论文标题
使用时间延迟的反馈控制器计算分段光滑软性影响系统的Lyapunov指数
Calculating the Lyapunov exponents of a piecewise-smooth soft impacting system with a time-delayed feedback controller
论文作者
论文摘要
Lyapunov指数是一种用于研究动态系统的广泛使用的工具。在计算带有时间延迟参数的分段平滑系统的Lyapunov指数时，人们面临缺乏变化问题的连续性。本文研究了如何在延迟非平滑系统的轨迹上建立一个差异方程，以有效地构建雅各布人。分段平滑系统的轨迹可能会遇到一个所谓的放牧事件，在该事件中，该轨迹以非转换方式在状态空间中接近状态空间中的不连续表面。在此事件中，我们开发了一个放牧点估计算法，以确保非线性和变分方程的轨迹的准确性。我们表明，由算法计算出的雅各布矩阵的特征值与数值集成方法的顺序一致，因此保证我们提出的数值方法的可靠性。最后，在延迟的反馈控制下，在定期强迫影响振荡器上证明了该方法。
Lyapunov exponents are a widely used tool for studying dynamical systems. When calculating Lyapunov exponents for piecewise-smooth systems with time-delayed arguments one faces a lack of continuity in the variational problem. This paper studies how to build a variational equation for the efficient construction of Jacobians along trajectories of the delayed nonsmooth system. Trajectories of the piecewise-smooth system may encounter a so-called grazing event where the trajectory approaches a discontinuity surface in the state space in a non-transversal manner. For this event we develop a grazing point estimation algorithm to ensure the accuracy of trajectories for the nonlinear and the variational equations. We show that the eigenvalues of the Jacobian matrix computed by the algorithm converge with an order consistent with the order of the numerical integration method, therefore guaranteeing the reliability of our proposed numerical method. Finally, the method is demonstrated on a periodically forced impacting oscillator under the time-delayed feedback control.
