Resonant behavior of a fractional oscillator with fluctuating frequency

The long-time behavior of the first moment for the output signal of a fractional oscillator with fluctuating frequency subjected to an external periodic force is considered. Colored fluctuations of the oscillator eigenfrequency are modeled as a dichotomous noise. The viscoelastic type friction kernel with memory is assumed as a power-law function of time. Using the Shapiro-Loginov formula, exact expressions for the response to an external periodic field and for the complex susceptibility are presented. On the basis of the exact formulas it is demonstrated that interplay of colored noise and memory can generate a variety of cooperation effects, such as multiresonances versus the driving frequency and the friction coefficient as well as stochastic resonance versus noise parameters. The necessary and sufficient conditions for the cooperation effects are also discussed. Particularly, two different critical memory exponents have been found, which mark dynamical transitions in the behavior of the system.

Figure 1

The amplitude $A$ of the mean value of the oscillator displacement $⟨X\left(t\right)⟩$ vs the driving frequency $\Omega$ at $\gamma =0.7$, $A_0=\omega =1$, and $a^2=0.4$. (1) Solid line: $\alpha =0.1$ and $\nu =0.1$; (2) dotted line: $\alpha =0.1$ and $\nu =0.3$; (3) dashed-dotted line: $\alpha =0.9$ and $\nu =1.0$; (4) dashed line: $\alpha =0.9$ and $\nu =0.1$.

Figure 2

A semilogarithmic plot of the phase diagram in the $\gamma -\alpha$ plane for the existence of a resonance of $A$ vs the driving frequency $\Omega$ at $A_0=\omega =1$, and $a^2=0.6$. In the unshaded region (0) the resonance is impossible. The light gray region (1) corresponds to the domain where the resonance has one peak. In the dark gray region (2) a multiresonance with two peaks occurs. The thin dashed line depicts the position of the critical memory exponent $\alpha ={\alpha }_1\approx 0.441$. The critical value of the switching rate ${\nu }_{cr}\left(0.6\right)\approx 0.494$. Panels: (a) $\nu =0$, (b) $\nu =0.25$, and (c) $\nu =0.5$.

Figure 3

Multiresonance of the response of the stochastic oscillator (1) to the driving frequency $\Omega$. Parameter values: $A_0=\omega =1$, $\alpha =0.25$, $\nu =0.01$, $\gamma =0.8$, and $a^2=0.9$. Panel (a): solid line: the imaginary part ${\chi }^{″}\left(\Omega \right)$ of the complex susceptibility; dashed line: the real part ${\chi }^{\prime }\left(\Omega \right)$ of the complex susceptibility; dotted line: the response function $A\left(\Omega \right)$. The inset depicts the behavior of ${\chi }^{″}$ vs $\Omega$ at small values of the driving frequency $\Omega$. Panel (b): the Cole-Cole plot of ${\chi }^{″}$ as a function of ${\chi }^{\prime }$. The system parameters are the same as in panel (a).

Figure 4

Dependence of the response function $A$ on the friction coefficient $\gamma$ at $A_0=\omega =1$. Panel (a) depicts the resonancelike behavior of the function $A^2\left(\gamma \right)$ at various values of the driving frequency $\Omega$. Other parameter values: $\nu =0.1$, $\alpha =0.1$, and $a^2=0.4$. Solid line: $\Omega =1.5$; dashed line: $\Omega =1.18$; dotted line: $\Omega =0.95$; dashed-dotted line: $\Omega =0.3$. Panel (b): $A^2$ vs $\gamma$ at different values of the memory exponent $\alpha$. Other parameter values: $\nu =0.1$, $a^2=0.4$, and $\Omega =0.95$. Solid line: $\alpha =0.15$; dashed line: $\alpha =0.2$; dotted line: $\alpha =0.5$; dashed-dotted line: $\alpha =0.9$. Note that the maximal value of $A^2$ increases as the memory exponent $\alpha$ decreases.

Figure 5

SR for the response function $A$ versus the noise amplitude $a$ at various values of the friction coefficient $\gamma$. Other parameter values: $A_0=\omega =1$, $\nu =0.1$, $\alpha =0.1$, and $\Omega =1.8$. (1) Solid line: $\gamma =1.3$; (2) dashed line: $\gamma =1.5$; (3) dotted line: $\gamma =2.85$; (4) dashed-dotted line: $\gamma =2.3$. Note that in the case of curve (4), $\gamma =2.3$, the phenomenon of stochastic resonance is absent. The inset emphasizes the nonmonotonic behavior of the curve (3).

Figure 6

A plot of the phase diagrams for SR in the $\gamma -\alpha$ plane at $A_0=\omega =1$. In the unshaded region the resonance of $A$ versus the noise amplitude $a$ is impossible. In the light gray region the function $A\left(a\right)$ exhibits a maximum at $a_m>a_{cr}$ [see Eq. (7)], i.e., at $a_m$ the first moment of the oscillator displacement $⟨X\left(t\right)⟩$ is unstable. In the dark gray domain a stochastic resonance for $A$ vs $a$ occurs (in the stability region). The thin dashed line depicts the position of the critical memory exponent ${\alpha }_c$, [Eq. (33)]. Panel (a): $\nu =0$, $\Omega =0.6$; panel (b): $\nu =1.0$, $\Omega =0.6$; panel (c): $\nu =0$, $\Omega =1.8$; panel (d); $\nu =1.0$, $\Omega =1.8$.

Figure 7

SR for $A^2$ versus the noise switching rate $\nu$, computed from Eq. (14) at various values of the friction coefficient $\gamma$. Other parameter values: $A_0=\omega =1$, $a^2=0.3$, $\alpha =0.3$, and $\Omega =0.8$. Solid line: $\gamma =0.04$; dashed line: $\gamma =0.05$; dotted line: $\gamma =0.06$; dashed-dotted line: $\gamma =0.1$. Note that in the case of $\gamma =0.1$ the phenomenon of stochastic resonance is absent.