Nonlinear resonance in systems with decaying perturbations and noise

Abstract

The effect of multiplicative white noise on the resonance capture in non-isochronous systems with time-decaying pumping is investigated. It is assumed that the intensity of perturbations decays with time, and its frequency is asymptotically constant. The occurrence of attractive solutions with an amplitude close to the resonant value and a phase synchronized with the excitation are considered. The persistence of such a regime in a stochastically perturbed system is analyzed. By combining the averaging method and the construction of suitable stochastic Lyapunov functions, conditions are derived that guarantee the stochastic stability of the resonant modes on infinite or asymptotically large time intervals. The proposed theory is applied to the Duffing oscillator with decaying parametric excitation and noise.

Figures (8)

• Figure 1: The evolution of $r(t)\equiv \sqrt{2U(x(t))+y^2(t)}$ for sample paths of solutions to system \ref{['Ex0']} with $s_0=3/2$, $n=2$, $p=1$, $\vartheta=2^{-5}$, $\mathcal{P}_0=\mathcal{Q}_1=0$, $\mathcal{B}_0=3.6$, $\mathcal{B}_1=0$ and different values of the parameters $\mathcal{Q}_0$, $\mathcal{P}_1$, $\varepsilon$ and initial data. The dashed curves correspond to $r(t)\equiv 3.6$.
• Figure 2: A typical phase portrait of the limiting system \ref{['limsys']} with $n=1$.
• Figure 3: The evolution of $r(t)$ and $\phi(t)\equiv \varphi(t)-S(t)$ for sample paths of solutions to system \ref{['Ex1']} with $s_0=1/2$, $p=1$, $\vartheta=1/4$, $\mathcal{Z}_0=\mathcal{Z}_1=\mathcal{B}_0=\mathcal{Q}_1=0$, $\mathcal{B}_1=1$, $\mathcal{Q}_0=-\varepsilon^2/8$, $\varepsilon=0.1$ and different values of initial data. The dashed curves correspond to $r(t)\equiv r_0$ and $\phi(t)\equiv \psi_0$, where $r_0=\sqrt 2$ and $\psi_0=\pi/4$.
• Figure 4: The evolution of $r(t)$ and $\phi(t)\equiv \varphi(t)-S(t)$ for sample paths of solutions to system \ref{['Ex1']} with $s_0=1/2$, $p=1$, $\vartheta=1/4$, $\mathcal{Z}_0=\mathcal{B}_1=\mathcal{Q}_1=0$, $\mathcal{B}_0=2$, $\mathcal{Z}_1=1/\sqrt{8}$, $\mathcal{Q}_0=-0.126$ and different values of $\varepsilon$. The dashed curves correspond to $r(t)\equiv r_0$ and $\phi(t)\equiv \psi_0$, where $r_0=\sqrt 2$ and $\psi_0=2\pi/3$.
• Figure 5: The evolution of $r(t)$ and $\phi(t)\equiv \varphi(t)-S(t)$ for sample paths of solutions to system \ref{['Ex1']} with $s_0=1/2$, $p=2$, $\vartheta=1/4$, $\mathcal{Z}_0=\mathcal{Q}_1=0$, $\mathcal{B}_0=\mathcal{B}_1=1$, $\varepsilon=0.1$ and different values of the parameters $\mathcal{Q}_0$ and $\mathcal{Z}_1$. The dashed curves correspond to $r(t)\equiv r_0$ and $\phi(t)\equiv \psi_0$, where $r_0=\sqrt 2$ and $\psi_0=3\pi/4$.

Theorems & Definitions (6)

• Theorem 1
• Lemma 1
• Lemma 2
• Theorem 2
• proof : Proof of Lemma \ref{['Lem2']}
• proof : Proof of Lemma \ref{['Lem3']}