Efficient and Accurate Methods for the Computation of Lyapunov Characteristic Exponents for n-Dimensional Continuous Dynamical Systems

N维动力系统Lyapunov特征指数的有效而准确的计算方法摘要这一建议旨在发展有效和准确地确定离散和连续动力系统的Lyapunov特征指数的计算方法。由于目前的方法是不充分的，我们的目标是开发新的计算技术，可以用于一般的n维系统。因此，将致力于发展求解非线性微分方程组的变分方程的新技术。将采用的方法将以数学分析为基础，因此开发的算法将既高效又准确。将对几种不同的方法进行调查和相互比较。将进行误差分析。信息的传播将包括开发可直接从互联网下载的计算机代码。在这项研究工作中开发的方法将在应用科学和工程的许多领域中得到应用。在许多结构系统中，如大跨度悬索桥，在生物力学系统中，如用于运动的中央模式发生器，以及在快速发展的保密通信系统领域，非线性动力学是一个重要特征。因此，这项研究的结果将对现实生活中的几个应用领域产生相当大的直接影响。

Efficient and Accurate Methods for the Computation of Lyapunov Characteristic Exponents for n-Dimensional Dynamical SystemsAbstractThis proposal aims to develop computational methods for the efficient and accurate determination of the Lyapunov characteristic exponents of discrete and continuous dynamical systems. Since current methods are inadequate, our goal is to develop new computational techniques that can be used for general, n-dimensional systems. Thus, the development of new techniques for solving the variational equation of a system of nonlinear differential equations will be undertaken. The methods to be employed will be underpinned by mathematical analysis so that the algorithms developed will be both efficient and accurate. Several different methods will be investigated and compared with one another. Error analysis will be undertaken. Dissemination of the information will include the development of computer codes that are directly downloadable from the internet. The methods developed in this research effort will have applications in numerous areas of applied science and engineering. Nonlinear dynamics is a key feature in many structural systems, such as long span suspension bridges; in biomechanical systems, such as Central Pattern Generators for locomotion; and in the rapidly developing field of secure communication systems. Hence, the results of this research will have a considerable and direct impact on several real-life areas of application.