Bifurcations and Sequences of Elements in Non-Smooth Systems Cycles

This article describes the implementation of a novel method for detection and continuation of bifurcations in non- smooth complex dynamic systems -- The method is an alternative to existing ones for the follow-up of associated phe- nomena, precisely in the circumstances in which the traditional ones have limitations (simultaneous impact, Filippov and first derivative discontinuities and multiple discontinuous boundaries) -- The topology of cycles in non-smooth sys- tems is determined by a group of ordered segments and points of different regions and their boundaries -- In this article, we compare the limit cycles of non-smooth systems against the sequences of elements, in order to find patterns -- To achieve this goal, a method was used, which characterizes and records the elements comprising the cycles in the order that they appear during the integration process -- The characterization discriminates: a) types of points and segments; b) direction of sliding segments; and c) regions or discontinuity boundaries to which each element belongs -- When a change takes place in the value of a parameter of a system, our comparison method is an alternative to determine topo- logical changes and hence bifurcations and associated phenomena -- This comparison has been tested in systems with discontinuities of three types: 1) impact; 2) Filippov and 3) first derivative discontinuities -- By coding well-known cy- cles as sequences of elements, an initial comparison database was built -- Our comparison method offers a convenient approach for large systems with more than two regions and more than two sliding segment