Bifurcations of emerging patterns in the presence of additive noise

A universal description of the effects of additive noise on super- and subcritical spatial bifurcations in one-dimensional systems is theoretically, numerically, and experimentally studied. The probability density of the critical spatial mode amplitude is derived. From this generalized Rayleigh distribution we predict the shape of noisy bifurcations by means of the most probable value of the critical mode amplitude. Comparisons with numerical simulations are in quite good agreement for cubic or quintic amplitude equations accounting for stochastic supercritical bifurcation and for cubic-quintic amplitude equation accounting for stochastic subcritical bifurcation. Experimental results obtained in a one-dimensional Kerr-like slice subjected to optical feedback confirm the analytical expression prediction for the supercritical bifurcation shape.

Figure 1

Growth rate as function of wave number $k$. Dashed, continuous, and dotted curves denote growth rate below, at, and above the spatial bifurcation. $k_c$ is the critical wave number.

Figure 2

Pattern precursors: spatiotemporal diagram observed in the Swift-Hohenberg equation (11) below the spatial bifurcation. The vertical and horizontal axis stand for time and space, respectively. Inset: the instantaneous profile of the Swift-Hohenberg equation.

Figure 3

Stationary structure function $S\left(k\right)$, obtained from the supercritical (a) and subcritical (b) Swift-Hohenberg model below the bifurcation point. Insets: The arrows indicate the corresponding values of the control parameter $\lambda$ with respect to this critical value ${\lambda }_c$. $k_c$ is the slowest mode.

Figure 4

Stationary probability distribution of the modulus $\left|A\right|$, from formula (17), for different bifurcation parameter values. From left to right: $\varepsilon =-1,0,0.25,0.5,1$. $\eta =0.1$.

Figure 5

Bifurcation diagrams of the amplitude of the critical mode obtained for different resolutions of Eq. (14). Namely the deterministic value $A$, the expectation value $|a_{\max }|$, the linear and nonlinear mean values ${\langle \left|a\right|\rangle }_{\mathrm{linear}}$ and ${\langle \left|a\right|\rangle }_{\mathrm{nonlinear}}$, respectively (see text for value definitions).

Figure 6

Bifurcation diagram of supercritical stochastic Swift-Hohenberg equation (31) for two different noise intensities. Continuous curves show the fits using the expression (18). The dashed curve is the corresponding bifurcation without noise.

Figure 7

Experimental bifurcation diagram for the optical feedback system. The continuous curve is the analytical expression and dots are experimental values. Error bars correspond to the full width at half maximum of the experimental probability distribution.

Figure 8

Stationary structure function $S\left(k\right)$, obtained from a vertically driven pendula chaim. Inset: The deterministic bifurcation diagram and the arrow indicate the corresponding value of the control parameter.

Figure 9

Stationary probability distribution for different values of the bifurcation parameter $\varepsilon$ and ${\sigma }_0=1.4$. From left to right: $\varepsilon =-6.26;-2.50;{\varepsilon }_{\min }=-0.35,0.0,black1.25,3.75$.

Figure 10

Bifurcation diagrams obtained from the different calculs of the amplitude (see text for definitions).

Figure 11

Evolution of the most probable value $|a_{\max }|$ versus the bifurcation parameter $\beta =\varepsilon /{\sigma }^{2/3}$. ${\beta }_{\min }$ stands for the minimum of the curve of $|a_{\max }|$ and then indicates the phase transition in the system.

Figure 12

Stationary probability distribution $P_s\left(\right|a\left|\right)$ for different values of the bifurcation parameter $\varepsilon$ and ${\eta }^{\prime }$. (a) ${\eta }^{\prime }=0.08>{\eta }_c^{\prime }$; (b) ${\eta }^{\prime }=0.009<{\eta }_c^{\prime }$.

Figure 13

Theoretical bifurcation diagram of the subcritical amplitude equation for different levels of noise. From right to left: ${\eta }^{\prime }=0$; ${\eta }^{\prime }=0.01<{\eta }_c$; ${\eta }^{\prime }={\eta }_c\equiv 0.07$; ${\eta }^{\prime }=0.1>{\eta }_c$. ${{\varepsilon }^{\prime }}_{\mathrm{down}}$ and ${{\varepsilon }^{\prime }}_{\mathrm{up}}$ account for the region of bistability for noise intensity level ${\eta }^{\prime }=0.01$.

Figure 14

Theoretical evolution of the bistability zone limits ${\varepsilon }_{\mathrm{down}}$ (continued curve above) and ${\varepsilon }_{\mathrm{up}}$ (continued curve below) with the level of noise. The intermediate curve represents the approximated evolution of the Maxwell point [Eq. (30)]. The dots are exact values of the Maxwell point numerically calculated. The shaded area represents the region of bistability. The insets represent, respectively, the generalized Rayleigh distribution in the respective places.

Figure 15

Numerical bifurcation diagram of the subcritical Swift-Hohenberg model, Eq. (31), for different levels of noise. Taking the curves of the left side from top to bottom ${\sigma }_0=0.1$ (blue $•$), ${\sigma }_0=0.05$ (cyan $▴$), ${\sigma }_0=0.02$ (green $⧫$), ${\sigma }_0=0.01$ (orange $\blacksquare$), and ${\sigma }_0=0.0$ (red $\circ$). The respective solid curves are obtained using the fitting formula (32).