Numerical continuation for fractional PDEs: sharp teeth and bloated snakes

Ehstand, Noémie; Kuehn, Christian; Soresina, Cinzia

doi:10.1016/j.cnsns.2021.105762

Cited by 9 publications

(3 citation statements)

References 55 publications

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“…Our main aim with these examples is to present interesting bifurcation problems and how they can be studied by numerical continuation and bifurcation. Other recent applications of pde2path are given in, e.g., [6,9,26,41,44,55,62,69,75,78,82,83,86]. In Table 1 we collect some acronyms frequently used in PDEs and bifurcation theory, and some specific for this paper.…”

Section: Introductionmentioning

confidence: 99%

Continuation and Bifurcation in Nonlinear PDEs – Algorithms, Applications, and Experiments

Uecker

2021

Jahresber. Dtsch. Math. Ver.

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Numerical continuation and bifurcation methods can be used to explore the set of steady and time–periodic solutions of parameter dependent nonlinear ODEs or PDEs. For PDEs, a basic idea is to first convert the PDE into a system of algebraic equations or ODEs via a spatial discretization. However, the large class of possible PDE bifurcation problems makes developing a general and user–friendly software a challenge, and the often needed large number of degrees of freedom, and the typically large set of solutions, often require adapted methods. Here we review some of these methods, and illustrate the approach by application of the package to some advanced pattern formation problems, including the interaction of Hopf and Turing modes, patterns on disks, and an experimental setting of dead core pattern formation.

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Section: Introductionmentioning

confidence: 99%

Continuation and Bifurcation in Nonlinear PDEs – Algorithms, Applications, and Experiments

Uecker

2021

Jahresber. Dtsch. Math. Ver.

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“…To this end, we have shown that the continuation software is easily applicable to different models and it can be adapted to treat systems that do not exactly belong to the class of problems for which it has been developed. For instance, another direction could be its extension to other non-standard diffusion processes such as fractional reaction-diffusion equations [21].…”

Section: Discussionmentioning

confidence: 99%

Numerical continuation for a fast-reaction system and its cross-diffusion limit

Kuehn

Soresina

2020

SN Partial Differ. Equ. Appl.

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In this paper we investigate the bifurcation structure of the triangular SKT model in the weak competition regime and of the corresponding fast-reaction system in 1D and 2D domains via numerical continuation methods. We show that the software pde2path can be adapted to treat cross-diffusion systems, reproducing the already computed bifurcation diagrams on 1D domains. We show the convergence of the bifurcation structure obtained selecting the growth rate as bifurcation parameter. Then, we compute the bifurcation diagram on a 2D rectangular domain providing the shape of the solutions along the branches and linking the results with the Turing instability analysis. In 1D and 2D, we pay particular attention to the fast-reaction limit by always computing sequences of bifurcation diagrams as the time-scale separation parameter tends to zero. We show that the bifurcation diagram undergoes major deformations once the fast-reaction systems limits onto the cross-diffusion singular limit. Furthermore, we find evidence for time-periodic solutions by detecting Hopf bifurcations, we characterize several regions of multi-stability, and improve our understanding of the shape of patterns in 2D for the SKT model.

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“…This package is an advanced continuation/bifurcation software based on the FEM discretization of the stationary elliptic problem exploiting the package OOPDE [40] for the FEM discretization. Since the software is quite flexible, it has also been used beyond its standard-setting, for instance, to treat cross-diffusion systems [31], and spectral fractional diffusion [13]. Moreover, thanks to powerful computer-assisted techniques [6,4] developed in the last three decades, and recently extended to treat non-linear diffusion terms [3], the approximated solutions found with pde2path can be validated rigorously a posteriori.…”

Section: Introductionmentioning

confidence: 99%

Hopf bifurcations in the full SKT model and where to find them

Soresina¹

2022

Preprint

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In this paper, we consider the Shigesada-Kawasaki-Teramoto (SKT) model, which presents cross-diffusion terms describing competition pressure effects. Even though the reaction part does not present the activator-inhibitor structure, cross-diffusion can destabilise the homogeneous equilibrium. However, in the full cross-diffusion system and weak competition regime, the cross-diffusion terms have an opposite effect and the bifurcation structure of the system modifies increasing the interspecific competition pressure. The major changes in the bifurcation structure, the type of pitchfork bifurcations on the homogeneous branch, as well as the presence of Hopf bifurcation points are here investigated. Through weakly nonlinear analysis, we can predict the type of pitchfork bifurcation. Increasing the additional cross-diffusion coefficients, the first two pitchfork bifurcation points from super-critical become sub-critical, leading to the appearance of a multi-stability region. The interspecific competition pressure also influences the possible appearance of stable time-period spatial patterns appearing through a Hopf bifurcation point.

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Numerical continuation for fractional PDEs: sharp teeth and bloated snakes

Cited by 9 publications

References 55 publications

Continuation and Bifurcation in Nonlinear PDEs – Algorithms, Applications, and Experiments

Continuation and Bifurcation in Nonlinear PDEs – Algorithms, Applications, and Experiments

Numerical continuation for a fast-reaction system and its cross-diffusion limit

Hopf bifurcations in the full SKT model and where to find them

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