Determining Lyapunov exponents of non-smooth systems: Perturbation vectors approach

•It is possible to easily estimate all Lyapunov Exponents of a discontinuous system.•Possibility of generating a trajectory from a selected point is the only requirement.•The presented method works for both: maps and continuous-time systems.•Application for mechanical systems, including impact and stick-slip, is presented. This paper presents a novel, simple method of Lyapunov Exponents (LEs) spectrum estimation for non-smooth and discontinuous systems. The presented algorithm works for continuous-time dynamical systems, as well as for discrete maps. Its simplicity and flexibility enables to estimate LEs of complex, discontinuous systems in an easy manner. The paper starts with a comprehensive review of the state-of-art methods of the LEs computation for non-smooth systems. Then, the novel algorithm is introduced. Further on, examples of its use are provided. The presented applications include non-smooth and discontinuous maps, as well as continuous-time mechanical systems with discontinuities induced by impacts or by dry friction. The paper confirms that the method is simple, effective and robust. The presented algorithm can facilitate research in a large class of discontinuous systems, including stick-slip oscillators and many others.