Open AccessJournal Article10.1016/J.PHYSD.2007.03.016

Computing absolute and essential spectra using continuation

- 15 May 2007

- Vol. 229, Iss: 2, pp 166-183

152

TL;DR: In this article, a continuation approach to the computation of essential and absolute spectra of dierential operators on the real line is presented, compared with direct eigenvalue computations for the discretized operator, are the ecient and accurate computation of selected parts of the spectrum (typically those near the imaginary axis) and the option to compute nonlinear travelling waves and selected eigenvalues or other stability indicators simultaneously.

About: This article is published in Physica D: Nonlinear Phenomena. The article was published on 15 May 2007. and is currently open access. The article focuses on the topics: Operator (computer programming) & Computation.

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A continuation approach to the computation of essential and absolute spectra of differential operators on the real line is presented.. The authors also discuss the implementation and usage of this approach with the software package AUTO and provide example computations for the FitzHugh–Nagumo and the complex Ginzburg–Landau equation.

Fig. 4. Details of the onset of instability of the isola shown in Fig. 3. (a) We plot an overlay of the critical parts of the isola for ≈ 0.00371011259 and ≈ 0.00371011266. (b) We plot the tangency coefficient λ‖ (see Section 3.4.3) as a function of , which corroborates that the onset occurs at zero wavenumber.

Fig. 5. (a) The isola from Fig. 3(b) in the unweighted space and the upper part of the same isola, now computed in a weighted space with Re ν ≈ −0.035. (b) Magnification of the isola in the weighted space for two different values of : the unstable isola to the right corresponds to a value of closer to the fold. Both isolas contain absolute spectrum, cf. [10].

Fig. 3. (a) ≈ 0.00371: The isola corresponding to the fold eigenvalue has merged with the isola at the origin. (b) ≈ 0.00371013: The merged isolas of the fast wave train have already crossed the imaginary axis before the fold point.

Fig. 2. (a) Bifurcation diagram of wave trains with period L = 200 to the FitzHugh–Nagumo equation in the ( , c) parameter plane. The inset shows the u-component of the fast wave train for = 0.0033. (b) The real part of the eigenvalue of L0 that changes sign at the fold bifurcation is plotted as a function of . Here the lower branch corresponds to the fast wave trains and the upper one to the slow wave trains.

Fig. 10. (a) The absolute spectrum is plotted for (α, β) = (−8, 1) and κ = −0.3. Note that Σ2abs is unstable, but that there are no branch points with index two. (b) Magnification of one of the critical regions.

Fig. 8. The generalized absolute spectrum Σ∗abs is plotted for (α, β) = (0.1, 0.2), where bullets correspond to branch points and numbers indicate the Morse index. (a) For the wavenumber κ = 0.1 the absolute spectrum, Σ2abs, is stable, and its rightmost point is a branch point at λ ≈ −0.0001. (b) For the wavenumber κ = 0.7 the absolute spectrum is unstable, and its rightmost points are branch points at λ ≈ 0.115± 0.036i (here we omitted two branch points of index one at λ ≈ 189 and λ ≈ 210).

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References

•Journal Article•10.1103/REVMODPHYS.65.851

Pattern formation outside of equilibrium

Michael Cross, +1 more

- 01 Jul 1993

- Reviews of Modern Physics

TL;DR: A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented in this article, with emphasis on comparisons between theory and quantitative experiments, and a classification of patterns in terms of the characteristic wave vector q 0 and frequency ω 0 of the instability.

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•Book

Geometric Theory of Semilinear Parabolic Equations

Daniel Henry

- 01 Feb 1981

TL;DR: The neighborhood of an invariant manifold near an equilibrium point is a neighborhood of nonlinear parabolic equations in physical, biological and engineering problems as mentioned in this paper, where the neighborhood of a periodic solution is defined by the invariance of the manifold.

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Monograph•10.1515/9780691213101

Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators

Lloyd N. Trefethen, +1 more

- 01 Jan 2005

TL;DR: In this article, the authors introduce pseudospectra and non-normal matrices, and describe the behavior of nonsymmetric eigenproblems in non-hermitian systems.

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AUTO-07p: Continuation and bifurcation software for ordinary differential equations

Eusebius J. Doedel, +9 more

- 01 Jan 2007

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AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont)

Eusebius J. Doedel, +3 more

- 01 Jan 1997

TL;DR: This is a guide to the software package AUTO for continuation and bifurcation problems in ordinary differential equations and the development of HomCont has much benefitted from various pieces of help and advice from, among others, W. W. Norton.

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Computing absolute and essential spectra using continuation

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Citations

Beyond Turing: The response of patterned ecosystems to environmental change

Rise and Fall of Periodic Patterns for a Generalized Klausmeier–Gray–Scott Model

Localized Pattern Formation with a Large-Scale Mode: Slanted Snaking

Multistability of model and real dryland ecosystems through spatial self-organization.

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

References

Pattern formation outside of equilibrium

Geometric Theory of Semilinear Parabolic Equations

Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators

AUTO-07p: Continuation and bifurcation software for ordinary differential equations

AUTO 2000 : CONTINUATION AND BIFURCATION SOFTWARE FOR ORDINARY DIFFERENTIAL EQUATIONS (with HomCont)

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