Fast and simple Lyapunov Exponents estimation in discontinuous systems

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]  A. Dabrowski,et al.  Estimation of the largest Lyapunov exponent from the perturbation vector and its derivative dot product , 2012 .

[2]  A. Dabrowski,et al.  The fastest, simplified method of Lyapunov exponents spectrum estimation for continuous-time dynamical systems , 2018, Nonlinear Dynamics.

[3]  James A. Yorke,et al.  Spurious Lyapunov Exponents Computed from Data , 2007, SIAM J. Appl. Dyn. Syst..

[4]  Śmiechowicz,et al.  Lyapunov Exponents of Early Stage Dynamics of Parametric Mutations of a Rigid Pendulum with Harmonic Excitation , 2019, Mathematical and Computational Applications.

[5]  I. Shimada,et al.  A Numerical Approach to Ergodic Problem of Dissipative Dynamical Systems , 1979 .

[6]  Jianping Li,et al.  Determining the spectrum of the nonlinear local Lyapunov exponents in a multidimensional chaotic system , 2017, Advances in Atmospheric Sciences.

[7]  Andrzej Stefański,et al.  Estimation of the largest Lyapunov exponent in systems with impacts , 2000 .

[8]  J. Wojewoda,et al.  Spectrum of Lyapunov exponents in non-smooth systems evaluated using orthogonal perturbation vectors , 2018 .

[9]  Marek Balcerzak,et al.  Determining Lyapunov exponents of non-smooth systems: Perturbation vectors approach , 2020 .

[10]  P. Müller.  Calculation of Lyapunov exponents for dynamic systems with discontinuities , 1995 .

[11]  G. Benettin,et al.  Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; A method for computing all of them. Part 2: Numerical application , 1980 .

[12]  Qishao Lu,et al.  A method for calculating the spectrum of Lyapunov exponents by local maps in non-smooth impact-vibrating systems , 2006 .

[13]  J. K. Hammond,et al.  The instantaneous Lyapunov exponent and its application to chaotic dynamical systems , 1998 .

[14]  G. Benettin,et al.  Kolmogorov Entropy and Numerical Experiments , 1976 .

[15]  Xingyuan Wang,et al.  A novel method based on the pseudo-orbits to calculate the largest Lyapunov exponent from chaotic equations. , 2019, Chaos.

[16]  J. Yorke,et al.  Chaotic behavior of multidimensional difference equations , 1979 .

[17]  Tomasz Kapitaniak,et al.  Estimation of the dominant Lyapunov exponent of non-smooth systems on the basis of maps synchronization , 2003 .

[18]  Marconi Kolm Madrid,et al.  A method for Lyapunov spectrum estimation using cloned dynamics and its application to the discontinuously-excited FitzHugh–Nagumo model , 2012 .

[19]  V. I. Oseledec.  A multiplicative ergodic theorem: Lyapunov characteristic num-bers for dynamical systems , 1968 .

[20]  Artur Dabrowski,et al.  The largest transversal Lyapunov exponent and master stability function from the perturbation vector and its derivative dot product (TLEVDP) , 2012 .

[21]  Iberê L. Caldas,et al.  Controlling chaotic orbits in mechanical systems with impacts , 2004 .

[22]  M. Oestreich,et al.  Bifurcation and stability analysis for a non-smooth friction oscillator , 1996 .

[23]  G. Benettin,et al.  Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory , 1980 .

[24]  Artur Dabrowski,et al.  Estimation of the largest Lyapunov exponent-like (LLEL) stability measure parameter from the perturbation vector and its derivative dot product (part 2) experiment simulation , 2014 .

[25]  F. Takens.  Detecting strange attractors in turbulence , 1981 .

[26]  A. Stefanski.  Determining thresholds of complete synchronization, and application , 2009 .

[27]  Ugo Galvanetto.  Numerical computation of Lyapunov exponents in discontinuous maps implicitly defined , 2000 .

[28]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .