TITLE:
Bifurcations and Sequences of Elements in Non-Smooth Systems Cycles
AUTHORS:
Ivan Arango, Fabio Pineda, Oscar Ruiz
KEYWORDS:
Bifurcation Sequences; Non-Smooth Systems; Limit Cycles; Dynamic Systems
JOURNAL NAME:
American Journal of Computational Mathematics,
Vol.3 No.3,
August
21,
2013
ABSTRACT: 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.