Noise-induced transitions in a double-well excitable oscillator

The model of a double-well oscillator with nonlinear dissipation is studied. The self-sustained oscillation regime and the excitable one are described. The first regime consists of the coexistence of two stable limit cycles in the phase space, which correspond to self-sustained oscillations of the point mass in either potential well. The self-sustained oscillations do not occur in a noise-free system in the excitable regime, but appropriate conditions for coherence resonance in either potential well can be achieved. The stochastic dynamics in both regimes is researched by using numerical simulation and electronic circuit implementation of the considered system. Multiple qualitative changes of the probability density function caused by noise intensity varying are explained by using the phase-space structure of the deterministic system.

Figure 1

Noise-free system (3) in the self-sustained oscillation regime in numerical simulation. (a) Phase space. Equilibrium points are shown by blue circles; the blue dashed line indicates the nullcline $\dot{y}=0$; the orange solid line shows the nullcline $\dot{v}=0$; the separatrix of the saddle at the origin is shown by the blue dotted line. Phase trajectories started from various initial conditions are shown by black arrowed lines. (b) Dependence of the dissipation, $q_1(y,v)$, on the system state [Eq. (5)]. Stable limit cycles corresponding to the self-sustained oscillation regimes 1 and 2 are marked by black closed curves. (c) Potential function, $U\left(y\right)$, corresponding to $q_2\left(y\right)$ [Eq. (5)]. Self-sustained oscillations in either potential well 1 and 2 are schematically shown. (d) Time traces of state variables in two coexisting self-sustained oscillatory regimes 1 and 2. Parameters are: $\varepsilon =0.01$, $c_1=1$, $c_3=9$, $c_5=22$, $a=3$, $b=50$, $D=0$.

Figure 2

Noise-induced transitions in the system (3) in the self-sustained oscillation regime (numerical simulations). The evolution of the PDF when the noise intensity increases (a) and the corresponding fragments of the phase trajectory (b): 1, $D=2.5\times {10}^{-7}$; 2, $D=5\times {10}^{-6}$; 3, $D=2\times {10}^{-5}$; 4, $D=1.2\times {10}^{-4}$; 5, $D=5\times {10}^{-4}$. On all panels: the blue dashed line indicates the nullcline $\dot{y}=0$; the orange solid line shows the nullcline $\dot{v}=0$. (c) Rice frequency, ${\omega }_R$, vs noise intensity, $D$. (d),(e) Evolution of the normalized power spectrum of the $y\left(t\right)$-oscillations (d) and $v\left(t\right)$-oscillations (e), $S_n\left(f\right)=S\left(f\right)/S_{max}\left(f\right)$, when the noise intensity increases: $D=2.5\times {10}^{-7}$ (the black solid curve), $D=5\times {10}^{-6}$ (the red solid curve), $D=2\times {10}^{-5}$ (the blue solid curve), $D=1.2\times {10}^{-4}$ (the black dashed curve), $D=5\times {10}^{-4}$ (the red dotted curve). (f) Normalized power spectra of the $y\left(t\right)$-oscillations [in (d)] illustrated in more details in the low frequency range. Other parameters are: $\varepsilon =0.01$, $c_1=1$, $c_3=9$, $c_5=22$, $a=3$, $b=50$.

Figure 3

Self-sustained oscillatory regime in analog experiment. (a)–(c) Evolution of the PDF when the noise intensity increases: (a) $D={10}^{-6}$; (b) $D=2.4\times {10}^{-5}$; (c) $D=6\times {10}^{-6}$. (d) Rice frequency, ${\omega }_R$, vs noise intensity, $D$. Parameters are: $\varepsilon =0.01$, $c_1=1$, $c_3=9$, $c_5=22$, $a=2.4$, $b=50$.

Figure 4

Excitable regime in numerical simulations [system (3)]. (a) Dependence of the dissipation, $q_1(y,v)$, on the system state [Eq. (5)]. Trajectories in a noise-free system ($D=0$) started from various initial conditions and coming to stable fixed points are marked by black arrowed lines. (b) Potential function, $U\left(y\right)$, corresponding to $q_2\left(y\right)$ [Eq. (5)]. Stability in either well is schematically shown. (c) Phase space of the system (3). Equilibrium points are shown by blue circles; the blue dashed line indicates the nullcline $\dot{y}=0$; the orange solid line shows the nullcline $\dot{v}=0$; the separatrix of the saddle at the origin is shown by the blue dotted line. Noise-induced oscillations ($D=2.5\times {10}^{-7}$) are shown by the black trajectory. (d) Time traces of state variables corresponding to the fragment (c). (e) Power spectra of the $v\left(t\right)$ noise-induced oscillations for various values of the noise intensity: $D=2\times {10}^{-7}$ (the black curve); $D={10}^{-6}$ (the red curve); $D={10}^{-5}$ (the blue curve). Parameters are: $\varepsilon =0.01$, $c_1=1$, $c_3=9$, $c_5=22$, $a=2.4$, $b=50$.

Figure 5

Excitable regime in numerical simulations [system (3)]. (a)–(e) Evolution of the PDF when the noise intensity increases: (a) $D=2.5\times {10}^{-7}$; (b) $D={10}^{-5}$; (c) $D=3\times {10}^{-5}$; (d) $D=2\times {10}^{-4}$; (e) $D=5\times {10}^{-4}$. (f) Rice frequency, ${\omega }_R$, vs noise intensity, $D$. (g) Evolution of the normalized power spectrum of the $y\left(t\right)$-oscillations, $S_n\left(f\right)=S\left(f\right)/S_{max}\left(f\right)$, when the noise intensity increases: $D=2.5\times {10}^{-7}$ (the black solid curve), $D={10}^{-5}$ (the red solid curve), $D=3\times {10}^{-5}$ (the blue solid curve), $D=2\times {10}^{-4}$ (the black dashed curve), $D=5\times {10}^{-4}$ (the red dotted curve). (h) Normalized power spectra of the $y\left(t\right)$-oscillations illustrated in more details in the low frequency range. Parameters are: $\varepsilon =0.01$, $c_1=1$, $c_3=9$, $c_5=22$, $a=2.4$, $b=50$.

Figure 6

Excitable regime in analog simulations on example of the system's (3) electronic model. (a)–(c) Evolution of the PDF when the noise intensity increases: (a) $D=1.2\times {10}^{-5}$; (b) $D=2.4\times {10}^{-5}$; (c) $D=3.78\times {10}^{-4}$. (d) Rice frequency, ${\omega }_R$, vs noise intensity, $D$. Parameters are: $\varepsilon =0.01$, $c_1=1$, $c_3=9$, $c_5=22$, $a=2.4$, $b=50$.