Cellular self-propulsion of two-dimensional dissipative structuresand spatial-period tripling Hopf bifurcation

Abstract

Instabilities arising in two-dimensional patterns are analyzed; in particular, we report on two generic instabilities of an ordered dissipative stationary structure. The first of these corresponds to several-point symmetry breakings, such as mirror and rotation symmetry, which cause the patterns to travel and may give rise to spirals or labyrinths. The second type manifests itself as a collective out-of-phase temporal oscillation, resulting in spatial-period tripling. We present both a numerical and an analytical analysis of the new emerging patterns.

A time-adaptive finite volume method for the Cahn–Hilliard and Kuramoto–Sivashinsky equations

“…The analysis of the Kuramoto-Sivashinsky equations has been mostly focused on the characterization of its dynamical properties, rather than on practical numerical simulation aspects, and thus most authors have used finite differences and spectral methods [40,19,42,18,50,44,1,30]. More recently, a discontinuous Galerkin method has been proposed for the one-dimensional KS equation [52].…”

“…Originally arisen in the context of phase separation after quenching of binary alloys [10], chemical phase turbulence [34] and flame front instabilities [43], the rich dynamical behaviour of the solutions of the Cahn-Hilliard and Kuramoto-Sivashinsky equations has inspired their use as a more general framework to model systems displaying complex spatiotemporal features, such as phase-ordering and coarsening [48,39,47], and pattern formation [42,18]. Under certain conditions, some of these equations may exhibit a chaotic behaviour, or weak turbulence [40,19,50,44].…”

Nonlocal interface equations in crystal growth

“…One phenomenon may serve as a case in point: period-tripling oscillations of hexagonal patterns were first discovered in an interface equation describing high-speed directional solidification [1] and in the damped Kuramoto-Sivashinsky equation [2] and only recently shown, by way of a phase-field simulation of the full dynamics, to be stable patterns in directional solidification at low speeds [3].…”

Hexagons and triangles in the Rayleigh–Bénard problem: quintic-order equations on a hexagonal lattice

“…A number in parentheses denotes the multiplicity of the foregoing eigenvalue. 7 On the hexagonal lattice, the 1 : ffiffi ffi 3 p resonance at the quadratic order was investigated by Daumont et al (1997) for the Swift-Hohenberg equation, whereas the 1 : 2 resonance at the quadratic order was examined by Fujimura (2008) for a two-layered RB problem.…”