Computing absolute and essential spectra using continuation

Rademacher, Jens D. M.; Sandstede, Björn; Scheel, Arnd

doi:10.1016/j.physd.2007.03.016

Cited by 115 publications

(176 citation statements)

References 12 publications

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“…43,60 It has been found that the sideband instability forms the stability boundary far beyond onset of the Turing(-Hopf) instability. 53,60 The continuation of breakup instabilities of striped patterns against perturbations with c 2 S 1 and ' 2 R can be similarly implemented by imposing constraints on kðc; 'Þ.…”

Section: Stability Of Striped Patternsmentioning

confidence: 99%

“…; whose spectrum is the union of spectra of the Blochoperators M per ðc; nÞ :¼ Mð@ n À logðcÞ=L; nÞ, with c on the unit circle so logðcÞ 2 ðÀpi; pi , posed on [0,1] with periodic boundary conditions. 43,48 Hence, the spectrum is determined by the family of eigenvalue problems M per ðc; nÞ À kId ¼ 0. is the dispersion relation analogous to the case of homogeneous steady states, which is holomorphic in k, c, and '. 43 The cases that are traced by numerical continuation in Sec.…”

Section: Appendix B: Dispersion Relations For Striped Patternsmentioning

confidence: 99%

“…43,48 Hence, the spectrum is determined by the family of eigenvalue problems M per ðc; nÞ À kId ¼ 0. is the dispersion relation analogous to the case of homogeneous steady states, which is holomorphic in k, c, and '. 43 The cases that are traced by numerical continuation in Sec. III C are c ¼ 61, the corresponding eigenfunctions (perturbations) have distinct periodicity properties.…”

Section: Appendix B: Dispersion Relations For Striped Patternsmentioning

confidence: 99%

See 2 more Smart Citations

Striped pattern selection by advective reaction-diffusion systems: Resilience of banded vegetation on slopes

Siero¹,

Doelman²,

Eppinga³

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

123

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For water-limited arid ecosystems, where water distribution and infiltration play a vital role, various models have been set up to explain vegetation patterning. On sloped terrains, vegetation aligned in bands has been observed ubiquitously. In this paper, we consider the appearance, stability, and bifurcations of 2D striped or banded patterns in an arid ecosystem model. We numerically show that the resilience of the vegetation bands is larger on steeper slopes by computing the stability regions (Busse balloons) of striped patterns with respect to 1D and transverse 2D perturbations. This is corroborated by numerical simulations with a slowly decreasing water input parameter. Here, long wavelength striped patterns are unstable against transverse perturbations, which we also rigorously prove on flat ground through an Evans function approach. In addition, we prove a "Squire theorem" for a class of two-component reaction-advection-diffusion systems that includes our model, showing that the onset of pattern formation in 2D is due to 1D instabilities in the direction of advection, which naturally leads to striped patterns. V C 2015 AIP Publishing LLC. This paper has been motivated by studies in one space dimension of a scaled phenomenological model for vegetation on possibly sloped planes in arid ecosystems. 53,60 One-dimensional patterns ideally represent striped patterns in two space dimensions by trivially extending them into a transversal direction. Such patterns are referred to as banded vegetation and have received considerable attention after reports of widespread observations. 8,58 Understanding the appearance and disappearance of vegetation bands may ultimately help prevent land degradation. The restriction to one space dimension may overestimate stability: patterns that are stable against 1D perturbations are not necessarily stable against all 2D perturbations. Natural questions to pose are:• Which of the 1D stable patterns extend to 2D stable striped patterns? • In case of destabilization by 2D perturbations, which mechanisms are responsible?In this paper, we answer these questions for the arid ecosystem model and determine the impact of slope induced advection of water. The influence of advection on striped pattern formation is studied in a more general setting. This approach provides a clear argumentation that is unobscured by model-specific details. Equally important, the results will be applicable to a wide range of models. Applicability to the arid ecosystem model is carefully checked though, assuring that the abstract requirements can in fact be met in practice.

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Section: Stability Of Striped Patternsmentioning

confidence: 99%

Section: Appendix B: Dispersion Relations For Striped Patternsmentioning

confidence: 99%

Section: Appendix B: Dispersion Relations For Striped Patternsmentioning

confidence: 99%

See 1 more Smart Citation

Striped pattern selection by advective reaction-diffusion systems: Resilience of banded vegetation on slopes

Siero¹,

Doelman²,

Eppinga³

et al. 2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

123

View full text Add to dashboard Cite

show abstract

“…However, this is a much more difficult numerical problem that was solved only recently (Sandstede & Scheel 2000;Bordiougov & Engel 2006;Rademacher et al 2007). Rademacher et al (2007) give a full but rather mathematically oriented account of the method; it involves applying AUTO to a boundary-value problem for the eigenfunctions, and the theory underlying numerical continuation of such problems is reviewed by Champneys & Sandstede (2007). A less technical summary, including computer programs written in a tutorial style, is available at www.ma.hw.ac.uk/wjas/ supplements/ptwreview/index.html.…”

Section: Mathematics Of Periodic Travelling Waves Ii: Wave Stabilitymentioning

confidence: 99%

Periodic travelling waves in cyclic populations: field studies and reaction–diffusion models

Sherratt

Smith

2008

J. R. Soc. Interface.

151

123

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Periodic travelling waves have been reported in a number of recent spatio-temporal field studies of populations undergoing multi-year cycles. Mathematical modelling has a major role to play in understanding these results and informing future empirical studies. We review the relevant field data and summarize the statistical methods used to detect periodic waves. We then discuss the mathematical theory of periodic travelling waves in oscillatory reactiondiffusion equations. We describe the notion of a wave family, and various ecologically relevant scenarios in which periodic travelling waves occur. We also discuss wave stability, including recent computational developments. Although we focus on oscillatory reactiondiffusion equations, a brief discussion of other types of model in which periodic travelling waves have been demonstrated is also included. We end by proposing 10 research challenges in this area, five mathematical and five empirical.

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“…BandSolve is used to solve the linear Schrödinger equation by using Bloch theory, as in solid state physics [2]. Using the Floquet parameter as a continuation parameter, Rademacher et al [23] have an algorithm which compares well with Hill's method in terms of speed and accuracy. However, they have not developed a general package: using their algorithm requires the user to implement it in AUTO [15].…”

Section: Remarksmentioning

confidence: 99%

SpectrUW: A laboratory for the numerical exploration of spectra of linear operators

Deconinck

Kiyak

Carter

et al. 2007

Mathematics and Computers in Simulation

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Spectra of linear operators play an important role in various aspects of applied mathematics. For all but the simplest operators, the spectrum cannot be determined analytically and as such it is difficult to build up any intuition about the spectrum. One way to obtain such intuition is to consider many examples numerically and observe emerging patterns. This is feasible using an efficient black-box numerical method, i.e., a method that requires no conceptual changes for different examples. Hill's method satisfies these requirements. It is the mathematical foundation of SpectrUW (pronounced "spectrum"), mathematical black-box software that serves as a laboratory for the numerical approximation of spectra of one-dimensional linear operators.

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Computing absolute and essential spectra using continuation

Cited by 115 publications

References 12 publications

Striped pattern selection by advective reaction-diffusion systems: Resilience of banded vegetation on slopes

Striped pattern selection by advective reaction-diffusion systems: Resilience of banded vegetation on slopes

Periodic travelling waves in cyclic populations: field studies and reaction–diffusion models

SpectrUW: A laboratory for the numerical exploration of spectra of linear operators

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