Vibrational resonance in an inhomogeneous medium with periodic dissipation

The role of nonlinear dissipation in vibrational resonance (VR) is investigated in an inhomogeneous system characterized by a symmetric and spatially periodic potential and subjected to nonuniform state-dependent damping and a biharmonic driving force. The contributions of the parameters of the high-frequency signal to the system's effective dissipation are examined theoretically in comparison to linearly damped systems, for which the parameter of interest is the effective stiffness in the equation of slow vibration. We show that the VR effect can be enhanced by varying the nonlinear dissipation parameters and that it can be induced by a parameter that is shared by the damping inhomogeneity and the system potential. Furthermore, we have apparently identified the origin of the nonlinear-dissipation-enhanced response: We provide evidence of its connection to a Hopf bifurcation, accompanied by monotonic attractor enlargement in the VR regime.

Figure 1

Effective potential $V_{\mathrm{eff}}\left(\chi \right)$ for (a) four values of $G$ with $k=1$ and $\mathrm{\Omega }=13$ and (b) four values of $k$ with $G=300$ and $\mathrm{\Omega }=13$.

Figure 2

Dependence of the response amplitude $Q$ on the control parameter $G$ for five values of $\lambda$ (from bottom to top against the ordinate axis, $\lambda =0,0.9,1.9,3.5,5.0)$. The other parameters are ${\gamma }_0=1.2,\varphi =0.1,k=1.0,F=0.1,\omega =0.65$, and $\mathrm{\Omega }=13$. Continuous curves represent the numerically computed $Q$ from Eq. (29) using Eq. (25), while the analytically calculated $Q$ from Eq. (24) are indicated by marker points and/or dashed lines.

Figure 3

Comparison between theory and numerics for the dependence of the response amplitude $Q$ on the coefficient of nonlinear dissipation $\lambda$ for $G=15$. The other parameters are ${\gamma }_0=1.2,\varphi =0.85,\omega =0.65$, and $\mathrm{\Omega }=13$. The red solid curve represents the numerically computed $Q$ from Eq. (29) using Eq. (25), while the theoretically calculated $Q$ from Eq. (24) is indicated by the blue dotted line with marker points.

Figure 4

Dependence of the response amplitude $Q$ on the control parameter $G$ for four values of phase shift, from bottom to top, $\varphi =0$, 0.1, 0.3, and 0.5. The other parameters are ${\gamma }_0=1.2,\lambda =0.9,k=1.0,F=0.1,\omega =0.65$, and $\mathrm{\Omega }=13$. The solid and dashed curves represent the numerically computed $Q$ from Eq. (29) using Eq. (25), while the corresponding analytically calculated $Q$ from Eq. (24) are indicated by marker points (closed shapes) of the same color.

Figure 5

Dependence of the response amplitude $Q$ on the control parameter $k$ for three amplitudes of fast forcing (a) $G=250$, (b) $G=350$, and (c) $G=650$. The other parameters are ${\gamma }_0=1.2,\lambda =0.9,\varphi =0.1,F=0.1,\omega =0.65$, and $\mathrm{\Omega }=13$. The solid curves represent the numerically computed $Q$ from Eq. (4), while the corresponding analytically calculated $Q$ from Eq. (24) is shown as open circle (marker) points.

Figure 6

Three-dimensional plot showing the dependence of the response amplitude $Q$ on the fast signal amplitude $G$ and the dissipation parameter $\lambda$ for a phase shift of $\varphi =0.85$. Notice that small values of $G$ also produce some low peaks in the regime marked in deep blue, while the largest peaks occur at higher $G$ and lower $\lambda$.

Figure 7

Bifurcation diagram (dotted red) of the displacement $x$, the corresponding response curve $Q$ (blue), and maximal Lyapunov exponent ${\lambda }_{\text{max}}$ (green) for $G=15$. The other parameters are set as follows: ${\gamma }_0=1.2,k=1.0,\varphi =0.85,V_0=1.0,F=0.1,\omega =0.65$, and $\mathrm{\Omega }=13$.

Figure 8

Transition from periodicity to quasiperiodicity accompanying the onset of VR with variation of the dissipation parameter $\lambda$: periodic orbit for $\lambda =0$ (red) and monotonic quasiperiodic attractor enlargement for $\lambda =1.35$ (green), $\lambda =1.4$ (blue), $\lambda =1.5$ (magenta), and $\lambda =2.0$ (cyan). The other parameters are set as $G=15,{\gamma }_0=1.2,k=1.0,\varphi =0.85,V_0=1.0,F=0.1,\omega =0.65$, and $\mathrm{\Omega }=13$.