Bifurcation to Rotating Waves in Equations with $O(2)$-Symmetry

Aston, Philip J.; Spence, A.; Wu, Wei

doi:10.1137/0152045

Cited by 16 publications

(15 citation statements)

References 21 publications

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“…The first are steady-state solutions satisfying h τ = 0, and the second are travelling waves satisfying h τ = ±ch X with an a priori unknown wave speed c; this is an additional degree of freedom that needs to be determined along with the shape of the film profile. A phase condition like the one described by Aston, Spence & Wu (1992) is used to isolate points on the group orbit of solutions resulting from the translational and reflectional symmetry of this problem (due to the periodic boundary conditions). 6.1.…”

Section: Numerical Solutions Of the Long-wave Equationmentioning

confidence: 99%

Nonlinear dynamics in horizontal film boiling

2000

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This paper uses thin-film asymptotics to show how a thin vapour layer can support a liquid which is heated from below and cooled from above, a process known as horizontal film boiling. This approach leads to a single, strongly-nonlinear evolution equation which incorporates buoyancy, capillary and evaporative effects. The stability of the vapour layer is analysed using a variety of methods for both saturated and subcooled film boiling. In subcooled film boiling, there is a stationary solution, a constant-thickness vapour film, which is determined by a simple heat-conduction balance. This is Rayleigh-Taylor unstable because the heavier liquid is above the vapour, but the instability is completely suppressed for sufficient subcooling. A bifurcation analysis determines a supercritical branch of stable, spatially-periodic solutions when the basic state is no longer stable. Numerical branch tracing extends this into the strongly-nonlinear regime, revealing a hysteresis loop and a secondary bifurcation to a branch of travelling waves which are stable under certain conditions. There are no stationary solutions in saturated film boiling, but the initial development of vapour bubbles is determined by directly solving the time-dependent evolution equation. This yields important information about the transient heat transfer during bubble development.

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Section: Numerical Solutions Of the Long-wave Equationmentioning

confidence: 99%

Nonlinear dynamics in horizontal film boiling

2000

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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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“…Before calculations, we recall some results from [15]. (27) has four static solution branches bifurcated from ðu; aÞ ¼ ð0; 2Þ and from ðu; aÞ ¼ ð0; 8Þ, which corresponds to the bifurcation solution branches of (28) for c 1 ¼ c 2 ¼ 0, denoted by C is suitable for small a value by numerical calculations later on.…”

Section: Numerical Examplementioning

confidence: 99%

“…And it also serves as a good example of bifurcation and chaos [13][14][15]17], etc. In [15], Aston et al ever considered the bifurcation to rotating waves in equation with Oð2Þ-symmetry, i.e., the spatial dimension is only one, where the basic method was exemplified for the K-S equation in one spatial dimension. In [17], steady-state bifurcations of the two-dimensional K-S equation were analyzed in details.…”

Section: Numerical Examplementioning

confidence: 99%

Symmetry-breaking bifurcation in O(2)×O(2)-symmetric nonlinear large problems and its application to the Kuramoto–Sivashinsky equation in two spatial dimensions

Yang²

2004

Chaos, Solitons & Fractals

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Bifurcation to Rotating Waves in Equations with $O(2)$-Symmetry

Cited by 16 publications

References 21 publications

Nonlinear dynamics in horizontal film boiling

Nonlinear dynamics in horizontal film boiling

Bifurcations of periodic orbits with spatio-temporal symmetries

Symmetry-breaking bifurcation in O(2)×O(2)-symmetric nonlinear large problems and its application to the Kuramoto–Sivashinsky equation in two spatial dimensions

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