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Entropy 2014, 16(4), 1949-1968; doi:10.3390/e16041949

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

1
Magma Ingeniería, Programa de Ingeniería Electrónica, Facultad de Ingeniería, Universidad del Magdalena, Santa Marta D.T.C.H., 2121630, Colombia
2
Mecatrónica, Programa de Ingenieria Mecánica, Facultad de Ingeniería, Universidad EAFIT, Medellín 7023, Colombia
*
Author to whom correspondence should be addressed.
Received: 2 September 2013 / Revised: 20 March 2014 / Accepted: 20 March 2014 / Published: 1 April 2014
(This article belongs to the Special Issue Dynamical Systems)

Abstract

In this paper, we propose a novel strategy for the synthesis and the classification of nonsmooth limit cycles and its bifurcations (named Non-Standard Bifurcations or Discontinuity Induced Bifurcations or DIBs) in n-dimensional piecewise-smooth dynamical systems, particularly Continuous PWS and Discontinuous PWS (or Filippov-type PWS) systems. The proposed qualitative approach explicitly includes two main aspects: multiple discontinuity boundaries (DBs) in the phase space and multiple intersections between DBs (or corner manifolds—CMs). Previous classifications of DIBs of limit cycles have been restricted to generic cases with a single DB or a single CM. We use the definition of piecewise topological equivalence in order to synthesize all possibilities of nonsmooth limit cycles. Families, groups and subgroups of cycles are defined depending on smoothness zones and discontinuity boundaries (DB) involved. The synthesized cycles are used to define bifurcation patterns when the system is perturbed with parametric changes. Four families of DIBs of limit cycles are defined depending on the properties of the cycles involved. Well-known and novel bifurcations can be classified using this approach.
Keywords: bifurcation theory; nonsmooth bifurcations; piecewise-smooth systems; limit cycles; topological equivalence bifurcation theory; nonsmooth bifurcations; piecewise-smooth systems; limit cycles; topological equivalence
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This is an open access article distributed under the Creative Commons Attribution License (CC BY 3.0).

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MDPI and ACS Style

Taborda, J.A.; Arango, I. Topological Classification of Limit Cycles of Piecewise Smooth Dynamical Systems and Its Associated Non-Standard Bifurcations. Entropy 2014, 16, 1949-1968.

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