In this paper, a new impact-to-impact mapping is constructed to investigate the stochastic response of a nonautonomous vibro-impact system. The significant feature lies in the choice of Poincaré section, which consists of impact surface and codimensional time. Firstly, we construct a new impact-to-impact mapping to calculate the one-step transition probability matrix from a given impact to the next. Then, according to the matrix, we can investigate the stochastic responses of a nonautonomous vibro-impact system at the impact instants. The new impact-to-impact mapping is smooth and it effectively overcomes the nondifferentiability caused by the impact. A linear and a nonlinear nonautonomous vibro-impact systems are analyzed to verify the effectiveness of the strategy. The stochastic P-bifurcations induced by the noise intensity and system parameters are studied at the impact instants. Compared with Monte Carlo simulations, the new impact-to-impact strategy is accurate for nonautonomous vibro-impact systems with arbitrary restitution coefficients.

中文翻译：

---

通过新的冲击对冲击映射的振动冲击系统的随机响应

在本文中，构建了一个新的冲击到冲击映射来研究非自主振动冲击系统的随机响应。显着的特点在于庞加莱截面的选择，它由冲击面和共维时间组成。首先，我们构建了一个新的影响到影响的映射来计算从给定影响到下一个影响的一步转移概率矩阵。然后，根据矩阵，我们可以研究非自主振动冲击系统在冲击瞬间的随机响应。新的影响到影响的映射是平滑的，它有效地克服了由影响引起的不可微性。对一个线性和一个非线性非自治振动冲击系统进行了分析，以验证该策略的有效性。研究了在冲击瞬间由噪声强度和系统参数引起的随机P分岔。与蒙特卡洛模拟相比，新的冲击对冲击策略对于具有任意恢复系数的非自主振动冲击系统是准确的。