Resonance in isochronous systems with decaying oscillatory and stochastic perturbations

TL;DR

The paper analyzes resonance phenomena in planar isochronous systems subjected to time-decaying multiplicative noise and oscillatory perturbations. By constructing a near-identity averaging transformation, it derives a truncated deterministic model that captures the leading resonant drift terms and identifies two asymptotic regimes: phase locking and phase drifting. Through Lyapunov methods and stochastic averaging, it proves the persistence and stochastic stability of resonant solutions and shows that decaying noise can shift the stability boundary; these results are illustrated with concrete examples. The work advances understanding of how decaying stochastic perturbations shape long-time resonant dynamics in isochronous oscillators, with implications for control and prediction in noisy resonant systems.

Abstract

The combined influence of oscillatory excitations and multiplicative stochastic perturbations of white noise type on isochronous systems in the plane is investigated. It is assumed that the intensity of perturbations decays with time and the excitation frequency satisfies a resonance condition. The occurrence and stochastic stability of solutions with an asymptotically constant amplitude are discussed. By constructing an averaging transformation, we derive a model truncated deterministic system that describes possible asymptotic regimes for perturbed solutions. The persistence of resonant solutions in the phase locking and the phase drifting modes is justified by constructing suitable Lyapunov functions for the complete stochastic system. In particular, we show that decaying stochastic perturbations can shift the boundary of stability domain for resonant solutions.

Figures (8)

• Figure 1: The evolution of $\varrho(t)\equiv \sqrt{x_1^2(t)+x_2^2(t)}$ for sample paths of solutions to system \ref{['Ex0']} with $s_0=2$, $n=2$, $p=1$, $A_1=0$, $B_0=-1$ and different values of the parameters $B_1$, $C_0$, $\varepsilon$ and initial data.
• Figure 2: Partition of the parameter plane $(B_0,Q_1)$ for system \ref{['Ex0']} with $n=2$, $p=1$, $s_0=2$ and $s_1=0$.
• Figure 3: The evolution of $\varrho(t)\equiv \sqrt{x_1^2(t)+x_2^2(t)}$ and $\psi(t)\equiv \varphi(t)-S(t)/2$, $\tan \varphi(t)=-x_2(t)/x_1(t)$ for solutions to system \ref{['Ex0']} with $s_0=2$, $s_1=0$, $A_1=0$, $B_0=-1$, $B_1=2.5$, $C_0=-0.2$, $\varepsilon=0.4$ and different values of initial data. The dashed curves correspond to $\varrho(t)\equiv \rho_0^+$ and $\psi(t)\equiv -\pi/4$, where $\rho_0^+ \approx 1.38$.
• Figure 4: The evolution of $\varrho(t)\equiv \sqrt{x_1^2(t)+x_2^2(t)}$ for solutions to system \ref{['Ex0']} with $s_0=2$, $s_1=8$, $A_1=0$, $B_0=1$, $B_1=1$, $C_0=-0.5$ and different values of $\varepsilon$ and initial data. The dashed curves correspond to $\varrho(t)\equiv \rho_0$, where $\rho_0=\sqrt{8/3+\varepsilon^2/2}$.
• Figure 5: The evolution of $\varrho(t)\equiv \sqrt{x_1^2(t)+x_2^2(t)}$ for sample paths of solutions to system \ref{['Ex2']} with $n=2$, $p=1$, $A_1=0$, $B_0=-1$, $\varepsilon=0$, and different values of the parameters $B_1$, $C_0$, and initial data.

Theorems & Definitions (15)

• Theorem 1
• Lemma 1
• Lemma 2
• Theorem 2
• Lemma 3
• Lemma 4
• Lemma 5
• Theorem 3
• proof : Proof of Lemma \ref{['Lem1']}
• proof : Proof of Lemma \ref{['Lem2']}