Theory of stochastic resonance for small signals in weakly damped bistable oscillators

The response of a weakly damped bistable oscillator to an external periodic force is considered theoretically. In the approximation of weak signals we can write a linearized equation for the signal and the corresponding nonlinear equation for the noise. These equations contain two unknown parameters: An effective stiffness and an additional damping factor. In the case of the weakly damped bistable oscillator, considered here, the two-dimensional Fokker-Planck equation corresponding to the equation for the noise can be solved approximately by changing to a slow variable (“energy”) and applying a method of successive approximation. This approach allows us to find the unknown parameters and to calculate the amplitude ratio of the output and input signals, i.e., the gain factor.

Figure 1

(Color online) Results of numerical simulations. (a) The signal-to-noise ratio $R$ as a function of noise intensity $K$. The dashed line corresponds to the input value, calculated numerically, whereas the circles correspond to theoretical values. The full line corresponds to the output $R$ calculated for $x\left(t\right)$; the dot-dashed line was calculated using the two-state approximation. (b) The mean switching frequency $⟨\omega ⟩$ is shown as a function of $K$. (c) The gain factor $Q$ and (d) the phase shift $\psi$ are shown as functions of noise intensity $K$. Full lines corresponds to numerical simulations of Eq. (1), dashed lines correspond to linear response theory, and dot-dashed lines correspond to results that were calculated numerically in the two-state approximation.

Figure 2

(Color online) Dependences of the (a) gain factor $Q$ and (b) phase shift $\psi$ on the noise intensity $K$ for $\delta =0.1$, $A=0.1$ and $\omega =0.5$, $\omega =1$, and $\omega =1.25$ [curves 1, 2, and 3, respectively]; (c) the plot of $K_m$ versus $\omega$ for maximum of $Q$; (d) the numerical dependence of the resonant frequency ${\omega }_r$ on $K$.

Figure 3

(Color online) Dependences on noise intensity $K$ of (a) the effective stiffness $c$, (b) the addition $b$ to the damping factor, and (c) the natural frequency ${\omega }_0$, for $\delta =0.1$, $A=0.1$ and $\omega =0.5$, $\omega =1$, and $\omega =1.25$ [curves 1, 2, and 3, respectively].

Figure 4

Dependence of the second moment $m_2$ on $K$.

Figure 5

(Color online) (a) Effective stiffness $c$, (b) the addition $b$ to the damping factor, and (c) the natural frequency ${\omega }_0$, are shown as functions of intensity $K$. Theoretical results in the first and second approximations are shown by dashed and full lines, respectively. The dependences found numerically are shown by crosses.

Figure 6

(Color online) (a) Gain factor $Q$ and (b) the phase shift $\psi$ are shown as functions of noise intensity $K$. Theoretical results in the first and second approximations are shown by dashed and full lines, respectively. The dependences found numerically are shown by crosses.