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In the real world there are many dynamical systems with discontinuities. Among them mechanical systems with dry friction are among most important ones. In such cases, the linearization of the equations of motion must be accompanied by a clear statement of the conditions and the transition functions while the trajectory is passing through the discontinuity. Another class of methods for the LEs calculation employs reduction of the dynamics of the phase flow determined in the N-dimensional phase space to a lower dimensional discrete map, e.g., a Poincaré map or an impact map, a local map etc. Then, the Lyapunov exponents of such a mapping are determined using classical approaches. The main application problem here lies in defining the Jacobi matrix of the mapping, where consecutive iterations are not explicitly defined by the known difference equation but they are reconstructed from the flow.
In this paper we apply the method of estimation of the Jacobi matrix using small perturbations of the initial vectors. For any N dimensional time-continuous system a N-1 dimensional map U(x) can be defined. For the discrete map system, direct calculation of Jacobian matrices is impossible, because transition functions from previous to next state are not known. Every next iteration is reconstructed numerically from the differential equations describing the investigated system. Therefore, Jacobian matrices have been evaluated with a method of small perturbations. Numerically, they may be obtained from the solution of linearized equations along the trajectories. The main problem here, accompanying the numerical implementation, is the requirement of very high precision of the trajectory simulation, especially in the presence of the discontinuity.
Summing up, the presented method allows us to determine the Lyapunov exponent of non-smooth friction oscillator using the method of evaluation of the Jacobi matrices from small vector perturbation. The developed method will be useful in quantifying, predicting and understanding chaos in nonsmooth discontinuous systems for which the straightforward calculation of the Lyapunov exponents is not possible. The approach presented in this paper can be generalized to the higher dimensional (N > 3) systems.