Title: Stability of asymptotically Hamiltonian systems with damped oscillatory and stochastic perturbations Show Chinese title

Title: 渐近哈密顿系统在阻尼振荡和随机扰动下的稳定性

Abstract: A class of asymptotically autonomous systems on the plane with oscillatory coefficients is considered. It is assumed that the limiting system is Hamiltonian with a stable equilibrium. The effect of damped multiplicative stochastic perturbations of white noise type on the stability of the system is discussed. It is shown that different long-term asymptotic regimes for solutions are admissible in the system and the stochastic stability of the equilibrium depends on the realized regime. In particular, we show that stable phase locking is possible in the system due to decaying stochastic perturbations. The proposed analysis is based on a combination of the averaging technique and the construction of stochastic Lyapunov functions. Show Chinese abstract

Abstract: 考虑了一类平面上具有振荡系数的渐近自治系统。假设极限系统是具有稳定平衡点的哈密顿系统。讨论了阻尼乘性白噪声型随机扰动对系统稳定性的影响。表明在系统中允许解的不同长期渐近行为，并且随机稳定性取决于实现的模式。特别是，我们表明由于衰减的随机扰动，系统中可能发生稳定的相位锁定。所提出的分析基于平均技术与随机李雅普诺夫函数的构造相结合。