We study analytically and numerically the noise-induced transition between an absorbing and an oscillatory state in a Duffing oscillator subject to multiplicative, Gaussian white noise. We show in a non-perturbative manner that a stochastic bifurcation occurs when the Lyapunov exponent of the linearised system becomes positive. We deduce from a simple formula for the Lyapunov exponent the phase diagram of the stochastic Duffing oscillator. The behaviour of physical observables, such as the oscillator's mean energy, is studied both close to and far from the bifurcation.

https://hal.archives-ouvertes.fr/hal-00014829/document

Stability analysis of a noise-induced Hopf bifurcation

The purpose of this review is to present a status report on the electrohydrodynamic convection in nematic liquid crystals. The considerable progress achieved in the past two years is emphasized.

Electrohydrodynamic Convection in Nematics

Boussinesq convection in rotating systems attracts a sustained attention of the fluid dynamics community, because it has intricate non-linear dynamics (Cross & Hohenberg 1993) and plays an important role in geophysical and astrophysical applications, such as the motion of the liquid outer core of Earth, the Red Spot in Jupiter, the giant cells in the Sun etc. (Alridge et al. 1990). A fundamental distinction between the real geo- and astrophysical problems and the idealized laboratory studies is that natural systems are inhomogeneous (Alridge et al. 1990). Heterogeneities modulate the flow and influence significantly the dynamics of convective patterns (Alridge et al. 1990; Hide 1971). The effect of modulations on pattern formation and transition to turbulence in Boussinesq convection is far from being completely understood (Cross & Hohenberg 1993; Aranson & Kramer 2002). It is generally accepted that in the liquid outer core of the Earth the transport of the angular momentum and internal heat occurs via thermal Rossby waves (Zhang et al. 2001; Kuang & Bloxham 1999). These waves been visualized in laboratory experiments in rotating liquid-filled spheres and concentric spherical shells (Zhang et al. 2001; Kuang & Bloxham 1999). The basic dynamical features of Rossby waves have been reproduced in a cylindrical annulus, a system much simpler than the spherical ones (Busse & Or 1986; Or & Busse 1987). For convection in a cylindrical annulus, the fluid motion is two-dimensional, and gravity is replaced by a centrifugal force, (Busse & Or 1986; Or & Busse 1987). Hide (1971) has suggested that the momentum and heat transport in the core might be influenced significantly by so-called bumps, which are heterogeneities on the mantle-core boundary. To model the effect of surface topography on the transport of momentum and energy in the liquid outer core of the Earth, Bell & Soward (1996), Herrmann & Busse (1998) and Westerburg & Busse (2001) have studied the nonlinear dynamics of thermal Rossby waves in a cylindrical annulus with azimuthally modulated height.

/pdf/the-effect-of-surface-topography-on-the-nonlinear-dynamics-h0504nj7y8.pdf

The Effect of Surface Topography on the Nonlinear Dynamics of Rossby Waves

We investigate analytically and numerically the effect of inhomogeneities on the nonequilibrium dynamics of wave patterns in the framework of a complex Ginzburg-Landau equation (CGLE) with parametric, nonresonant forcing periodic in space and time. It is found that the forcing results in occurrence of traveling waves with different dispersion properties. In the limiting case of forcing with very large wavelength, the waves have essentially anharmonic spatial structure. We consider the influence of modulations on the development of an intermittent chaos and show that the parametric forcing may completely suppress the appearance of chaotic patterns. The relations between this and other pattern-forming systems are discussed. The results obtained are applied to describe the dynamics of thermal Rossby waves influenced by surface topography.

Influence of parametric forcing on the nonequilibrium dynamics of wave patterns.

Multiplicative noise in spatially extended systems produces different effects depending upon whether the noise is spatially homogeneous or spatially varying. Whereas in previous work a stochastic distribution was treated, here we consider the spatially periodic case, which is more amenable to an experimental approach, in particular in the electrically driven instabilities of nematic liquid crystals. We shall principally be interested in the threshold for the onset of symmetry breaking instabilities controlled by bifurcations in several stochastic partial differential equations. For the Ginzburg-Landau and Swift-Hohenberg equations we calculate the behavior of the threshold for all moments to second order in the noise strength, allowing one to reconstruct the full probability distribution. For a system of two coupled equations which mimics electroconvection in nematic liquid crystals (the ``one-dimensional model''), we calculate the first two moments up to second order and estimate the threshold for convection. The general conclusion of our work is that spatially periodic noise induces a reduction in the threshold similar to the stochastically distributed case. We propose that this reduction be independent of the periodicity of the noise to first order in the noise strength, the dependence on period appearing only at second order. This is in contrast to spatially homogeneous noise where threshold shifts may be entirely absent.

Linear stability analysis of bifurcations with a spatially periodic, fluctuating control parameter

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Linear stability analysis of bifurcations with a spatially periodic, fluctuating control parameter