Abstract
This paper covers application of the novel method of Lyapunov exponents (LEs) spectrum estimation in non smooth mechanical systems. In the presented method, LEs are obtained from a Poincaré map. By analysing the map instead of the full trajectory, problems with transition of perturbations through discontinuities can be avoided. However, the explicit formula of the map is usually not known. Therefore, the Jacobi matrix of the map is estimated using small perturbations of the initial point. In such a manner, direct calculation of the Jacobi matrix can be avoided. The article provides a detailed description of the method accompanied by clear schemes. The algorithm of Jacobi matrix estimation is elaborated and an example is given. Efficiency of the method is confirmed by a numerical experiment. The mechanical oscillator with impact has been simulated. Bifurcation diagrams and Lyapunov exponents graphs have been generated. It has been shown that the method provides values of the whole Lyapunov exponents spectrum with high accuracy.
Highlights
Lyapunov exponents (LEs) are one of the most useful criteria for determining the stability of solutions of dynamical systems in local [1] and global sense [2]
It can be noticed that the bifurcation diagram presented in the Fig. 3 is consistent with the Lyapunov exponents graph
As one can notice in the graph, the sum of both Lyapunov exponents from the spectrum is equal to
Summary
Lyapunov exponents (LEs) are one of the most useful criteria for determining the stability of solutions of dynamical systems in local [1] and global sense [2]. In order to apply this method, linearization of the equations of motion must be accompanied by a clear statement of the transition conditions while the trajectory is passing through the discontinuity Another class of methods for the LEs calculation employ reduction of the dynamics of the phase flow in the k-dimensional phase space to a discrete map of a lower dimension, such as a Poincaré map [12], an impact map [22], a local map [23] or a transcendental map [24]. The main problem in application of these algorithms is determination of the Jacobi matrix of the mapping, where consecutive iterations are not explicitly defined by a known difference equation, but they are reconstructed from the flow The example of such a map-based algorithm is the method by Galvanetto [28], who applied implicitly defined maps for calculation of the two largest LEs of the 2-DoF stick–slip system.
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