Bifurcations and Sequences of Elements in Non-Smooth Systems Cycles

Abstract

<p class="MsoNormal" style="text-align:justify;"> <span lang="EN-US">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 phenomena, 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 systems 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 topological 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 cycles 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 segments.</span> </p

Topological Classification of Limit Cycles of Piecewise Smooth Dynamical Systems and Its Associated Non-Standard Bifurcations

“…Characteristic points on DB can be identified using the Singular Point Tracking (SPT) method explained in previous works [17][18][19]. The set of characteristic points in a limit cycle constitutes the topological point set while the sequence of characteristic points in a limit cycle constitutes the topological point sequence ( ).…”

“…In [18] tools to discriminate 42 singular and special points including the segments of orbit belonging to different regions or DBs were characterized and developed. Second, in [19] an operative sketch of a numeric tool to implement the methodology in order to work with systems having simultaneous the three types of discontinuity present in Piecewise Smooth Dynamical Systems-impact, Filippov and first derivative discontinuities-was presented. The results offer a convenient approach for large systems with more than two regions and more than two sliding segments.…”