Resonances in nonlinear systems with a decaying chirped-frequency excitation and noise

Abstract

The influence of multiplicative white noise on the resonance capture of strongly nonlinear oscillatory systems under chirped-frequency excitations is investigated. It is assumed that the intensity of the perturbation decays polynomially with time, and its frequency grows according to a power low. Resonant solutions with a growing amplitude and phase, synchronized with the excitation, are considered. The persistence of such a regime in the presence of stochastic perturbations is discussed. In particular, conditions are described that guarantee the stochastic stability of the resonant modes on infinite or asymptotically large time intervals. The technique used is based on a combination of the averaging method, stability analysis and construction of stochastic Lyapunov functions. The proposed theory is applied to the Duffing oscillator with a chirped-frequency excitation and noise.

Figures (5)

• Figure 1: The level lines of $H(x_1,x_2)\equiv x_1^4/4+(x_2^2-x_1^2)/2$. The solid curves correspond to $H(x_1,x_2)>0$.
• Figure 2: The evolution of $\rho(t)\equiv [H(x_1(t),x_2(t))]^{1/4}$ for sample paths of solutions to system \ref{['Ex']} with $\alpha=\beta=1/3$ and $p=0$.
• Figure 3: The evolution of $\rho(t)$ and $\Theta(t)=\varphi(t)-\varkappa^{-1}S(t)$ for sample paths of the resonant solutions $(\varkappa=1)$ to system \ref{['Ex1']} with $\mathcal{Q}=4$ and $s=1/2$. The black dashed curves correspond to $\rho(t)\equiv z_0 t^{1/3}$ and $\Theta(t)\equiv \theta_0$, where $z_0\approx 0.56$, $\theta_0\approx -0.127$. In this case, $\mathcal{Q}_1\approx 2.04$.
• Figure 4: The evolution of $\rho(t)$ and $\Theta(t)=\varphi(t)-\varkappa^{-1}S(t)$ for sample paths of the resonant solutions $(\varkappa=2)$ to system \ref{['Ex2']} with $\mathcal{Q}=8$ and $s=1$. The black dashed curves correspond to $\rho(t)\equiv z_0 t^{1/2}$ and $\Theta(t)\equiv \theta_0$, where $z_0\approx 0.63$, $\theta_0\approx -0.132$. In this case, $\mathcal{Q}_2\approx 2.101$.
• Figure 5: The evolution of $\rho(t)$ and $\Theta(t)=\varphi(t)-\varkappa^{-1}S(t)$ for sample paths of the resonant solutions $(\varkappa=2)$ to system \ref{['Ex3']} with $\mathcal{Q}=5$ and $s=1$. The black dashed curves correspond to $\rho(t)\equiv z_0 t^{1/2}$ and $\Theta(t)\equiv \theta_0$, where $z_0\approx 0.63$, $\theta_0\approx -0.132$. In this case, $\mathcal{Q}_2\approx 2.101$.

Theorems & Definitions (11)

• Lemma 1
• Lemma 2
• Theorem 1
• Lemma 3
• Lemma 4
• Theorem 2
• proof : Proof of Lemma \ref{['Lem1']}
• proof : Proof of Lemma \ref{['Lem2']}
• proof : Proof of Lemma \ref{['LTh2']}
• proof : Proof of Lemma \ref{['LTh3']}