Detecting Sliding Areas in Three-Dimensional Filippov Systems using an Integration-Free Method IVÁN ARANGO Universidad EAFIT Department of Mechanics Engineering Kra 42d 50 sur 40 Medellín Colombia JOHN ALEXANDER TABORDA Universidad Nacional de Colombia Department Electronics Engineering Campus La Nubia Manizales Colombia Abstract: In this paper, we detect sliding areas in three-dimensional (3D) Filippov systems using an integration-free method denominated Singular Point Tracking (SPT). Many physical applications in engineering can be modelled as Filippov systems. Sliding dynamics due to nonsmooth phenomena as friction, hysteresis or switching are in- herent to Filippov systems. The analysis of sliding dynamics has many mathematical and numerical difficulties. Several well-known numerical problems can be avoid using integration-free methods. In this paper, we extend the SPT method to 3D Filippov systems. In comparison with the 2D case, the evaluation of the vector fields on the discontinuity boundary (DB) should be reformulated and new dynamics on DB should be characterized. Key–Words: Bifurcation theory, numerical analysis, non-smooth bifurcations, Filippov systems. 1 Introduction Nonsmooth characteristics as sliding, switching or impact cause many mathematical and numerical dif- ficulties in modeling, simulation and analysis stages [1], [2], [3]. The bifurcation theory and the piecewise- smooth approach have been used widely to analyze the dynamics of nonsmooth systems as power con- verters [4], [5], [6], [7], friction oscillator [8], [9], [3], [10], [11] or impact oscillators [1], [12], [13], [14]. In this paper we concentrate in Filippov systems of three dimensions (3D). A lot of papers have been restricted to 2D Filippov systems or Filippov systems not involving sliding motion because of the analysis is more simplified [9]. When the sliding motion ex- ist and the Filippov system is 3D then the analysis is more complicated. The analysis of sliding dynamics has many math- ematical and numerical difficulties. The number of specialized software in nonsmooth dynamics is re- duced [15], [16]. In [17] and [18], two toolboxes are presented for analysis and continuation of nonsmooth bifurcations in Filippov systems. Several well-known numerical problems can be avoid using integration- free methods. In this paper, we analyze sliding dy- namics in three-dimensional (3D) Filippov systems using an integration-free method denominated Singu- lar Point Tracking (SPT). We use the evaluation of the vector fields on the discontinuity boundary (DB) to analyze the dynam- ics of the Filippov systems without integration of the ODE sets or integrating only in the points computed for the SPT algorithm. We apply a classification of points and events on DB recently proposed in [19], [20], [21]. In comparison with the 2D case, the evaluation of the vector fields on the discontinuity boundary (DB) should be reformulated and new dynamics on DB should be characterized. The existence conditions of the crossing areas, sliding areas and singular sliding lines are formulate using Boolean-valued functions B(.) based on integration-free geometric criterions. These conditions are easily programmable and they can be used directly in the detection of nonsmooth bi- furcations. The Boolean-valued functions B(.) return TRUE or FALSE when their arguments are evaluated. The logical functions are composed of logical connectives: AND, OR and NOT denoted by ∧, ∨ and ¬, respec- tively. The paper is organized as follows. In section II we present the background concepts of Filippov sys- tems and the SPT numerical method. The type of ar- eas on DB and singular lines on DB are summarized in the sections III and IV, respectively. The basic dy- namics on DB are presented in the section V while an illustrative example is presented in section VI. Finally, the conclusions are presented in the section VII. 2nd EUROPEAN COMPUTING CONFERENCE (ECC’08) Malta, September 11-13, 2008 ISSN:1790-5109 160 ISBN: 978-960-474-002-4