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We study the modifications induced in the behavior of the quorum percolation model on neural networks with Gaussian in-degree by taking into account an uncorrelated Gaussian thresholds variability. We derive a mean-field approach and show its relevance by carrying out explicit Monte Carlo simulations. It turns out that such a disorder shifts the position of the percolation transition, impacts the size of the giant cluster, and can even destroy the transition. Moreover, we highlight the occurrence of disorder independent fixed points above the quorum critical value. The mean-field approach enables us to interpret these effects in terms of activation probability. A finite-size analysis enables us to show that the order parameter is weakly self-averaging with an exponent independent on the thresholds disorder. Last, we show that the effects of the thresholds and connectivity disorders cannot be easily discriminated from the measured averaged physical quantities.

We investigate the selection of structures generated by diffusive instabilities in asymmetric bistable systems. It is shown that the nonlinear coupling between the zero mode and the different spatial modes emanating from the two homogeneous states can give rise to self-parametric instabilities favoring the occurrence of stable resonant rhombic and superlattice patterns.

Attractive Bose-Einstein condensates are investigated with numerical continuation methods capturing stationary solutions of the Gross-Pitaevskii equation. The branches of stable (elliptic) and unstable (hyperbolic) solutions are found to meet at a critical particle number through a generic Hamiltonian saddle node bifurcation. The condensate decay rates corresponding to macroscopic quantum tunneling, two and three body inelastic collisions, and thermally induced collapse are computed from the exact numerical solutions. These rates show experimentally significant differences with previously published rates. Universal scaling laws stemming from the bifurcation are derived. [S0031-9007(99)08550-6]

The competition between patterns of different symmetries is studied for the generalized Swift-Hohenberg model in the presence of a large quadratic coupling in one dimension (1D) and 2D. It is shown that hexagons with different phase relations may coexist for some values of the control parameter. Numerical experiments exhibiting the effects of linear spatial ramps of the control parameter on this selection are presented. The analogy with recent patterns obtained experimentally in open chemical reactors is also discussed.

Physico-chemical systems driven away from thermal equilibrium usually undergo various types of instabilities leading to the formation of spatio-temporal patterns on macroscopic time and space scales. In two-dimensional geometries, patterns of different symmetries may be simultaneously stable. We study here the phase stability of hexagonal planforms in the framework of amplitude equations and reduced dynamical models close to the instability points.

The imploding dynamics of {sup 85}Rb condensates is obtained in terms of the nonadiabatic dynamical characteristics of the homoclinic bifurcation giving rise to the collapse of the cloud. The agreement with the recent experimental observation of the imploding dynamics is excellent.

Non-Isotropic Attractive Bose-Einstein condensates are investigated with Newton and inverse Arnoldi methods. The stationary solutions of the Gross-Pitaevskii equation and their linear stability are computed. Bifurcation diagrams are calculated and used to find the condensate decay rates corresponding to macroscopic quantum tunneling, two-three body inelastic collisions and thermally induced collapse. Isotropic and non-isotropic condensates are compared. The effect of anisotropy on the bifurcation diagram and the decay rates is discussed. Spontaneous isotropization of the condensates is found to occur. The influence of isotropization on the decay rates is characterized near the critical point., Comment: revtex4, 11 figures, 2 tables. Submitted to Phys. Rev. A

Pattern selection in reaction-diffusion systems exhibiting bistability of homogeneous steady states is discussed. In agreement with recent experimental results, we obtain new bifurcation diagrams involving large amplitude structures that arise from the coupling of the spatial critical modes with a quasi-neutral homogeneous mode.

The stationary pattern selection problem in one and two dimensions is discussed on the generalized Swift-Hohenberg model, in the presence of a large quadratic coupling term.