Figure 1

Typical signatures of stochastic resonance. In the top-left graph we display the power spectrum of the energy of the FPU chain for the two noise strengths $D=0.03$ and $D=0.09$ at which the signal-to-noise ratio (SNR) shows the marked peaks reported in the top-right graph. The forcing signal, with carrier frequency $\Omega =2.05$ and modulation $W=0.003$, is plotted in the central graph. The time evolution of the energy for the noise intensity corresponding to the first peak of the SNR plot is shown in the bottom graph: it displays the characteristic bistable features of stochastic resonance.Reuse & Permissions

Figure 2

In the top-left plot the energy of the approximate solutions (10) of the FPU chain, obtained using our semicontinuum approach is displayed as a function of the driving amplitude $A$. The intersections of the vertical solid line at $A=0.09$ with the different branches of the curve correspond to multiple solutions of the consistency relation (12). Type 1 ($E_1\simeq 0.1$) and 2 ($E_2\simeq 0.6$) solutions, which turn out to be stable, are compared with numerical simulations (points) in the top-right and bottom-left panels. A type 3 ($E_3\simeq 1.6$) solution, which is metastable (see text), is represented in the bottom-right panel. $A_1=0.18$, $A_2=0.31$, and $A_3=0.44$ are threshold values for the excitation of the different solutions: $A_1$ is the driving amplitude for which the transition from a type 1 stationary state to a type 2 stationary state occurs, while $A_2$ marks the transition from type 2 to type 3 (metastable) solutions, and so on. In the figure, we also display a feature which is present because of the damping and is not taken into account by our approximate solutions: an additional lower threshold $L$ is present (vertical dashed line), below which all the solutions collapse to the type 1 lowest energy state.Reuse & Permissions

Figure 3

Time evolution of the energy of the FPU chain for a constant driving amplitude $A=0.18$ and a linearly varying noise intensity. Transition from a type 1 to a type 2 state occurs when the noise intensity reaches the threshold value $D=0.021$, while the transition to a type 3 states takes place at $D=0.055$. In this simulation, damping is present and the driving is not modulated, $W=0$.Reuse & Permissions

Figure 4

Time evolution of the signal (top) and system’s energy (central and bottom) at the noise intensities $D=0.05$ (corresponding to the point between first two peaks of the SNR in Fig. 1) and $D=0.09$ (second peak of the SNR), respectively.Reuse & Permissions