Methods in Equivariant Bifurcations and Dynamical Systems

Chossat, Pascal; Lauterbach, Reiner

doi:10.1142/4062

Cited by 218 publications

(379 citation statements)

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“…In the presence of a spatial Z 2 symmetry, the saddle-node bifurcation becomes a pitchfork. And if the symmetry group is O(2), we obtain a pitchfork of revolution (Golubitsky et al 1988;Iooss & Adelmeyer 1998;Chossat & Lauterbach 2000).…”

Section: Symmetries and The Half-period-flip Mapmentioning

confidence: 99%

“…When the symmetries are purely spatial in nature (e.g. reflections, translations, rotations), these consequences have been extensively studied (see, for example, Golubitsky & Schaeffer 1985;Golubitsky, Stewart & Schaeffer 1988;Crawford & Knobloch 1991;Cross & Hohenberg 1993;Chossat & Iooss 1994;Iooss & Adelmeyer 1998;Chossat & Lauterbach 2000;Golubitsky & Stewart 2002). The system may also be invariant to the action of spatio-temporal symmetries.…”

Section: Introductionmentioning

confidence: 99%

“…centrifugal, buoyancy or shear instability, in different situations. The point is that the symmetries of the system govern the types of possible bifurcations that may occur, as well as the symmetry properties of the bifurcating solutions themselves, an idea that is formalised in the equivariant branching lemma (see, for example Golubitsky, Stewart & Schaeffer 1988;Chossat & Lauterbach 2000).…”

Section: Introductionmentioning

confidence: 99%

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Symmetry breaking of two-dimensional time-periodic wakes

2005

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A number of two-dimensional time-periodic flows, for example the Kármán street wake of a symmetrical bluff body such as a circular cylinder, possess a spatio-temporal symmetry: a combination of evolution by half a period in time and a spatial reflection leaves the solution invariant. Floquet analyses for the stability of these flows to three-dimensional perturbations have in the past been based on the Poincaré map, without attempting to exploit the spatio-temporal symmetry. Here, Floquet analysis based on the half-periodflip map provides a comprehensive interpretation of the symmetry breaking bifurcations.

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Section: Symmetries and The Half-period-flip Mapmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Symmetry breaking of two-dimensional time-periodic wakes

2005

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“…We further assume that a finite-dimensional (not necessarily compact) Lie group (G, •) is given that acts on X via a representation in GL(X); that is, we have a homomorphism (cf. [6,Ch. 4…”

Section: Equivariant Evolution Equationsmentioning

confidence: 99%

“…The main applications we have in mind are reaction-diffusion systems on unbounded domains Ω ⊂ R d such as the semilinear system Up to now there is a well-developed bifurcation theory for equivariant dynamical systems that covers the infinite-dimensional case of PDEs and certain aspects of noncompact Lie groups; see the monographs [12], [6]. In particular, we refer to [13], [23], and the remarkable series of papers [10], [28], [29], [11].…”

Section: Introductionmentioning

confidence: 99%

Freezing Solutions of Equivariant Evolution Equations

Beyn¹,

Thümmler²

2004

SIAM J. Appl. Dyn. Syst.

155

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Abstract. In this paper we develop numerical methods for integrating general evolution equations ut = F (u), u(0) = u0, where F is defined on a dense subspace of some Banach space (generally infinitedimensional) and is equivariant with respect to the action of a finite-dimensional (not necessarily compact) Lie group. Such equations typically arise from autonomous PDEs on unbounded domains that are invariant under the action of the Euclidean group or one of its subgroups. In our approach we write the solution u(t) as a composition of the action of a time-dependent group element with a "frozen solution" in the given Banach space. We keep the frozen solution as constant as possible by introducing a set of algebraic constraints (phase conditions), the number of which is given by the dimension of the Lie group. The resulting PDAE (partial differential algebraic equation) is then solved by combining classical numerical methods, such as restriction to a bounded domain with asymptotic boundary conditions, half-explicit Euler methods in time, and finite differences in space.We provide applications to reaction-diffusion systems that have traveling wave or spiral solutions in one and two space dimensions.

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Landau Damping to Partially Locked States in the Kuramoto Model

Dietert

Fernandez

Gérard-Varet

2018

Comm Pure Appl Math

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In the Kuramoto model of globally coupled oscillators, partially locked states (PLS) are stationary solutions that capture the emergence of partial synchrony when the interaction strength increases. While PLS have long been considered, existing results on their stability are limited to neutral stability of the linearized dynamics in strong topology or to specific invariant subspaces (obtained via the so‐called Ott‐Antonsen (OA) ansatz) with specific frequency distributions for the oscillators. In the mean‐field limit, the Kuramoto model shows various ingredients of the Landau damping mechanism in the Vlasov equation. This analogy has been a source of inspiration for stability proofs of regular Kuramoto equilibria. In addition, the major mathematical issue with PLS asymptotic stability is that these states consist of heterogeneous and singular measures. Here we establish an explicit criterion for their spectral stability and prove their local asymptotic stability in weak topology for a large class of analytic frequency marginals. The proof strongly relies on a suitable functional space that contains (Fourier transforms of) singular measures, and for which the linearized dynamics is well under control. For illustration, the stability criterion is evaluated in some standard examples. We confirm in particular that no loss of generality results in assuming the OA ansatz. To the best of our knowledge, our result provides the first proof of Landau damping to heterogeneous and irregular equilibria in the absence of dissipation. © 2018 Wiley Periodicals, Inc.

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Methods in Equivariant Bifurcations and Dynamical Systems

Abstract: This book is printed on acid-free paper.

Cited by 218 publications

References 0 publications

Symmetry breaking of two-dimensional time-periodic wakes

Symmetry breaking of two-dimensional time-periodic wakes

Freezing Solutions of Equivariant Evolution Equations

Landau Damping to Partially Locked States in the Kuramoto Model

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