Fast and simple Lyapunov Exponents estimation in discontinuous systems
- PDF / 773,265 Bytes
- 15 Pages / 481.89 x 708.661 pts Page_size
- 19 Downloads / 208 Views
https://doi.org/10.1140/epjst/e2020-900275-x
THE EUROPEAN PHYSICAL JOURNAL SPECIAL TOPICS
Regular Article
Fast and simple Lyapunov Exponents estimation in discontinuous systems M. Balcerzaka , T. Sagan, A. Dabrowski, and A. Stefanski Division of Dynamics, Lodz University of Technology, ul. Stefanowskiego 1/15 Lodz, Poland Received 6 December 2019 / Accepted 8 June 2020 Published online 28 September 2020 Abstract. Typically, to estimate the whole spectrum of n Lyapunov Exponents (LEs), it is necessary to integrate n perturbations and to orthogonalize them. Recently it has been shown that complexity of calculations can be reduced for smooth systems: integration of (n-1 ) perturbations is sufficient. In this paper authors demonstrate how this simplified approach can be adopted to non-smooth or discontinuous systems. Apart from the reduced complexity, the assets of the presented approach are simplicity and ease of implementation. The paper starts with a short review of properties of LEs and methods of their estimation for smooth and non-smooth systems. Then, the algorithm of reduced complexity for smooth systems is shortly introduced. Its adaptation to non-smooth systems is described in details. Application of the method is presented for an impact oscillator. Implementation of the novel algorithm is comprehensively explained. Results of simulations are presented and validated. It is expected that the presented method can simplify investigations of non-smooth dynamical systems and support research in this field.
1 Introduction Lyapunov Exponents (LEs) are the measures of the sensitivity of a dynamical system to a perturbation of its initial conditions: they indicate average, exponential rates of expansion or contraction of an infinitesimal disturbance of the initial state [1]. As such, they are widely used in analysis of systems’ dynamics. The negative sum of all the LEs in the spectrum is a necessary and sufficient condition for existence of an attractor. Positive value of the largest Lyapunov Exponent (LLE) indicates that the attractor is chaotic, whereas two positive LEs are the sign of hyper-chaos. If the whole spectrum of LEs is negative, then the limit set is a stable equilibrium point. When one LE equals 0 and the rest of them are negative, a stable limit cycle is present. Finally, existence of K LEs equal 0, with the remaining ones negative, confirms that the attractor is a K -torus [1]. Last but not least, values of the LEs are connected with the fractal dimension of the attractor [2]. Note that the Lyapunov Exponents being the topic of this paper are often called global [3], as they predict behavior of the system as the time goes to infinity. Their values remain the same for any a
e-mail: [email protected]
2168
The European Physical Journal Special Topics
initial conditions within the same basin of attraction [3]. In contrast, local Lyapunov exponents predict behavior of the system in a finite time [3] and thus may attain different values for various initial conditions located in the same basi
Data Loading...
