Abstract In a smooth dynamical system the characteristics of a given reference trajectory can be determined, to lowest order, by examining the linearised system about the reference trajectory. In other words, we can approximate the deviations of trajectories after a given time, with starting points in a neighbourhood of the reference trajectory, by multiplying the initial deviations by the corresponding fundamental matrix solution. This form of analysis cannot be used directly in nonsmooth systems as the vector field is either not everywhere differentiable or the flow function is not continuous. To account for this, one can derive the zero-time discontinuity mapping (ZDM) associated with the discontinuity boundary. The Jacobian of this mapping is known as the saltation matrix and its properties can tell us how the crossing of the discontinuity boundary affects the deviations of trajectories from a reference trajectory. In particular, this matrix can be composed with the fundamental matrix solutions of the individual flows on either side of the discontinuity boundary in order to determine the overall fundamental matrix solution of a trajectory that crosses the boundary. In this paper we derive a saltation matrix for a piecewise-smooth dynamical system in which the position of the discontinuity boundary oscillates according to a mean-reverting stochastic process. The derived saltation matrix contains the entire effect of both the discontinuity and the uncertainty introduced into the system by the noisy boundary, and is composable with the deterministic fundamental matrix solutions of the individual flows to give the overall fundamental matrix solution of a crossing trajectory. We also present some simple examples of piecewise-smooth systems with stochastically varying boundaries, analysed using the derived noisy saltation matrix. In particular we focus on the analysis of a discontinuous variant of the Chua circuit. In this case we apply noise to the system’s discontinuity boundaries which are generated by the piecewise-linear nature of the voltage–current response of the Chua diode. We find that our method allows us to analyse the effects of boundary noise on periodic attractors close to bifurcation points. In particular we show that we can use the method to accurately predict the noise amplitudes required to destroy or merge periodic attractors.

中文翻译：

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估计具有噪声边界的系统的动力学

摘要 在光滑动态系统中，通过检查关于参考轨迹的线性化系统，可以确定给定参考轨迹的特性，最低阶。换句话说，我们可以通过将初始偏差乘以相应的基本矩阵解，以参考轨迹的邻域为起点来近似给定时间后轨迹的偏差。这种形式的分析不能直接用于非光滑系统，因为矢量场不是处处可微的，或者流函数不是连续的。为了解决这个问题，可以推导出与不连续边界相关联的零时间不连续映射 (ZDM)。这种映射的雅可比矩阵被称为跃移矩阵，它的性质可以告诉我们不连续边界的交叉如何影响轨迹与参考轨迹的偏差。特别地，该矩阵可以与不连续边界两侧的各个流的基本矩阵解组合，以确定跨越边界的轨迹的整体基本矩阵解。在本文中，我们推导出分段平滑动力系统的跃迁矩阵，其中不连续边界的位置根据均值回复随机过程振荡。导出的跃迁矩阵包含由噪声边界引入系统的不连续性和不确定性的全部影响，并且可以与各个流的确定性基本矩阵解组合，以给出交叉轨迹的整体基本矩阵解。我们还展示了一些具有随机变化边界的分段平滑系统的简单示例，使用派生的噪声跃迁矩阵进行分析。我们特别关注 Chua 电路的不连续变体的分析。在这种情况下，我们将噪声应用到系统的不连续边界，这是由蔡二极管的电压-电流响应的分段线性特性产生的。我们发现我们的方法允许我们分析边界噪声对靠近分叉点的周期性吸引子的影响。特别是，我们表明我们可以使用该方法准确预测破坏或合并周期性吸引子所需的噪声幅度。