American Journal of Computational Mathematics, 2013, 3, 222230 http://dx.doi.org/10.4236/ajcm.2013.33032 Published Online September 2013 (http://www.scirp.org/journal/ajcm) Bifurcations and Sequences of Elements in NonSmooth Systems Cycles Ivan Arango1, Fabio Pineda1, Oscar Ruiz2 1Mechatronics and Machine Design Group, Universidad EAFIT, Medellin, Colombia 2Laboratorio de CAD/CAM/CAE, Universidad EAFIT, Medellin, Colombia Email: oruiz@eafit.edu.co Received May 24, 2013; revised July 1, 2013; accepted July 12, 2013 Copyright © 2013 Ivan Arango et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 followup of associated phe nomena, 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 nonsmooth sys tems 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 nonsmooth 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 topo logical 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 wellknown cy cles as sequences of elements, an initial comparison database was built. Our comparison method offers a convenient ap proach for large systems with more than two regions and more than two sliding segments. Keywords: Bifurcation Sequences; NonSmooth Systems; Limit Cycles; Dynamic Systems 1. Introduction Physical systems can often operate in different modes, and as the time of the transition from one mode to an other mode is small, the transition is considered as in stantaneous [1]. Events such as impact, dry friction, back lash, hysteresis, saturation and commutation carry a dis continuity or sudden change. Therefore, they can be mod eled declaring at least two modes. Each mode is repre sented by differential equation or mixes of differential and difference equations. The mathematical modeling of these systems switches between different modes and they are classified as piecewisesmooth or nonsmooth system. Piecewisesmooth systems may be classified according to the degree of discontinuity that the orbits and vector fields present [1]. An updated classification by [2] dis cusses systems with three degrees of smoothness. In the zero level, one has jumps in the state variables. They are typically systems with impact, where the phenomenon is modeled assuming no deformation and a negligible im pact time [3]. In the first degree of smoothness, we have systems described by differential equations with discon tinuous right hand terms (Filippov systems) [4]. In these cases the vector field is discontinuous in the switching Boundary, as usual in mechanical systems with dry fric tion [5]. The second degree of smoothness, includes sys tems with continuous vector fields but discontinuities in the first derivative of the vector field. As an example for second degree, we might consider a mechanical system with a single mass, spring, damping element and limiting elastic support [6]. In general, a discontinuity in the ith derivative implies that the system is classified as being i + 1 degree of smoothness. Nonstandard bifurcations in nonsmooth systems have been intensively studied [69]. But, there are only mathe matical tools to analyze phenomena in 2D or 3D systems with two vector fields and one discontinuity boundary [10,11]. The names assigned to the bifurcations vary ac cording to the researcher. For example, [2] is used Graz ing, Switching, Crossing and Multisliding. For the same C opyright © 2013 SciRes. AJCM
I. ARANGO ET AL. 223 bifurcations, in [12] is used Touching, Bucking, Crossing and Adding. Other sliding bifurcation types, recently re ported in [8], have been characterized in systems with two DBs. Those bifurcations have been called Exchang ing, Sticking Disappearance and Nonsmooth Fold. Article Outline. This article is organized as follows. Section 2 explains the notation and symbols used. Sec tion 3 summarizes the solutions for the types of nonsmooth systems. Section 4 describes the wellknown bifurcations as sequences of elements. Section 5 analyzes the proce dure of identification and comparison of the elements of the cycles versus the elements of an integration. Section 6 concludes the article. 2. Notation and Symbology for Points in the DB The study of Nonsmooth systems includes more infor mation than a smooth system. The proposed method is based on the information of each element of the cycle. Therefore, we had to introduce a notation to see all the information of the points, segments and orbits. The in formation should be fully contained inside the textual or graphical symbols assigned to each element. Some distin guished symbols follow. x: State variable vector, with () 12 ,,, n xx x=. Zi: th smooth region of the space state. i α: Parameter of the physical system . () α ∈ () , i α Fx : Vector field on region i . DB: Discontinuity Boundary. Σij: Discontinuity Boundary between regions i and . () :, n ij ijij ZZ xH α Σ== ∈=x0 : : . () , ij α x ( , ij H α xx H Smooth scalar function defining the between regions and . . DB ij () 1 ,: nn ij H α +→x ),( ) Gradient of xHij. () () () 1 ,,HH αα ∂∂xx ,,, ij ij ij n Hxx α = ∂∂ xx. : i − Ω th component of i before impact. : i + Ω th component of i after impact. γ: Impact restitution coefficient I γ −+ =Ω Ω . : i x G Point at the end of th integration step. i () , ij α x i : Vector field that acts on the DB between regions and , for sliding. j Cycle equations include indicators, separators and ele ments (for cycles: points or segments). Cycles are identi fied with a letter C accompanied by a subscript number (e.g. 4: 4th cycle). If the cycle contains sliding seg ments they appear as superscript preceding the C letter (e.g. 5: cycle 5 has sliding segments). In the equations, the symbol Φ is used to represent a com posed segment, determined by a sequence of points of a common type (e.g. 5: a composed segment in region 5). The points are identified with the letter with superindices (− or +) indicating whether the point is an C S SC Φ Ω initial (−) or endpoint (+) of a sliding segment S. The symbol/notes a separator between consecutive elements. The indicator shows that the elements of the equa tion in an evolution are continuously repeated (e.g. : segment i in region is continuously repeated). Equations that describe the elements of Bifurcations (cy cles) are identified by the symbol Ò Ò i Φ∕ Φi . Sliding bifurca tions are identified with a superscript that precedes the S symbol and an alphabetic subscript that indi cates the bifurcation type (e.g. S c is a sliding crossing bifurcation). 3. Background of the NonSmooth Solution Typically, Nonsmooth systems are modeled as piece wisesmooth systems (PWS) where the state space contains four kinds of spaces: Smooth Zones, undefined Zones associated to regions behind of impact boundaries, Di s co ntin uity Boundaries with dynamics represented by convex combinations of the solution of the ODEs of each vector field and Impact boundaries with dynamic repre sented by algebraic equations. Equation (1) shows the statespace representation of the simplest nonsmooth system with the three types of dynamics. () () () {} () () {} () () () {} 1 1 ,, ,if :,0 ,if :(,)0 ,if :,0 ,if :, n ii n jj n ij n I ijk ZH ZH H H αα αα αα αα − − ∈=∈ > ∈=∈< =∈Σ =∈= ∈Σ= ∈= Fx xxx Fx xxx xGxx xx Ixxxx 0 (1) In Equation (1), i and F are smooth vector fields; i and are the corresponding regions and is a parameter. The state space regions are determined by the smooth scalar function and the boundary of impact of 1 α ∈ ( , α x ) H i or regions is determined by the scalar function . () α ,x I H 3.1. Zero Degree of Smoothness Systems In electromechanical Nonsmooth systems the impact phe nomena is highly dynamical, then can be declared using an algebraic relation due to the impact time is negligible in relation with the time constant of mechanical systems. In this relation, is the restitution coefficient and () − Ω , () + Ω are respectively the approximation and bounce speed. () () () () () ,II I I α γ +− + Ω=Ω =Ω=Ω x − (2) The first row of Equation (2) expresses that the posi tion before and after the impact are identical. The second one expresses that the rebound velocity (+) equals the Copyright © 2013 SciRes. AJCM
I. ARANGO ET AL. 224 impact velocity (−) multiplied by the restitution coeffi cient . 3.2. First Degree of Smoothness Systems Filippov systems, a set of firstorder ordinary differential equations with a discontinuous righthand side are a sub class of discontinuous dynamical systems. The trajectory of a sliding orbit remaining partially inside the disconti nuity boundary may be calculated by the Filippov convex method as in [4]. Systems with multiple regions and DBs are treated in [13], where an extended equation for Filippov systems is described in order to deal with the intersection of several discontinuity surfaces. In Filippov systems, between i and in the dis continuity boundary, we assume that there is a region ij , which are a vector field of dimension con formed by three types of points: crossing C, sliding and singular ( , and each one with subtypes. The scalar function is used to determine the point type, according to the geometric condition of the vectors in the point of analysis. Equation (3) de scribes the geometric conditions of an sliding point. Equation (4) helps to determine which is the nature of the point, according to the value of and the neighbor ing vector fields at . Σ ( S Ω 1n− ) ( σ x ( Ω ) ) ) SO Ω ( σ x x ) x ()()( )()( ) ,,, , xix j HF HF σα =xxxxx α (3) () () () () ()() () () ,: 0 0, 0 0 ij C SOx ji S HF F σ σ σ ∈Σ Ω> Ω=∧− = Ω< x x xxxx x ) ) (4) Crossing points ( C, characterized by , are points which the evolution of the trajectory will not remain in the . Instead, it crosses from the region in which has been previously evolving to the other. Ω () 0 σ >x DB Singular sliding points , characterized by σ (x) = 0, are points having the associated vectors with the normal component ( SO Ω () , i Fx equal to 0. This is because the vectors are tangential to the DB or vanishes. At such points: a) i and F F are tangent to the DB; b) either i or F F F vanishes while the other is tangent to the DB; or c) i and F vanish. To avoid the lack of definition of the Filippov solution for these points, in the examples, we adopt the methods presented in [14] which coincide with the topology of the normal forms VV, VI and II presented in [12]. Sliding points are characterized by . When a sliding motion is presented in the discontinuity boundary, the Filippov method gives as a solution a tan gent vector to the DB which is a convex combination , of the vector fields and ( S Ω ,ij ∈Σx (Equation (5)). ()()( )( ,,1 i GF F αλ αλα =+−xx x ) , j (5) () () ()()() ,, ,, , xj xji HF HF F α λαα =− xx xx x (6) is a scalar function defined through the projections of the vector fields in the direction of the normal vector to the discontinuity boundary. According to the direction of the normal components of the vectors, the sliding points are stable (or attractor) , or unstable (or repulsive) (Equation (7)). () ( xx ) ) ) H ( SS Ω ( SU Ω () () () () () () () () , ,0 , :,0 , SSx ixj ij SUx ixj HH x HH Ω>∧ < ∈Σ Ω<∧ > xF xF xFxF 0 0 (7) From Equation (4) the crossing set is open but the sliding set is closed, it is the union of the sliding seg ments, singular points and isolated or special sliding points. In this paper, the terms special points or isolated points refer to points whose neighbor points belong to a different class. Special points define important dynamics in the sliding segments of 2d systems or areas in 3D systems. These points are: a) Equilibria points, in which both vectors i and are attractive, transversal to the and are at the end of two sliding segments pointing each other. b) Quasiequilibria points with both vectors i and DB F attractive transversal or anti collinear and which are at the start of two sliding segments pointing away each other. The contrary case have also quasi equilibria points: repulsive, transversal points which are at the end of two sliding segments pointing each other. c) Boundary equi libria points, in which one of the vector i or vanishes. d) Tangent points, in which one of the vectors i or is tangent to the DB. [15] is done a more strict classification giving the characterization of 42 types of points with the objective of differentiate topo logies in order to detect bifurcations. 3.3. Second Degree of Smoothness Systems The second degree of smoothness systems are represent ed as variable structure systems having different dynam ics in each zone or region. The dynamics of the system does not allow sliding or stops on the boundary zone, all points are crossing and hence, there is not a particular dynamics defined in the limit zone, instead there is a change of the region equations set. ) ) () 0 σ <x ( ,G α xi F F at a point 4. Sequences of Well Known Bifurcations In this and the following sections, we will present the Copyright © 2013 SciRes. AJCM
I. ARANGO ET AL. Copyright © 2013 SciRes. AJCM 225 ) () ( 123 ss GCC C β = ) s (11) cycles of the most referenced sliding bifurcations as se quences of elements. In each cycle are presented the con stituent elements assuming that its presence was detected, in the same order, in the evolution of a dynamical system. In next equations the symbol is used to represent segments composed by the same type of point. Arrows indicate the direction of the sliding segments related to the DB. Φ Impact systems also present grazing bifurcations. An orbit that is evolving in a region, due to a change in a parameter, makes contact with a boundary in only one point. This point has approximation speed equal to zero. Consequently, the bouncing speed is also zero. If the physical parameter continues changing, the approxima tion and rebound points separate. The corresponding cy cles are: 4.1. Grazing Bifurcation The Grazing Bifurcation ( s G occurs in the following sequence of changes. First, there is an orbit of a limit cycle 1 evolving in only one of the regions or , without hitting the boundary, as shown in Figure 1(a). Ci j () () () 1 2 3 Ò Ò Ò i i iI i iI I C C C +− +− =Φ =Φ Ω =Φ ΩΩ (12) 1Ò i C=Φ (8) 4.2. Switching Bifurcation Then, when the parameter changes, for example, from 1 to 2, the cycle grows or moves toward the discontinuity and has a tangent contact with the last point of a sliding segment α α α () + Ω. The structure presented cor responds to a type cycle. 2 sC The sequence of changes for a Switching Bifurcation ( s S ) is as follows: the sliding piece of a limit cycle of type 3 grows until it reaches the first point of the sliding segment. See Figure 1(d). The type of struc ture presented, corresponds to a cycle 4. In general, the second cycle always characterizes the bifurcation type and it is only presented for one value of the para meter or a very narrow range in the numerical calculation terms. SC () S − Ω sC () 2Ò s is C+ =Φ Ω (9) Subsequently, as the parameter is moved further, the limit cycle changes again as is depicted in Figure 1(c). The structure presented corresponds to a type cycle. 3 sC () () 4Ò s iss s C− → =Φ ΩΦΩ () 3Ò s is s C+ → =Φ ΦΩ (10) + (13) With a further change in the parameter, the orbit has now three segments: two of them, i and Φ Φ are in two different regions separated by the discontinuity bound ary, and the third piece is on the sliding region moving to the right. See Figure 1(e). The structure presented cor responds to a type cycle. 5 sC The orbit of the limit cycle 3 has now two differ ent pieces: one without touching the discontinuity bound ary and the other one, corresponding to a sliding segment sC → Φ that starts in any intermediate point of the discon tinuity and ends at a tangent point () + Ω. The equation describing the sequence of cycles is: Figure 1. Grazing (a)(c), Switching (c)(e) and Crossing (e)(g) bifurcations.
I. ARANGO ET AL. 226 () , 5Ò ij s iCj ss C+ → =Φ ΩΦΦΩ (14) The equation describing the sequence of cycles is: ()( 345 ssss SCCC β = ) ) (15) 4.3. Crossing Bifurcation A Crossing Bifurcation ( s C occurs when the sliding piece of a cycle 5 gets smaller and smaller. At a parameter value 6, the piece of trajectory SC α Φ hits the sliding region just at the last point of the sliding segment () + Ω. See Figure 1(f). The structure presented corre sponds to a type cycle. 6 sC () , 6Ò ij s iC js C+ =Φ ΩΦΩ (16) As the parameter further changes at some value 7, the limit cycle has now two pieces without sliding. The structure presented corresponds to a type cycle. α 7 sC ,, 7Ò ij ji iC jC C=Φ ΩΦΩ (17) The equation describing the sequence of cycles is: () ( 567 ssss CCCC β = ) (18) 4.4. Adding or Multisliding Bifurcation The sequence of changes for the Adding or Multisliding bifurcation is related to the addition or destruction of a second sliding segment in the discontinuity boundary as is described in [12]. Other sliding bifurcations recently reported are those including more than two discontinuity boundaries that are moving due to variations of a para meter. Those ones were introduced in [8] using an exam ple. 5. The Implementation of the Sequences as a Method of Comparison Next we will describe the tool which were developed to get the results obtained in the previous section. Addi tional to the numerical integrator, there are some data bases, procedures and methods running in parallel. They perform the evaluation of information collected previ ously, and the information acquired in real time, when the system is evolving. These tools are: 5.1. Collection of Points The collection of the values of the points is done in a vector, called vector of states. The new point includes the values of the states, the amount of time since the integration started and the data of the vector fields involved in the dynamics. As shown in Figure 2(a), after each iteration of the numerical integration, one point is added to the vector of states and the graphic of the space states. 5.2. Database of Point Characteristics Each point, additional to the characterization given by the states is classified by the region or DB it belongs. The orientation of the two vector fields for points in the DB determines types as anticollinear, transversal, tangent, also the attractiveness or repulsiveness and the direction relative to the DB. The magnitude of the vectors might tend to zero. The Equation (4) determines if is a crossing or sliding point. Finally the Equation (1), that represents its dynamics indicates if is an impact point. All points and their characteristics are listed in a 2 × 2 array called matrix of points, where the first column is the list of points and each row are the list of attributes that each point should to fulfill [15]. Other points presenting them selves in the evolution belonging only to one region, are the nodes and focus, stable and unstable. 5.3. Recognition of Points From the states of the points and vector fields involved, secondary information is estimated. For a point in the DB it is evaluated if it is impacting or normal. Then it is evaluated if the point is crossing or sliding. If a point is crossing, it is evaluated to which vector field the evolu tion will move. The evolution of sliding points has direc tion tangent to the DB, spanning 42 possible subtypes [15]. Summarizing, each point should match all attributes listed in a row of the point matrix. The detected points are stored in vector of elements (Figure 3). • While the vector of elements is being filled out other functions are debugging the information. Each point in a cell of the vector of elements is compared with the point that was met immediately before. Data of points having equal identity are removed from the vector. Instead, the repetition of points turns the first point in the repetition into a piece of curve of the same type. This procedure is carried out with the objective of avoiding a situation in which the vector is filled or saturated with the same data. • While picking elements for the matrix, events with wrong result can be found and should be corrected. For example, it is impossible to accept the sequence ij ΦΦ∕ because implies a change of region i to . In the change, a crossing point must be found, and an admissible sequence would be ij . Thus, a function to correct the sequences of elements is necessary. In [16] are listed 51 rules to correct errors. ijΦΩ Φ∕∕ 5.4. Database of Cycle Elements Each cycle as presented in the previous section, has a set of elements which could be points or segments of points. Copyright © 2013 SciRes. AJCM
I. ARANGO ET AL. 227 Figure 2. Implementation of Cycle Bifurcation. (a) Process of filling the vector with elements appearing in the numeric integration; (b) Searching process for a specific cycle; (c) Cycle tracking process for bifurcations detection; (d) Cycle continuation process. The order of the elements also determines the cycle. In order to have a wider data base all papers in the literature should be analyzed and the cycles presented must be converted in sequences of elements. The information is stored in a bidimensional array, called matrix of cycles, in which each row are the identities of the elements of a cycle. 5.5. Comparison of Cycles In this step the comparison between the matrix of cycles and the vector of elements is performed. We wish to know whether inside the vector of elements there exists a subvector of consecutive and ordered elements that matches with some row of the matrix of cycles. The se quence in the appearance of cycles (and other dynamics) in this step is recorded in a vector called vector of cycles. The result in the vector of cycles, for a given set of pa rameters, admits the presence of a) equilibrium points, b) limit cycles, and c) chaotic behavior. For timevarying parameters, the system evolution might be a sequence of n cycle types, whose order is dictated by the system nature (Figures 2(b) and (c)). To prevent that a repetition of a cycle be mistaken as a single cycle, a function running in parallel with the inte grator performs the evaluation and the correction. When a subsequence of the vector of elements, beginning in the position 1, is equal to the subsequence beginning in the position 21 nb nbnb l=+ and l is the number of elements of the cycle, it is concluded that a cycle is repeating. A cycle is completed when a sequence of elements is continuously repeated and the time to repeat becomes constant. Let us assume, as illustration, a sequence with a grazing cycle s . After some time , the matrix of elements would contain a cycle with the sequence , which is not correct. () + ΦΩ∕ i ΦΩ∕ () is ++ ∕ 3Γ () is ΦΩ∕∕ () is ΦΩ∕ + If the search is for a specific cycle, the procedure is slightly different. In this case, the number of elements in the cycle under consideration is a date and then it is reserved the same amount of cells to store the elements during the integration process. When a new element Copyright © 2013 SciRes. AJCM
I. ARANGO ET AL. 228 Figure 3. General method of comparison of sequences of cycles and bifurcations. appears, a comparison is carried out until all the elements of the stored cycle are identical to the elements that are picked up from the integration (Figure 2(b) ). 5.6. Change in Parameter and Storing of Cycles When a cycle is already stored in vector of cycles and it is continuously repeating, a programmed disturbance is introduced in a physical parameter, to continue searching the bifurcations. The previous processes are repeated, and recorded in vector of cycles. 5.7. Database of Cycles Sequence Each bifurcation is constituted by three ordered cycles, the first and third are presented for a wide range of the parameter but the second is only presented for a value of the parameter. The information of the bifurcations is then stored in a bidimensional array, called matrix of bifur cations, in which each row are the identities of the three cycles of the bifurcation. 5.8. Comparison of Cycles Sequence The objective of the comparison is to identify if inside the vector of cycles there is a subvector of three con secutive and ordered cycles which matches a row of the bifurcation matrix (Figure 2(c)). Here we are looking for a specific sequence that corresponds to a known bifurca tion. To achieve this, a double comparison must be per formed: the first part is the comparison of elements that forms cycles, and the other part is referred to the com parison of the behavior of cycles in a specific sequence, until a full match is detected. When the phenomenon is poorly understood, the comparison could be used to iden tify sequences of cycles which occur when a parameter is modified within a range. For this purpose, the integrator uses the vector of cycles to store information regarding the cycles which have been found during the time that the method has been active. Each time the integrator de tects a repeated sequence of elements, stores the infor mation of the cycle, and changes the parameter value in order to continue with the next identification. 5.9. Continuation To continue a bifurcation the parameters are adjusted corresponding to the central cycle of a previously de tected bifurcation. Next, two additional parameters are slightly changed as per the rules of continuation. The first parameter is disturbed and the second changes ac Copyright © 2013 SciRes. AJCM
I. ARANGO ET AL. 229 cordingly, to keep the dynamics of the central cycle. This controlled disturbance of the two parameters is repeated, such that it determines a trajectory in a continuationplot. The change of parameters could be done using methods like predictorcorrector described in [17] or [18]. In this cases, the predictive function is the cycle that generates the bifurcation, and the previous and posterior cycles to the bifurcation are used for correction. Figure 2(d) shows an example of how is used the method of comparison. The first step is a sensibility analysis that indicates to which cycle, the system evolves when the parameters are increased or decreased. For example, the bifurcation 2 has a sequence of cycles 5 . Assume that a direct proportional sensibility exists for parameter 1. This implies that a small increment in the parameter value tends to change the cycle into and a small decrement tends to change the cycle into 3. Changing 1, the cycle 4 is obtained. Then, the second parameter 2 is decreased (in this case the initial point has a high value). After the change in parameter 2, the cycle 4 changes to 3 or to 5. In the first case, the continuation algo rithm increases 1 until the cycle type 4 is found again. In the second case, the algorithm acts conversely. The process is continuously iterated until the prescribed final value of parameter is reached. SC α () ( 34 SSS CCC 5 SC SC SCSC α ) α α SC α SC S α C 2 Two objectives of an application for automatic bifur cation detection are: 1) to perform the detection task without a close supervision; and 2) to track bifurcations through continuation. The procedures developed here can be used to achieve these goals. 6. Conclusions This article presents an alternative method for detecting bifurcations of limit cycles in nonsmooth systems. We focused on complex systems, which defy boundaryvalue methods. The comparison method, reported in this article, is not intended to focus in the same achievements of other methods. Instead, it addresses open issues left by them, such as multiple sliding segments and discontinu ity boundaries (DB). The comparison method differs from other approaches in the identification and manipu lation of the system information. While the methods in [10,11] consider a system as one entity to be solved by a group of equations, the comparison method uses previ ously collected information in a data base of points, cy cles and bifurcations. This information allows compari sons and decision making. To enable the method for nonsmooth systems, the cases when the evolution crosses the DBs of systems having simultaneously the three degrees of smoothness (impact, Filippov and first derivative discontinuities) was analyzed. To achieve the goal was used the method that characterizes and records the elements comprising the cycles in the order they ap pear in the integration process. The cycles were charac terized as sequences of elements (points and segments). It must be noticed that the sequence of cycles has the topological changes (e.g. bifurcations) implicit. Some of the types of data considered as topological characteristic and collected during the evolution are: a) number of ele ments of the cycle; b) order in which the cycle elements are generated; c) position of the sliding elements in the sequence of cycle generation; d) way (e.g. extreme or interior) in which the cycle reaches and leaves the sliding segment; e) discontinuity boundary to which the element belongs; f) direction (CW, CCW) in which the cycle evolves. In this article we also report a textual notation to describe the elements of the cycles. The comparison method is also able to handle continuation of sliding bifurcations. The method of comparison could be implemented us ing tools of the sequence theory, suffixtrees and string matching, which offer procedures to drive a large number of elements and allow us to discriminate subsets with low computing time investment. The procedure of compari son fulfills the two tasks required by an application for automatic bifurcations detection: perform the detection task without a closed supervision and track bifurcations through continuation. REFERENCES [1] M. Amorin, S. Divenyi, L. Franca and H. I. Weber, “Nu merical and Experimental Investigations of the Nonlinear Dynamics and Chaos in NonSmooth Systems,” Journal of Sound and Vibration, Vol. 301, No. 12, 2007, pp. 59 73. doi:10.1016/j.jsv.2006.09.014 [2] I. Arango, “Singular Point Tracking: A Method for the Analysis of Sliding Bifurcations in Nonsmooth Sys tems,” PhD Thesis, Universidad Nacional de Colombia, 2011. [3] I. Arango and J. A. 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