Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof

Zgliczyński, Piotr

doi:10.3934/jcd.2015.2.95

Cited by 16 publications

(18 citation statements)

References 20 publications

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“…Similar approach has been 2 successfully applied for validated computation of bifurcations of equilibria for ODEs and steady states for PDEs -see for example [8,30,31]. A different, geometric method for validation of bifurcations of steady states in the Kuramoto-Sivashinsky PDE is also proposed in [55]. Our algorithm for proving the existence of period-tupling and touch-and-go bifurcation is similar in the spirit to that proposed in [51,55], but exploits the presence of a reversing symmetry of the system.…”

Section: Introductionmentioning

confidence: 99%

“…A different, geometric method for validation of bifurcations of steady states in the Kuramoto-Sivashinsky PDE is also proposed in [55]. Our algorithm for proving the existence of period-tupling and touch-and-go bifurcation is similar in the spirit to that proposed in [51,55], but exploits the presence of a reversing symmetry of the system. After fixing an appropriate coordinate system in a neighbourhood of an apparent bifurcation points, we perform validated Lyapunov-Schmidt reduction.…”

Section: Introductionmentioning

confidence: 99%

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Validated numerics for period-tupling and touch-and-go bifurcations of symmetric periodic orbits in reversible systems

Walawska

Wilczak

2019

Communications in Nonlinear Science and Numerical Simulation

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We propose a general framework for computer-assisted verification of the presence of symmetry breaking, period-tupling and touch-and-go bifurcations of symmetric periodic orbits for reversible maps. The framework is then adopted to Poincaré maps in reversible autonomous Hamiltonian systems.In order to justify the applicability of the method, we study bifurcations of halo orbits in the Circular Restricted Three Body Problem. We give a computer-assisted proof of the existence of wide branches of halo orbits bifurcating from L 1,2,3 -Lyapunov families and for wide range of mass parameter. For two physically relevant mass parameters we prove, that halo orbits undergo multiple period doubling, quadrupling and third-order touch-and-go bifurcations.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Validated numerics for period-tupling and touch-and-go bifurcations of symmetric periodic orbits in reversible systems

Walawska

Wilczak

2019

Communications in Nonlinear Science and Numerical Simulation

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“…In preceding works only the existence or the uniqueness of equilibria are mainly focused (e.g. [13,14]). Such methods do not give us precise information of dynamics around equilibira, like stability, except the special case.…”

Section: Discussionmentioning

confidence: 99%

Rigorous numerics for stationary solutions of dissipative PDEs - Existence and local dynamics -

Matsue

2013

NOLTA

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Abstract:We show a criterion for verifying the locally unique existence of stationary solutions of a class of partial differential equations and the local dynamics around them with computer assistance. Our method is designed so that, in near future, it can be extended to further analysis of dynamics, like connecting orbits between equilibria. Several sample examples for verifying hyperbolic equilibria of a concrete PDE are also shown.

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“…We are not the first to validate bifurcation points. Reference [39] validates bifurcation points using dynamical systems/Conley index-type arguments. The method relies on having coordinate-specific information.…”

Section: Introductionmentioning

confidence: 87%

“…One can easily exchange the nonlinearity f , and choose considerably larger dimensions n. For example, we validated branches of solutions for two mosaic solutions in dimension n = 100 with nonlinearity f (u) = sin(πu)/π. This merely required a small change of the validation code-one has to exchange the function definition and adapt certain nonlinearity estimates for f ; see (39). After these changes, validation succeeded as easily as in the previous situation.…”

Section: Robustness and Bifurcation Of Equilibriamentioning

confidence: 99%

Validated Saddle-Node Bifurcations and Applications to Lattice Dynamical Systems

Sander¹,

Wanner²

2016

SIAM J. Appl. Dyn. Syst.

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The use of rigorous verification methods is a powerful tool which permits progress in the analysis of dynamical processes that is not possible using purely analytical techniques. In this paper we develop a set of tools for branch validation, which allows for the rigorous verification of branch behavior, bifurcation, and solution index on branches generated through a saddle-node bifurcation. While the presented methodology can be applied in a variety of settings, we illustrate the use of these tools in the context of materials science. In particular, lattice models have been proposed as a more realistic reflection of the behavior of materials than traditional continuum models. For example, unlike their continuum counterparts, lattice models can account for phenomena such as pinning, and a significant body of work has been developed to study traveling waves. However, in a variety of other contexts such as bifurcation theory, questions about lattice dynamical systems are significantly harder to answer than those for continuum models. In the present paper, we show that computer-assisted proof techniques can be used to answer some of these questions. We apply these tools to the discrete Allen-Cahn equation, giving us results on the existence of branches of mosaic solutions and their robustness as it relates to grain size. We also demonstrate that there are situations in which classical continuation methods can fail to identify the correct branching behavior.

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Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof

Abstract: Abstract. We apply the method of self-consistent bounds to prove the existence of multiple steady state bifurcations for Kuramoto-Sivashinski PDE on the line with odd and periodic boundary conditions.

Cited by 16 publications

References 20 publications

Validated numerics for period-tupling and touch-and-go bifurcations of symmetric periodic orbits in reversible systems

Validated numerics for period-tupling and touch-and-go bifurcations of symmetric periodic orbits in reversible systems

Rigorous numerics for stationary solutions of dissipative PDEs - Existence and local dynamics -

Validated Saddle-Node Bifurcations and Applications to Lattice Dynamical Systems

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