Pattern dynamics near inverse homoclinic bifurcation in fluids (original) (raw)

Localized States in Bistable Pattern-Forming Systems

Physical Review Letters, 2006

We present an unifying description close to a spatial bifurcation of localized states, appearing as large amplitude peaks nucleating over a pattern of lower amplitude. Localized states are pinned over a lattice spontaneously generated by the system itself. We show that the phenomenon is generic and requires only the coexistence of two spatially periodic states. At the onset of the spatial bifurcation, a forced amplitude equation is derived for the critical modes, which accounts for the appearance of localized peaks. During the last years emerging localized structures in dissipative systems have been observed in different fields, such as domains in magnetic materials [1], chiral bubbles in liquid crystals [2], current filaments in gas discharge experiments [3], spots in chemical reactions [4], localized 2D states in fluid surface waves [5], oscillons in granular media [6], isolated states in thermal convection [7], solitary waves in nonlinear optics [8], just to mention a few.

Pattern formation induced by nonequilibrium global alternation of dynamics

Physical Review E, 2002

We recently proposed a mechanism for pattern formation based on the alternation of two dynamics, neither of which exhibits patterns. Here we analyze the mechanism in detail, showing by means of numerical simulations and theoretical calculations how the nonequilibrium process of switching between dynamics, either randomly or periodically, may induce both stationary and oscillatory spatial structures. Our theoretical analysis by means of mode amplitude equations shows that all features of the model can be understood in terms of the nonlinear interactions of a small number of Fourier modes.

A Transition to Spatiotemporal Chaos under a Symmetry Breaking in a Homeotropic Nematic System

Journal of the Physical Society of Japan, 2008

This Letter reports a transition to spatiotemporal chaos (STC) in an electroconvective system of a homeotropic nematic liquid crystal under a symmetry breaking. The homeotropic system has a continuous symmetry which leads to the occurring of STC at a threshold for convection. When the symmetry is broken by an external field, a periodic pattern appears first and then STC appears at a higher transition point. The transition from the periodic pattern to STC is investigated by order parameters which were obtained from spectra of convective patterns. The results are compared with the recent theoretical researches for the damped Nikolaevskiy equation.

Global bifurcations close to symmetry

Journal of Mathematical Analysis and Applications, 2016

Heteroclinic cycles involving two saddle-foci, where the saddle-foci share both invariant manifolds, occur persistently in some symmetric differential equations on the 3-dimensional sphere. We analyse the dynamics around this type of cycle in the case when trajectories near the two equilibria turn in the same direction around a 1-dimensional connection-the saddlefoci have the same chirality. When part of the symmetry is broken, the 2-dimensional invariant manifolds intersect transversely creating a heteroclinic network of Bykov cycles. We show that the proximity of symmetry creates heteroclinic tangencies that coexist with hyperbolic dynamics. There are n-pulse heteroclinic tangencies-trajectories that follow the original cycle n times around before they arrive at the other node. Each n-pulse heteroclinic tangency is accumulated by a sequence of (n+1)-pulse ones. This coexists with the suspension of horseshoes defined on an infinite set of disjoint strips, where the first return map is hyperbolic. We also show how, as the system approaches full symmetry, the suspended horseshoes are destroyed, creating regions with infinitely many attracting periodic solutions.