Dabrowski Artur, Sagan Tomasz, Denysenko Volodymyr, Balcerzak Marek, Zarychta Sandra, Stefanski Andrzej
Division of Dynamics, Lodz University of Technology, Stefanowskiego 1/15, 90-001 Lodz, Poland.
Materials (Basel). 2021 Nov 25;14(23):7197. doi: 10.3390/ma14237197.
Controlling stability of dynamical systems is one of the most important challenges in science and engineering. Hence, there appears to be continuous need to study and develop numerical algorithms of control methods. One of the most frequently applied invariants characterizing systems' stability are Lyapunov exponents (LE). When information about the stability of a system is demanded, it can be determined based on the value of the largest Lyapunov exponent (LLE). Recently, we have shown that LLE can be estimated from the vector field properties by means of the most basic mathematical operations. The present article introduces new methods of LLE estimation for continuous systems and maps. We have shown that application of our approaches will introduce significant improvement of the efficiency. We have also proved that our approach is simpler and more efficient than commonly applied algorithms. Moreover, as our approach works in the case of dynamical maps, it also enables an easy application of this method in noncontinuous systems. We show comparisons of efficiencies of algorithms based our approach. In the last paragraph, we discuss a possibility of the estimation of LLE from maps and for noncontinuous systems and present results of our initial investigations.
控制动力系统的稳定性是科学与工程领域最重要的挑战之一。因此，持续需要研究和开发控制方法的数值算法。表征系统稳定性最常应用的不变量之一是李雅普诺夫指数（LE）。当需要有关系统稳定性的信息时，可以基于最大李雅普诺夫指数（LLE）的值来确定。最近，我们已经表明，可以通过最基本的数学运算从向量场属性估计LLE。本文介绍了用于连续系统和映射的LLE估计新方法。我们已经表明，应用我们的方法将显著提高效率。我们还证明了我们的方法比常用算法更简单、更高效。此外，由于我们的方法适用于动态映射，它也使得该方法能够轻松应用于非连续系统。我们展示了基于我们方法的算法效率比较。在最后一段中，我们讨论了从映射和非连续系统估计LLE的可能性，并展示了我们初步研究的结果。
