P. Coullet

A simple global characterization for normal forms of singular vector fields

We derive a new global characterization of the normal forms of amplitude equations describing the dynamics of competing order parameters in degenerate bifurcation problems. Using an appropriate scalar product in the space of homogeneous vector polynomials, we show that the resonant terms commute with the group generated by the adjoint of the original critical linear operator. This leads to a very efficient constructive method to compute both the nonlinear coefficients and the unfolding of the normal form. Explicit examples, and results obtained when there are additional symmetries, are also presented.

Strong resonances of spatially distributed oscillators: a laboratory to study patterns and defects

Abstract This paper describes phenomena which arise in a continuous system of coupled oscillators subject to a strongly resonant forcing. We show that the temporal forcing induces a new type of instability, leading to the formation of spatial patterns such as stripes, hexagons and spatio-temporal intermittent states, reminiscent of chemical and magnetical systems. We also report the existence of localized states, where the instability is confined in the core of a defect, leading to an increased mobility and diffusive behaviour of the defect.

Normal form of a Hopf bifurcation with noise

Abstract It is proved that the stochastic normal form of a general system undergoing a Hopf bifurcation contains through stochastic resonance new types of terms which did not appear in the deterministic normal form.

A form of turbulence associated with defects

Abstract We show by means of numerical simulations of complex Ginzburg-Landau equations that phase instability leads to the spontaneous nucleation of topological defects, which disorganize the system.

Horizon effects with surface waves on moving water

Surface waves on a stationary flow of water are considered in a linear model that includes the surface tension of the fluid. The resulting gravity-capillary waves experience a rich array of horizon effects when propagating against the flow. In some cases, three horizons (points where the group velocity of the wave reverses) exist for waves with a single laboratory frequency. Some of these effects are familiar in fluid mechanics under the name of wave blocking, but other aspects, in particular waves with negative co-moving frequency and the Hawking effect, were overlooked until surface waves were investigated as examples of analogue gravity (Schutzhold R and Unruh W G 2002 Phys. Rev. D 66 044019). A comprehensive presentation of the various horizon effects for gravity-capillary waves is given, with emphasis on the deep water/ short wavelength case kh1, where many analytical results can be derived. A similarity of the state space of the waves to that of a thermodynamic system is pointed out.

Resonance and phase solitons in spatially-forced thermal convection

Abstract The competition between external and internal length scales is studied theoretically on a two-dimensional amplitude evolution model of a fluid layer heated from below. A spatially-periodic excitation is applied to the stress-free, isothermal boundaries at a wavenumber close to the critical wavenumber for the onset of the instability. Near resonance, the large-scale phase dynamics give rise to stationary phase solitons at high Prandtl numbers and propagating sine-Gordon solitons at low Prandtl numbers. The propagating solitons are associated with the external breaking of the Galilean invariance of the problem.

Sources and sinks of wave patterns

Abstract We analyze in this paper the behavior of sources and sinks of traveling waves in one-dimensional systems. It is shown that the presence of these defects induces a selection of the wavenumber in the bulk of the pattern itself. This selection mechanism can drive the pattern into the Benjamin-Fier-Eckhaus unstable regime generating spatio-temporal chaos with space-time dislocations. In the convectively unstable regime, the source of group perturbation becomes a noise amplifier.

Spatial Forcing of 2D Wave Patterns

The effect of two-dimensional spatial modulations on spatio-temporal Hopf bifurcations is investigated. In particular, in the case of a spatial forcing corresponding to triangular or hexagonal planforms, a strong resonance between triplets of travelling waves may occur, provided that the wavelength of the forcing is nearly equal to one-third of the critical one. In this case a periodic spatio-temporal pattern of triangular symmetry may be stabilized in regions of the parameter space where it is otherwise unstable. Such a phenomenon may present the onset of turbulence in forced hexagonal vortex lattices.

Dynamics of Bloch Walls in a Rotating Magnetic Field: a Model

We both analytically and numerically show the existence of a drift of Bloch walls when submitted to a uniform parallel-to-the-wall-plane rotating magnetic field. The drift velocity changes sign with Bloch wall handedness and is proportional to the amplitude square of the magnetic field, when the latter is small.

Spatio-temporal intermittency in a 1D convective pattern: theoretical model and experiments

Abstract We describe the occurence of spatio-temporal intermittency in a one-dimensional convective system that first shows time-dependent patterns. We recall experimental results and propose a model based on the normal form description of a secondary Hopf bifurcation of a stationary periodic structure. Numerical simulations of this model show spatio-temporal intermittent behaviors, which we characterize briefly and compare to those given by the experiment.