RESUMO
Mobility properties of spatially localized structures arising from chaotic but deterministic forcing of the bistable Swift-Hohenberg equation are studied and compared with the corresponding results when the chaotic forcing is replaced by white noise. Short structures are shown to possess greater mobility, resulting in larger root-mean-square speeds but shorter displacements than longer structures. Averaged over realizations, the displacement of the structure is ballistic at short times but diffusive at larger times. Similar results hold in two spatial dimensions. The effects of chaotic forcing on the stability of these structures is also quantified. Shorter structures are found to be more fragile than longer ones, and their stability region can be displaced outside the pinning region for constant forcing. Outside the stability region the deterministic fluctuations lead either to the destruction of the structure or to its gradual growth.
RESUMO
Driven dissipative many-body systems are described by differential equations for macroscopic variables which include fluctuations that account for ignored microscopic variables. Here, we investigate the effect of deterministic fluctuations, drawn from a system in a state of phase turbulence, on front dynamics. We show that despite these fluctuations a front may remain pinned, in contrast to fronts in systems with Gaussian white noise fluctuations, and explore the pinning-depinning transition. In the deterministic case, this transition is found to be robust but its location in parameter space is complex, generating a fractal-like structure. We describe this transition by deriving an equation for the front position, which takes the form of an overdamped system with a ratchet potential and chaotic forcing; this equation can, in turn, be transformed into a linear parametrically driven oscillator with a chaotically oscillating frequency. The resulting description provides an unambiguous characterization of the pinning-depinning transition in parameter space. A similar calculation for noise-driven front propagation shows that the pinning-depinning transition is washed out.
RESUMO
The properties of a hinged floating elastic sheet of finite length under compression are considered. Numerical continuation is used to compute spatially localized buckled states with many spatially localized folds. Both symmetric and antisymmetric states are computed and the corresponding bifurcation diagrams determined. Weakly nonlinear analysis is used to analyze the transition from periodic wrinkles to singlefold and multifold states and to compute their energy. States with the same number of folds have energies that barely differ from each other and the energy gap decreases exponentially as localization increases. The stability of the different competing states is studied and the multifold solutions are all found to be unstable. However, the decay time into solutions with fewer folds can be so slow that multifolds may appear to be stable.