Equivariant bifurcation theory and symmetry breaking

Abstract We examine micromagnetic pattern formation in chiral magnets, driven by the competition of Heisenberg exchange, Dzyaloshinskii–Moriya interaction, easy-plane anisotropy and thermodynamic Landau potentials. Based on equivariant bifurcation theory, we prove existence of lattice solutions branching off the zero magnetization state and investigate their stability. We observe in particular the stabilization of quadratic vortex–antivortex lattice configurations and instability of hexagonal skyrmion lattice configurations, and we illustrate our findings by numerical studies.

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Equivariant Bifurcation Theory: Dynamics

Dynamics and Symmetry - ICP Advanced Texts in Mathematics ◽

10.1142/9781860948541_0005 ◽

2007 ◽

pp. 137-200

Keyword(s):

Bifurcation Theory ◽

Equivariant Bifurcation ◽

Equivariant Bifurcation Theory

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On the Stability of Equivariant Bifurcation Problems and Their Unfoldings

Canadian Mathematical Bulletin ◽

10.4153/cmb-1992-034-x ◽

1992 ◽

Vol 35 (2) ◽

pp. 237-246 ◽

Cited By ~ 3

Author(s):

Ali Lari-Lavassani ◽

Yung-Chen Lu

Keyword(s):

Bifurcation Theory ◽

Local Level ◽

Equivariant Bifurcation ◽

Bifurcation Problems ◽

The Stability ◽

Equivariant Version

AbstractIn their book Singularities and Groups in Bifurcation Theory M. Golubitsky, I. Stewart and D. Schaeffer have introduced an equivariant version of Martinet's notion of V (for variety)-equivalence with parameter. In this paper we give a unified proof that, in this context, infinitesimal stability is equivalent to stability at the local level of germs and that stability in the unfolding category is equivalent to versality.

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Generic Spontaneous Symmetry Breaking in SU(n) — Equivariant Bifurcation Problems

The Physics of Structure Formation - Springer Series in Synergetics ◽

10.1007/978-3-642-73001-6_31 ◽

1987 ◽

pp. 394-401

Author(s):

C. Geiger ◽

W. Güttinger

Keyword(s):

Symmetry Breaking ◽

Spontaneous Symmetry Breaking ◽

Equivariant Bifurcation ◽

Bifurcation Problems

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Huygens’ clocks revisited

Royal Society Open Science ◽

10.1098/rsos.170777 ◽

2017 ◽

Vol 4 (9) ◽

pp. 170777 ◽

Cited By ~ 7

Author(s):

Allan R. Willms ◽

Petko M. Kitanov ◽

William F. Langford

Keyword(s):

Normal Form ◽

Physical System ◽

Bifurcation Theory ◽

Phase Synchronization ◽

Unified Framework ◽

System Parameters ◽

Experimental Systems ◽

Weakly Coupled ◽

Equivariant Bifurcation Theory ◽

Phase Motion

In 1665, Huygens observed that two identical pendulum clocks, weakly coupled through a heavy beam, soon synchronized with the same period and amplitude but with the two pendula swinging in opposite directions. This behaviour is now called anti-phase synchronization. This paper presents an analysis of the behaviour of a large class of coupled identical oscillators, including Huygens' clocks, using methods of equivariant bifurcation theory. The equivariant normal form for such systems is developed and the possible solutions are characterized. The transformation of the physical system parameters to the normal form parameters is given explicitly and applied to the physical values appropriate for Huygens' clocks, and to those of more recent studies. It is shown that Huygens' physical system could only exhibit anti-phase motion, explaining why Huygens observed exclusively this. By contrast, some more recent researchers have observed in-phase or other more complicated motion in their own experimental systems. Here, it is explained which physical characteristics of these systems allow for the existence of these other types of stable solutions. The present analysis not only accounts for these previously observed solutions in a unified framework, but also introduces behaviour not classified by other authors, such as a synchronized toroidal breather and a chaotic toroidal breather.

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