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Journal Articles Physical Review E Year : 2008

Universal shape law of stochastic supercritical bifurcations: Theory and experiments


A universal analytical expression for the supercritical bifurcation shape of transverse one-dimensional 1D systems in the presence of additive noise is given. The stochastic Langevin equation of such systems is solved by using a Fokker-Planck equation, leading to the expression for the most probable amplitude of the critical mode. From this universal expression, the shape of the bifurcation, its location, and its evolution with the noise level are completely defined. Experimental results obtained for a 1D transverse Kerr-type slice subjected to optical feedback are in excellent agreement. In nature, most physical systems are subjected to fluctuations. For a long time, the effects of these fluctuations were either considered as a nuisance degradation of the signal-to-noise ratio or ignored because it was not known how to handle them. For three decades, a wealth of theoretical and experimental research has shown that fluctuations can have rather surprisingly constructive and counterintuitive effects in many physical systems and that they can be figured out with the help of different analysis tools. These situations occur when there are mechanisms of noise amplification or when noise interacts with nonlinearities or driving forces on the system. The most well-known examples in zero-dimensional systems are noise-induced transition 1 and sto-chastic resonance 2. More recently, examples of spatially extended systems are noise-induced phase transition, noise-induced patterns see 3 and references therein, noise-sustained structures in convective instability 4, stochastic spatiotemporal intermittency 5, noise-induced traveling waves 6, noise-induced ordering transition 7, and front propagation 8. Among these effects, a direct consequence of noise effects is the modification of the deterministic bifur-cation shapes where the critical points and the physical mechanisms are masked by fluctuations. It is important to remark that the critical points generically represent a change of balance between forces. Hence, the characterization of noisy bifurcations is a fundamental problem due to the ubiquitous nature of bifurcations. For instance, the supercritical bifurcations transform into smooth transitions between the two states and the subcritical bifurcations experience hyster-esis size modifications. In the absence of noise, the shape of a bifurcation and its characteristics are given by the analytical solution of the deterministic amplitude equation of the critical mode 9. On the other hand, in the presence of noise, no such analytical expression can be obtained from the sto-chastic amplitude equation. In this latter situation, the below and above bifurcation point regimes are usually treated separately , but without continuity between their respective solutions. For instance, in noisy spatially extended systems in which the systems are characterized by the appearance of pattern precursors below the bifurcation point and by established patterns that fluctuate above this point 10, the precursor amplitude 11, obtained from the linear study of the stochastic equation, diverges at the bifurcation point and does not connect to the " mean " amplitude of the fluctuating pattern, obtained from the deterministic equation. To our knowledge, no universal analytical expression of the critical mode amplitude, describing the complete transition from below to above the bifurcation point, exists for the supercritical bifurcations in the presence of noise. In this paper, we propose a universal description of the supercritical bifurcation shapes of one-dimensional 1D transverse systems either uniform or very slowly varying in space in the presence of noise that is also valid for the second-order bifurcations of temporal zero-dimensional systems. More precisely, we give a unified analytical expression for the most probable amplitude describing the super-critical bifurcations in the presence of noise, including the noise level and the deterministic bifurcation point location. The systems under study are described by stochastic partial differential equations SPDEs of the Langevin type 12 first order in time and with linear noise terms involving additive white noise. First, we reduce the SPDE to an ordinary differential equation ODE for the amplitude of the critical mode. Second, we solve the Langevin ODE describing the stochastic dynamics by using a Fokker-Planck equation for the probability density of the critical mode amplitude. Then, from the stationary distribution of this amplitude, we deduce the bifurcation shape by means of the most probable value of the pattern amplitude. Finally, the comparison with experimental results obtained in a Kerr-type slice subjected to 1D optical feedback is given and leads to an excellent agreement. Let us consider a 1D extended system that exhibits a su-percritical spatial bifurcation described by t u = f u , x , + 0 x,t, 1 where u x , t is a field that describes the system under study, f is the vector field, is a set of parameters that characterizes the system, 0 is the noise level intensity, and x , t is a white Gaussian noise with zero mean value and correlation i x , t j x , t = ij t− tx − x.


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hal-01730538 , version 1 (13-03-2018)



Gonzague Agez, Marcel G Clerc, Eric Louvergneaux. Universal shape law of stochastic supercritical bifurcations: Theory and experiments. Physical Review E , 2008, 77 (2), pp.026218. ⟨10.1103/PhysRevE.77.026218⟩. ⟨hal-01730538⟩
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