Stochastic resonance across bifurcation cascades

The classical setting of stochastic resonance is extended to account for parameter variations leading to transitions between a unique stable state, bistability, and multistability regimes, across singularities of various kinds. Analytic expressions for the amplitude and the phase of the response in terms of key parameters are obtained. The conditions for optimal responses are derived in terms of the bifurcation parameter, the driving frequency, and the noise strength.

Figure 1

Bifurcation diagram corresponding to the potential in Eq. (3) with $u=1/2$, ${\lambda }_1=2$, and $\alpha =3$.

Figure 2

Dependence on the bifurcation parameter $\lambda$ in the absence of forcing, of the potential barrier $\mathrm{\Delta }U$ (a); and of factors $U^{″}\left(x_1\right)$ (full line) and $U^{″}\left(x_0\right)$ (dashed line) (b). Parameter values as in Fig. 1

Figure 3

Amplitude of the response (left ordinate) and corresponding values of the transition rates (right ordinate) versus the bifurcation parameter $\lambda$, in the subcritical region, Eq. (15b), (a) and the supercritical region, Eq. (17b), (b) for $\omega ={10}^{-4}$ (thin full line), $\omega ={10}^{-3}$ (dashed line), and $\omega ={10}^{-2}$ (fat full line). Other parameters as in Fig. 1 with $q^2={10}^{-2}$.

Figure 4

As in Fig. 3 with smaller values of the forcing frequency and $q^2=0.025$.

Figure 5

Optimal values of the noise strength for which the response is maximised as a function of the bifurcation parameter $\lambda$ in the supercritical region (a) and the subcritical region (b) for different values of the forcing frequency. Parameter values as in Fig. 1.

Figure 6

Asymptotic probability masses of the corresponding attraction basins as obtained from direct numerical integration of the Fokker-Planck equation (5) in the supercritical case (a) and the subcritical case (b). Parameter values as in Fig. 1 and $q^2=\omega =\epsilon ={10}^{-2}$.

Figure 7

One-sided average of variable $x$ in the domain of attraction of state $x_1$ versus the noise strength $q^2$ at the critical point $\lambda =0$ separating the two-state and three-state regimes.