A simple global characterization for normal forms of singular vector fields

Elphick, Christian; Tirapegui, E.; Brachet, Marc; Coullet, P.; Iooss, Gérard

doi:10.1016/0167-2789(87)90049-2

Cited by 381 publications

(363 citation statements)

References 7 publications

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“…The nonlinear resonant components of the normal form vector field must commute with (15) for 7 E R [26]. An arbitrary monomial term ziziz2Z2 in the normal form must therefore obey the constraint…”

Section: Reduction To a Finite-dimensional Systemmentioning

confidence: 99%

Oscillatory bifurcation with broken translation symmetry

Landsberg

Knobloch

1996

Phys. Rev. E

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The effect of distant end walls on the bifurcation to traveling waves is considered. Previous approaches have treated the problem by assuming that.it is a weak perturbation of the translation invariant problem. When the problem is formulated instead in a finite box of length L and the limit L --+ 00 is taken, one obtains amplitude equations that differ from the usual Ginzburg-Landau description by the presence of an additional nonlinear term. This formulation leads to a description in terms of the amplitudes of the primary box modes, which are odd and even parity standing waves. For large L, the equations that result take the form of a Hopf bifurcation with approximate D4 symmetry. These equations are able to describe, qualitatively, not only traveling and "blinking" states, but also asymmetrical blinking states and "repeated transients," all of which have been observed in binary fluid convection experiments. PACS number(s): 47.20. Bp, 47.20.Ky, 03.40.Kf

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Section: Reduction To a Finite-dimensional Systemmentioning

confidence: 99%

Oscillatory bifurcation with broken translation symmetry

Landsberg

Knobloch

1996

Phys. Rev. E

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“…According to the above results, the inner-field solution takes the value (w) inner (x = 0) = −w 0 (x = 0) = −1, see again (13), at the border between the two zones. The matching condition thus implies C = 16, and therefore we find…”

Section: A Inner Zone Analysismentioning

confidence: 99%

Accumulation of embedded solitons in systems with quadratic nonlinearity

Malomed

Wagenknecht

Champneys

et al. 2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

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Previous numerical studies have revealed the existence of embedded solitons (ESs) in a class of multi-wave systems with quadratic nonlinearity, families of which seem to emerge from a critical point in the parameter space, where the zero solution has a fourfold zero eigenvalue. In this paper, the existence of such solutions is studied in a three-wave model.An appropriate rescaling casts the system in a normal form, which is universal for models supporting ESs through quadratic nonlinearities. The normal-form system contains a single irreducible parameter ε, and is tantamount to the basic model of type-I second-harmonic generation. An analytical approximation of WKB type yields an asymptotic formula for the distribution of discrete values of ε at which the ESs exist. Comparison with numerical results shows that the asymptotic formula yields an exact value of the scaling index, −6/5, and a fairly good approximation for the numerical factor in front of the scaling term. In this work, we resolve these issues, by developing an asymptotic analysis and verifying the predictions against numerical results. As a result, we derive a universal asymptotic approximation (normal form) for models with quadratic nonlinearity that support embedded solitons. The normal form amounts to a system of two second-order equations, which is well known as a basic model for ordinary solitons in second-harmonic-generating systems. There was a rather common belief that all soliton solutions had 3 Accumulation of embedded solitons been found in this much-studied system. Nevertheless, in this work we are able to predict the existence of an infinite series of previously unknown embedded solitons in the model. This is done in an analytical form, by realizing that the embedded solitons feature a broad inner zone, the solution in which must be matched to exponentially decaying ones in outer zones. Further, a method (Bohr-Sommerfeld quantization rule) borrowed from quantum mechanics is applied to the solution in the inner zone. The most essential part of the eventual analytical result is an asymptotic distribution law for values of the single control parameter of the normal-form system at which the embedded solitons exist. Comparison with numerical results clearly shows that the analysis has produced an exact value of the respective scaling index.

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“…Suppose that f is Cequivariant. By analyzing the normal form reduction procedure, Elphick et al 39 proved that C-equivariant coordinate changes can be chosen so that each g k (y) is C Â Sequivariant.…”

Section: Poincar E-birkhoff Normal Formmentioning

confidence: 99%

Recent advances in symmetric and network dynamics

Golubitsky

Stewart

2015

Chaos: An Interdisciplinary Journal of Nonlinear Science

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We summarize some of the main results discovered over the past three decades concerning symmetric dynamical systems and networks of dynamical systems, with a focus on pattern formation. In both of these contexts, extra constraints on the dynamical system are imposed, and the generic phenomena can change. The main areas discussed are time-periodic states, mode interactions, and non-compact symmetry groups such as the Euclidean group. We consider both dynamics and bifurcations. We summarize applications of these ideas to pattern formation in a variety of physical and biological systems, and explain how the methods were motivated by transferring to new contexts Rene Thom's general viewpoint, one version of which became known as "catastrophe theory." We emphasize the role of symmetry-breaking in the creation of patterns. Topics include equivariant Hopf bifurcation, which gives conditions for a periodic state to bifurcate from an equilibrium, and the H/K theorem, which classifies the pairs of setwise and pointwise symmetries of periodic states in equivariant dynamics. We discuss mode interactions, which organize multiple bifurcations into a single degenerate bifurcation, and systems with non-compact symmetry groups, where new technical issues arise. We transfer many of the ideas to the context of networks of coupled dynamical systems, and interpret synchrony and phase relations in network dynamics as a type of pattern, in which space is discretized into finitely many nodes, while time remains continuous. We also describe a variety of applications including animal locomotion, Couette-Taylor flow, flames, the Belousov-Zhabotinskii reaction, binocular rivalry, and a nonlinear filter based on anomalous growth rates for the amplitude of periodic oscillations in a feed-forward network. Symmetry is a feature of many systems of interest in applied science. Mathematically, a symmetry is a transformation that preserves structure; for example, a square looks unchanged if it is rotated through any multiple of a right angle, or reflected in a diagonal or a line joining the midpoints of oppose edges. These symmetries also appear in mathematical models of real-world systems, and their effect is often extensive. The last thirty to forty years has seen considerable advances in the mathematical understanding of the effects of symmetry on dynamical systems-systems of ordinary differential equations. In general, symmetry leads to pattern formation, via a mechanism called symmetry-breaking. For example, if a system with circular symmetry has a time-periodic state, repeating the same behavior indefinitely, then typically this state is either a standing wave or a rotating wave. This observation applies to the movement of a flexible hosepipe as water passes through it: in the standing wave, the hosepipe moves to and fro like a pendulum; in the rotating wave, it goes round and round with its end describing a circle. This observation also applies to how flame fronts move on a circular burner. We survey some basic mathematical ideas that have ...

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A simple global characterization for normal forms of singular vector fields

Cited by 381 publications

References 7 publications

Oscillatory bifurcation with broken translation symmetry

Oscillatory bifurcation with broken translation symmetry

Accumulation of embedded solitons in systems with quadratic nonlinearity

Recent advances in symmetric and network dynamics

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