A new way to compute the Lyapunov characteristic exponents for non-smooth and discontinues dynamical systems

Abstract

One of the most important problems of nonlinear dynamics is related to the development of methods concerning the identification of the dynamical modes of the corresponding systems. The classical method is related to the calculation of the Lyapunov characteristic exponents (LCEs). Usually, to implement the classical algorithms for the LCEs calculation, the smoothness of the right-hand sides of the corresponding equations is required. In this work, we propose a new algorithm for the LCEs computation in systems with strong nonlinearities (these nonlinearities cannot be linearized) including the hysteresis. This algorithm uses the values of the Jacobi matrix in the vicinity of singularities of the right-hand sides of the corresponding equations. The proposed modification of the algorithm is also can be used for systems containing such design hysteresis nonlinearity as the Preisach operator. (Thus, this modification can be used for all members of the hysteresis family.) Moreover, the proposed algorithm can be successfully applied to the well-known chaotic systems with smooth nonlinearities. Examples of dynamical systems with hysteresis nonlinearities, such as the Lorentz system with hysteresis friction and the van der Pol oscillator with hysteresis block, are considered. These examples illustrate the efficiency of the proposed algorithm.