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Continuation of Nonsmooth Bifurcations in Filippov Systems using Singular Point Tracking
by
Ivan Arango
EAFIT University
Coauthors: John Alexander Taborda
National University of Colombia
In this article, we propose a novel method for continuation of nonsmooth bifurcations in discontinuous piecewise smooth autonomous systems or Filippov systems which we are denominated: Singular Points Tracking (SPT). Information of the discontinuity boundary (DB) is used to find relation between points that compose the discontinuity and the dynamics of the system. We use a classification recently proposed of points and events on DB to characterize the system. This classification has allowed to identify the characteristics of general behavior on the discontinuity boundary without to run a simulation or integration algorithms. Using the SPT method the changes in the parameters of the vectorial fields are done and then the existence of bifurcations on the discontinuity boundary is determined. With the results, it is possible to establish procedures to make continuation processes especially by the borders of areas or volumes defined by the singular points. Thirty-five different singular points are considered. The SPT method can be extended to impact systems and continuos nonsmooth systems. The SPT method has two principal advantages: one, to determine the general behavior of the system without necessity of big number of integrations with different initial conditions, i.e., the method does not require the construction of the phase portrait. The other advantage is low computing time in the continuation problem. First in this article we present a summary of the results of the classification of the types of points on the discontinuity boundary, afterward relationship among sets of points, its order and the existence of each dynamics. Next it is presented the continuation method to find areas of existence of different phenomena. Finally, the TSP method is proven and compared with the results of published works.
Date received: December 15, 2007
Copyright © 2007 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cavk-52.