Symmetry Breaking Hopf Bifurcations in Equations with O(2) Symmetry with Application to the Kuramoto–Sivashinsky Equation

Amdjadi, Faridon; Aston, Philip J.; Plecháč, Petr

doi:10.1006/jcph.1996.5599

Cited by 8 publications

(6 citation statements)

References 31 publications

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“…He shows that the resulting bifurcation problem is Σ-equivariant, where Σ is the isotropy subgroup of symmetries of the relative equilibrium, and, building on work of Field (1980), provides a group theoretic method for determining whether or not the bifurcating solutions drift. Aston et al (1992), and Amdjadi et al (1997) develop a technique for numerically investigating bifurcations of relative equilibria in O(2)-equivariant partial differential equations, and apply their method to the Kuramoto-Sivashinsky equation. Their approach isolates one solution on a group orbit, while still keeping track of any constant drift along the group orbit.…”

Section: Introductionmentioning

confidence: 99%

Bifurcations of periodic orbits with spatio-temporal symmetries

Rucklidge

Silber

1998

Nonlinearity

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Abstract. Motivated by recent analytical and numerical work on two-and three-dimensional convection with imposed spatial periodicity, we analyse three examples of bifurcations from a continuous group orbit of spatio-temporally symmetric periodic solutions of partial differential equations. Our approach is based on centre manifold reduction for maps, and is in the spirit of earlier work by Iooss (1986) on bifurcations of group orbits of spatially symmetric equilibria. Two examples, two-dimensional pulsating waves (PW) and threedimensional alternating pulsating waves (APW), have discrete spatio-temporal symmetries characterized by the cyclic groups Z n , n = 2 (PW) and n = 4 (APW). These symmetries force the Poincaré return map M to be the n th iterate of a map G: M = G n . The group orbits of PW and APW are generated by translations in the horizontal directions and correspond to a circle and a two-torus, respectively. An instability of pulsating waves can lead to solutions that drift along the group orbit, while bifurcations with Floquet multiplier +1 of alternating pulsating waves do not lead to drifting solutions. The third example we consider, alternating rolls, has the spatio-temporal symmetry of alternating pulsating waves as well as being invariant under reflections in two vertical planes. When the bifurcation breaks these reflections, the map G has a "two-symmetry," as analysed by Lamb (1996). This leads to a doubling of the marginal Floquet multiplier and the possibility of bifurcation to two distinct types of drifting solutions.

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Section: Introductionmentioning

confidence: 99%

Bifurcations of periodic orbits with spatio-temporal symmetries

Rucklidge

Silber

1998

Nonlinearity

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“…Spatial period-multiplying instabilities have arisen in a variety of contexts, in both one [17][18][19][20] and two lateral directions [1,[6][7][8]16,[21][22][23]. Most of these situations involved relatively simple groups; part of the difficulty and interest here has been the size of the symmetry group, enlarged because of the number of translations broken by the new pattern.…”

Section: Discussionmentioning

confidence: 99%

“…The issue of spatial period-multiplying instabilities is an interesting one that has arisen in a variety of experimental and theoretical contexts. Period-multiplying bifurcations in one lateral direction have arisen in convection problems [16], magnetoconvection [17], Taylor-Couette experiments [18] and in numerical solutions of the Kuramoto-Sivashinsky equations [19,20]. Much less is known about spatial period-multiplying bifurcations in two directions.…”

Section: Introductionmentioning

confidence: 99%

Spatial period-multiplying instabilities of hexagonal Faraday waves

Tse

Rucklidge

Hoyle

et al. 2000

Physica D: Nonlinear Phenomena

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A recent Faraday wave experiment with two-frequency forcing reports two types of `superlattice' patterns that display periodic spatial structures having two separate scales [1]. These patterns both arise as secondary states once the primary hexagonal pattern becomes unstable. In one of these patterns (so-called `superlattice-II') the original hexagonal symmetry is broken in a subharmonic instability to form a striped pattern with a spatial scale increased by a factor of 2sqrt{3} from the original scale of the hexagons. In contrast, the time-averaged pattern is periodic on a hexagonal lattice with an intermediate spatial scale (sqrt{3} larger than the original scale) and apparently has 60 degree rotation symmetry. We present a symmetry-based approach to the analysis of this bifurcation. Taking as our starting point only the observed instantaneous symmetry of the superlattice-II pattern presented in [1] and the subharmonic nature of the secondary instability, we show (a) that the superlattice-II pattern can bifurcate stably from standing hexagons; (b) that the pattern has a spatio-temporal symmetry not reported in [1]; and (c) that this spatio-temporal symmetry accounts for the intermediate spatial scale and hexagonal periodicity of the time-averaged pattern, but not for the apparent 60 degree rotation symmetry. The approach is based on general techniques that are readily applied to other secondary instabilities of symmetric patterns, and does not rely on the primary pattern having small amplitude.Comment: 24 pages, 4 figures, LaTex2e (elsart.cls), to appear in Physica D. Postscript version (with bigger figures) available on http://www.damtp.cam.ac.uk/user/alastair/papers/TRHS_faraday_abs.shtm

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“…The K-S equation models pattern formation in different physical contents [Benny, 1966;Kuramoto, 1978;Kuramoto & Tsuzuki, 1975, 1976Michelson et al, 1977;Rost & Krug, 1995;Sivashinsky, 1977Sivashinsky, , 1980, and they have been studied in recent years, within the contexts of inertial manifolds and finitedimensional attractors as well as in numerical simulations of dynamical behavior [Amdjadi et al, 1997;Foias et al, 1988;Li & Yang, 1998;Nicolaenko et al, 1985;Temam, 1988].…”

Section: Introductionmentioning

confidence: 99%

Bifurcation From an Equilibrium of the Steady State Kuramoto–sivashinsky Equation in Two Spatial Dimensions

Chen

2002

Int. J. Bifurcation Chaos

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The paper deals with the steady state bifurcations of the Kuramoto-Sivashinsky (K-S) equation in two spatial dimensions with zero mean and periodic boundary value conditions. Applying the perturbation method, asymptotic expressions of the steady state solution branches that have bifurcated from the equilibrium are obtained. Furthermore, stability of the bifurcated solution branches is discussed. Int. J. Bifurcation Chaos 2002.12:103-114. Downloaded from www.worldscientific.com by MEMORIAL UNIVERSITY OF NEWFOUNDLAND on 02/02/15. For personal use only.

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Symmetry Breaking Hopf Bifurcations in Equations with O(2) Symmetry with Application to the Kuramoto–Sivashinsky Equation

Cited by 8 publications

References 31 publications

Bifurcations of periodic orbits with spatio-temporal symmetries

Bifurcations of periodic orbits with spatio-temporal symmetries

Spatial period-multiplying instabilities of hexagonal Faraday waves

Bifurcation From an Equilibrium of the Steady State Kuramoto–sivashinsky Equation in Two Spatial Dimensions

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