction-diffusion systems appear as relevant models for such processes and are at the core of the mathematical analysis of pattern formation. This review presents an introduction to the basic mathematical 'tools' by which an understanding of complex pattern dynamics can be built. For simplicity, it focuses on 2-component reaction-diffusion systems in one spatial dimension. In this setting the techniques by which pattern formation can be studied-both 'near onset' as well as 'far from equilibrium'-can be presented in such a way that an explicit study of a given model can be set up. Moreover, the text provides furthers steps by which more complex patterns in more complex systems than considered here may be understood.

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Spectra and Stability of Spatially Periodic Pulse Patterns: Evans Function Factorization via Riccati Transformation

Cited by 28 publications

References 39 publications

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

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

Destabilization Mechanisms of Periodic Pulse Patterns Near a Homoclinic Limit

Pattern formation in reaction-diffusion systems — an explicit approach

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