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IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 9, NO. 2, APRIL 2001

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Local and Global Bifurcations in Simple Takagi–Sugeno Fuzzy Systems Federico Cuesta, Enrique Ponce, and Javier Aracil, Senior Member, IEEE

Abstract—The relevance of bifurcation analysis in Takagi–Sugeno (T–S) fuzzy systems is emphasized mainly through examples. It is demonstrated that even the most simple cases can show a great variety of behaviors. To understand the richness of asymptotic dynamics, one can find in T–S systems, a methodology is proposed by invoking bifurcation theory. Several local and global bifurcations (some of them, degenerate) are detected and summarized in the corresponding bifurcation diagrams. It is claimed that this kind of careful analysis could help to cope with some criticism raised regarding the blind use of fuzzy systems. Index Terms—Bifurcations, parametric robustness, piecewise smooth systems, stability, T–S fuzzy systems.

I. INTRODUCTION

T

HE class of Takagi–Sugeno (T–S) fuzzy systems has been widely used since the 1980s [15]. These systems are defined by a set of rules whose consequents are deterministic nonlinear dynamical systems. Nonlinear systems can display complex dynamical behaviors [3] and, what is more important, after (even small) changes in parameters or in their system structure, one can observe qualitative changes in their behavior modes, (see, for instance [16]). This is the realm of bifurcation theory [4], [12], which supplies tools to study the points where these changes are produced and the archetypical forms of the state portrait changes in these points. Bifurcation theory can be a valuable tool for understanding the behavior richness of nonlinear systems. Roughly speaking, when after a small change in the value of system parameters one observes a qualitative change in the system response (to be deduced from a phase portrait, for instance) it is said that the system undergoes a bifurcation phenomenon. These phenomena can lead to a different number of stationary solutions (equilibrium points), to the appearance of oscillations, or even more complex behaviors (chaos, for instance). Thus, from a certain point of view, eventual bifurcations are related to robustness issues since only far from the bifurcation points the system displays behaviors that are structurally stable. We refer to some comprehensive works [3], [4], [12] for a thorough introduction to the subject. Even when there are not actual parameters in the system to study, it is interesting from the point of view of bifurcation analysis to assume that some parameters are involved and, by studying the possible bifur-

Manuscript received July 6, 2000; revised November 18, 2000. This work was supported in part by the Spanish Agency CICYT under Grants TAP97-0553 and TAP99-0926-C04-01. The authors are with the Escuela Superior de Ingenieros, Universidad de Sevilla, 41092 Spain (e-mail: {fede, aracil}@cartuja.us.es; [email protected]). Publisher Item Identifier S 1063-6706(01)02821-1.

cations that can arise, one obtains valuable information about the behavior corresponding to the actual values of parameters. Anyway, after a bifurcation analysis one can split the parameter space on several regions with different asymptotic dynamics (see, for instance [17]). These questions are quite relevant for fuzzy systems, where, through the fuzzification–defuzzification process, some “plastic” changes can occur (depending on the concrete method used in that process). The changes could be associated to bifurcations, whose consequences should be known by the system user and, maybe have not attracted enough attention yet [7], [8]. It should be clearly understood that through the fuzzification-defuzzification process, a fuzzy system with a component of linguistic rules has been converted into a mathematical object with well-known properties that one cannot forget when using such a system. As will be seen in the current paper, some of the qualitative characteristics of these complex behaviors can be captured through a bifurcation analysis. In this paper, some elementary examples of T–S fuzzy dynamical systems have been chosen to illustrate the relevance of the bifurcation analysis for fuzzy systems. For the sake of simplicity, in all cases, only one parameter is assumed, the so-called bifurcation parameter and the state–space will be supposed bi-dimensional. Also, it will be only considered piecewise linear membership functions, in order to deal with an easier analysis. It is amazing to realize the variety and richness of behaviors such very elementary fuzzy dynamical systems can display. This fact cannot be underestimated by those interested both in theoretical research in fuzzy systems and in practical applications, mainly in the context of fuzzy control, as pointed out in [10] and [19]. This paper is organized as follows. In the next section, some concepts on fuzzy systems and bifurcation theory are introduced. Sections III–V are devoted to present illustrative examples of the application of bifurcation theory to the analysis of T–S fuzzy systems. The paper closes with Section VI, conclusions and, finally, references. II. T–S FUZZY SYSTEMS AND BIFURCATION THEORY An affine T–S fuzzy system is composed by if

is

rules such as

then

for , where is an -dimentional membership function and can be represented as the nonlinear dynamical system

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(1)

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where

(2)

, representing are nonlinear functions of the state vector in the dynamic of the the weight or contribution of each rule system. The mathematical object of (1) belongs to the class of the non, where stands for linear dynamical systems the parameters of the system. In a fuzzy dynamical system, the parameter space is formed, among other things, by the parameters appearing in the membership functions . However, we will not consider explicitly that possibility in this paper, and we focus in the parameters appearing in the consequents instead. As indicated before, the main concern of bifurcation theory is the analysis of the qualitative changes which take place in the behavior modes of a nonlinear dynamical system as the parameters are varied. Thus, since in fuzzy systems parametric functions are involved, bifurcation theory is well suited to analyze the parametric robustness of fuzzy systems. The first point to realize when dealing with nonlinear dynamical systems is that they can have more than one attractor. As each attractor has its own attraction basin, the landscape of such systems can be very complex with several basins bounded by separatrices. The shape of this landscape can suffer qualitative changes for some critical values of the parameters. These values are called bifurcation points. Elementary bifurcations give the simplest ways in which the qualitative structure of the state portrait changes. Happily, with a few of these elementary bifurcations, many practical situations can be dealt with. Bifurcation phenomena can be classified in two main classes, namely local and global ones. Local bifurcations are due to changes in the dynamics of a small region of the phase space, typically, a neighborhood of an equilibrium point. If the eigenvalues of the linearization of the system at one equilibrium point have no zero real part, the equilibrium is called hyperbolic. As is well known, hyperbolic equilibria with eigenvalues of negative real parts are asymptotically stable, whereas if there are some eigenvalues with a positive real part, then the equilibrium is unstable. Thus, starting for instance from a stable equilibrium, a crossing of eigenvalues through the imaginary axis will involve a local bifurcation. The most basic bifurcations are the corresponding to a zero eigenvalue (saddle-node, pitchfork, and transcritical bifurcations [3], [4]) or to a pair of pure imaginary eigenvalues (Hopf bifurcation). For instance, by a supercritical (also called soft [4]) Hopf bifurcation a stable limit cycle is born from a point attractor that becomes unstable. Hopf bifurcation can also be detected by means of harmonic balance [11], [14]. Furthermore, there are global bifurcations. In that case, the bifurcation is produced in such a way that it involves phenomena not reducible to locality. That happens, for instance, when the interaction of a limit cycle with a saddle point is produced. In such a case, an attractor can suddenly appear or disappear. Apart from the quoted situation, which is called a homoclinic or saddle con-

Fig. 1. State–space partition induced by piecewise linear membership functions.

Fig. 2.

Membership functions.

nection [3], [4], other global bifurcations are, for instance, the saddle mode of periodic orbits (two periodic orbits of different stability character collide to disappear or vice versa) and the Hopf bifurcation from infinity, where a periodic orbit of great amplitude comes from (goes to) infinity [5]. Global bifurcations cannot be detected by local analysis and normally need of numerical or approximate global methods as the harmonic balance [9], [10], [13]. The relevance of qualitative analysis and bifurcations in T–S systems has been already pointed out (see [18] and references therein). However, their complicated structure from the mathematical point of view explains the lack of theoretical results up to this moment. In this sense, one of the aims of this paper is to stimulate the research in the field. To lower the aforementioned difficulties, one can resort to piecewise linear membership functions with local support. These functions clearly induce a partition in the state–space of T–S systems (see Fig. 1). Thus, the state–space is divided , where only one rule and the corinto operating regions responding affine dynamic is active and interpolation regions in between them where several dynamics are present. Nevertheless, the problem remains nontrivial to analyze (bifurcation theory normally assumes a high degree of smoothness for the system). For the sake of simplicity, in this paper we will deal with the and simplest T–S systems, that is, we will take in (2). As indicated before, in order to facilitate the analysis,

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we will assume normalized piecewise linear membership functions; even for this case we will find nontrivial behaviors. To be more precise, all the examples analyzed in this paper have the following structure: is is

(3)

where the linguistic terms and are represented by the normalized trapezoidal membership functions shown in Fig. 2, resulting in the following dynamical system: (4)

degenerate bifurcations. It should be emphasized that these degenerate bifurcations might not persist after some smoothing of the piecewise linear membership functions, even when the perturbations were small ones. That will not be the case for the rest of bifurcations we will find in the following examples. The first two have homogeneous consequents and the last one is composed by affine consequents. III. EXAMPLE 1: HOMOGENEOUS CONSEQUENTS As our first example, consider the fuzzy system is

where for for for for for for

is

(5) .

, and that there will Note that appear two operating regions and only one interpolating region. which is Equations (3)–(5) define a dynamical system in governed by a piecewise smooth vector field (it is only of class for and ). Of course, the system is well posed and for every initial conditions one can assume the existence and uniqueness of solutions. Therefore, the dynamical system is equivalent to for for

(7)

where is the parameter, which can vary (to be assumed the and are repbifurcation parameter). The linguistic terms resented by normalized trapezoidal membership functions (see Fig. 2). As indicated before, the phase space can be partitioned in two operating regions and one interpolating region according to the membership functions, such that one gets for

for

(6)

and gives rise to a quadratic system in the interpolation region, . that is, for In this setting, the bifurcation analysis can take advantage of the state–space partition induced by the membership functions. From the point of view of the dynamics that is intrinsic to every operating region there is no problem in the analysis. In fact, as in every operating region, the system becomes purely affine. All one has to do is to characterize the corresponding linearization, and to locate the evenwhich is given by the matrix , tual equilibrium points. It should be noted that even virtual equi, , that are not in libria, that is, solutions of the corresponding region, do govern the dynamics in the region. In the interpolating region, the analysis of its intrinsic dynamics is more involved, as we are dealing with a quadratic system, but special difficulties should not be expected. The problem arises when the region dynamics are merged into a unique state space. For instance, the merging of only two affine regions is enough to produce limit cycles see [6] what is impossible for each separate region. The interaction of trajectories in different regions can produce interesting global phenomena that one might not be aware of given the linear nature of the two involved systems. As will be seen, these elementary T–S systems can display local and global bifurcations, some of them due to the structure of the regions and other originated by the loss of differentiability. In this last case (low differentiability), we will speak of

for

(8)

The global dynamics in the phase space is the composition of the three regional dynamics. Therefore, all the trajectories can be determined by gluing (not only continuously but also with continuous derivative) trajectories in each region. To start with, it is easily concluded that the dynamics in the left operating re) is a linear dynamics that is independent of the gion ( bifurcation parameter. For this region, the origin is the only equilibrium governing the dynamics (in fact, one unstable focus), but note that it is out of the region and so it constitutes a virtual equilibrium. Trajectories enter this region from the middle region at for and they always return to the middle for . region at Now, considering the right operating region, the corresponding dynamics, which is also linear, depends on the value , the only equilibrium governing the of . Again, for dynamics is the origin (a virtual equilibrium) and by linear analysis one gets the following assertions: the virtual equilibrium at the origin is a saddle; • for there appear a continuum of equilibria at the • for axis; for these points are real equilibrium points and the dynamics is rather degenerate as all trajectories are

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horizontal straight lines (entering the region for and leaving it for ); the dynamics is of stable focus type; • for for trajectories enter the region at and always return to the middle region at for ; the dynamics is governed by an improper stable • for for node; trajectories enter the right region at and always return to the middle region at for . At for trajectories leave the region coming from the point at infinity; the dynamics is governed by a stable node with • for a behavior of trajectories similar to the previous case. The analysis of the middle (interpolating) region dynamics is somehow more involved as we are dealing with a quadratic system. First of all, apart from the equilibrium point at the origin there is another equilibrium (now, a real equilibrium) if point at (9) and a real one for . which is a virtual equilibrium for clearly Note that for the complete system, the value represents a bifurcation value since the dynamics changes at this the system has one equilibrium point. At value. When there appears a half-straight line of equilibrium points , with ) from which the system inherits one new ( ) for . This bifurcation can be equilibrium point at ( thought as a degenerate saddle-node (DSN) bifurcation. The linearization of the equilibrium point at the origin is

= Homoclinic Connection; = Degenerated Saddle Node;

Fig. 3. Bifurcation diagram of example 1 (HC H Subcritical Hopf; T Transcritical; DSN GC Global Center).

= =

=

To study the character of the second equilibrium point, it suffices to compute from (8) the corresponding linearization, namely

(12)

and it should be remarked that (10) (13) such that

trace

det

(11)

the origin is unstable (a saddle point). For it is an unstable nonhyperbolic equilibrium (what is a necessary condition for a bifurcation to occur; the character of this bifurcation will be determined later). The origin is also , becoming stable unstable (node or focus) for . Again, when this equilibrium is nonhyperfor bolic since its linearization has a pair of pure imaginary eigenvalues, what might be associated to a Hopf bifurcation. To detect the character of this Hopf bifurcation, an additional analysis is required, as shown in the Appendix. From this Appendix, it is there is a global nonlinear center (GC) concluded that for , but small, does not give rise to any periodic that for orbit. For

) is a saddle point for . When Thus, the point ( the system undergoes a bifurcation since this equilibrium and the equilibrium at the origin coalesce. Analyzing the sign of the trace in (13), one concludes that it and for , where , is positive both for are the roots of the quadratic , that is (14) With this information it is not difficult to see that at a transcritical bifurcation takes place (two equilibrium points collide interchanging their stability properties). since the trace Also, another bifurcation arises when changes its sign with a positive determinant. In fact, the system undergoes a subcritical Hopf bifurcation (to be denoted as H ) ) (an unstable focus for ) becomes a as the point ( , with one unstable limit cycle around it. stable focus for As predicted by bifurcation theory, the amplitude of this limit cycle evolves with the bifurcation parameter and it can be ap-

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(a)

(b)

(a)

(b)

(e)

=0

Fig. 4. Phase portraits of example 1 for different values of . (a) For 6, there is an unstable limit cycle stating the attraction basin of one stable equilibrium. (b) For = 5:6 there are no stable equilibrium points (at = the systems undergoes a subcritical Hopf bifurcation within the interpolating region). (a) = 6 < and (b) = 5:6 > . (c) For = 0:9 the system has one unstable equilibrium at the origin. (d) For = 1 the system has a global center. (c) = 0:9 and (d) = 1. (e) = 1:1.

0

0

0

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proximated by an expression that is for small. Therefore, the unstable limit cycle will remain in the inif is sufficiently small, terpolating region for ) [see Fig. 4(a)]. stating the attraction basin of the point ( bigger (with ) the above unstable limit Letting cycle begins to enter the left operating region, also approaching the origin from its right-hand side (RHS). Then, for certain value of , a global bifurcation appears when the limit cycle becomes a loop connecting one branch of the unstable manifold of the origin with one branch of its stable manifold (saddle connection or homoclinic bifurcation HC). After this critical value, the relative positions of these two manifold branches change, giving as result the disappearance of the limit cycle. Now, the attraction ) is no longer bounded and, depending basin of the point ( on the initial conditions situation with respect to the stable manifold of the origin, trajectories go to the infinity or to the stable ). point ( the system exhibits a standard behavior Clearly, for with no sensitivity to initial conditions and only one equilibthe system also rium, which is globally stable. When possesses a stable equilibrium, but we can say that it is not robust since its attraction basin is limited. For the intermediate , the system is unstable values of , that is, in the range and so it can be considered useless. Thus, we can find strongly different system behaviors depending on the actual value of . It must be emphasized that the identification of the adequate value of turns out to be a critical issue. To summarize the whole analysis, we present the bifurcation diagram of Fig. 3. Also, the corresponding phase portraits for several values of are sketched in Fig. 4. IV. EXAMPLE 2: MODIFYING THE RIGHT CONSEQUENT In this example, let us change with respect to previous one only the second rule, to use instead is

(15)

taking the same normalized trapezoidal membership functions (5) as in Example 1. Now, the system becomes for

This equilibrium is a saddle point for and a stable . For , there appoint (focus or node) for pears a continuum of equilibria making up the half straight line with (for they constitute virtual equilibria). Trajectories enter the middle interpolating region from the right region, coming from the point infinity,, at for and they return to the right region at for approaching asymptotically one of the continuum of equilibria. Concerning the interpolating region, the origin is always a there are another two real equilibrium, and for equilibrium points, namely

(17) and

(18) at the point both coalescing for The linearization matrix at

(19) with null trace and null determinant. This point is so a nonhyperbolic equilibrium with two zero eigenvalues. The corresponding bifurcation is called a Bogdanov–Takens bifurcation (see [4]; this bifurcation needs two parameters to see all the possible beis a cuspidal haviors nearby). Thus, the point point, and by moving we will see a one-dimensional (1-D) section of the unfolding of the Bogdanov–Takens bifurcation. is a real equilibrium both for and The point and a virtual equilibrium for . On for is a real equilibrium only for the other hand, the point (a virtual equilibrium for ). represents a bifurcation value for the whole Clearly, (with small), the only real equisystem. When there appears a half-straight librium is the origin. At , with ) from line of equilibrium points ( which the system inherits a new equilibrium point at for in a bifurcation DSN. The linearization of the equilibrium point at the origin is

for

for

. is

(20) (16)

Repeating the above steps, the first remark is that the dynamics in the left operating region is identical to that of Example 1. For the right operating region the linear dynamics is . governed by a virtual equilibrium at the origin for

such that trace

det

Thus, the origin is stable for ). For and a focus for

(21) (a node for , it is a nonhyperbolic

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(a)

(b)

(c)

(d)

(e)

=0

Fig. 5. Phase portraits of Example 2 for different values of . (a) For 0:4 the origin is the global attractor of the system. (b) For = 0 the system has a global center. (a) = 0:4 and (b) = 0. (c) For = 0:45 there is a stable limit cycle surrounding the origin which is unstable (at = 0 the system undergoes a supercritical Hopf bifurcation at infinity). (d) For = 0:5 the system has a half line of stable equilibrium points at x = 2x with x 1. (c) = 0:45 and (d) = 0:5. (e) = 0:6.

0

0



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equilibrium and it is unstable for (a focus for , and a node for ). a Hopf bifurcation could be expected. It is interFor is the same than that esting to note that the system for . From Lemma A-1 in the Appendix, at of Example 1 for there is a global nonlinear center. However, for , and small, there now appears a stable limit cycle, as a result of a supercritical Hopf bifurcation at the infinity (the existence of this bifurcation can be confirmed by using the techniques introduced in [5]). The amplitude of the stable limit cycle decreases as the bifurcation parameter grows. Furthermore, it is possible to show that the periodic orbit disappears suddenly at in a degenerate global bifurcation, due to the appearance of the continuum of equilibria in the right region (see Fig. 5). , the linearization process In the case of the point results in

(22)

and so (23)

(24) and The trace (23) is positive both for (being zero for and ). On the other hand, the . Therefore, it is easy determinant (24) is positive for is an unstable to conclude that the equilibrium point ; a nonhyperbolic equilibrium for ; focus for (recall that the equilibrium does not and a saddle point for , being a virtual one for ). exist for Similarly, the linearization at the equilibrium point is

(25)

Fig. 6. Phase portrait of example 2 for equilibrium point of the system.

= 05; the origin is the only stable

could speak of a saddle-focus bifurcation), which is, in fact, a section of the more general Bogdanov–Takens bifurcation. with small, the only equilibrium Thus, for of the system is the origin, which is globally stable. However, the system exhibits three equilibria, due to the apfor pearance of two new equilibria (an unstable focus and a saddle point), making in principle the stability of the origin to be only local (see Fig. 6). Finally, the whole analysis can be summarized in the bifurcation diagram shown in Fig. 7. The main conclusion is that the origin is the only (quasi-) global attractor (the for does not preclude this presence of other equilibria for assertion, as excepting these equilibrium points all trajectories tend to the origin). V. EXAMPLE 3: AFFINE CONSEQUENTS The last example deals with an affine T–S fuzzy system is is

(28)

giving (26)

where is the bifurcation parameter, and the membership functions are the same than in previous examples. Therefore, system (28) can be expressed as for

(27) and it is zero for Now, the trace (26) is positive for . Also, the determinant (27) is null only for . Thus, is a saddle point for ,a the equilibrium point and a virtual unstable nonhyperbolic equilibrium for . node for the complete system undergoes a bifurTherefore, at cation similar to the saddle-node bifurcation of equilibria (we

for for

(29)

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gion. For and small, the center disappears, leaving one unstable limit cycle (which arises from the outermost periodic orbit of the center, in a bifurcation analogous to the one studied in [2]), whose amplitude grows as decreases, so living partially in the middle region and coexisting with the stable limit and small, the center disappears without giving cycle. For rise to new limit cycles but the stable limit cycle persists. With respect to the interpolating region, apart from the origin, another two equilibria could exist for

(assuming also that

=

), namely

Fig. 7. Bifurcation diagram of example 2 (SF Saddle Focus from a cuspidal point; H Supercritical Hopf at infinity; DSN Degenerated Saddle Node; GC Global Center).

=

=

=

In the left operating region, the dynamics corresponds to an affine system, being independent of the bifurcation parameter. The only equilibrium in this region should be

(32) and

a stable focus, which is out of the region, being a virtual equilibrium point. Trajectories enter this region from the middle (infor and they return to terpolating) region at for . the middle region at On the other hand, the right operating region is governed by an affine system depending on the bifurcation parameter. Thus, in this region, the only equilibrium in given by (30) , which is a real equilibrium of system (29) for , and a virtual equilibrium elsewhere. Trajectories enter for this region from the interpolating region at , and they return to the middle region at for . The trace and determinant of the linearized system at the equiare librium point (31) , the equilibrium is a stable focus; when Therefore, for , it is a nonhyperbolic equilibrium and, for , it is , the trace vanishes, with a positive an unstable focus. At determinant, what suggest the existence of a Hopf bifurcation (indeed, as affine systems cannot undergo Hopf bifurcations, one must better speak of a Hopf-like bifurcation). Analyzing this bifurcation, it can be concluded that there appears a linear center around the point (2, 0), which is restricted to the right operating region. The outermost periodic orbit of this center is tangent to the and it is a semistable border of the region at periodic orbit (stable from its inner side but unstable from its outer side). Trajectories starting near this semistable periodic orbit from its outer side, tend to another stable periodic orbit of greater amplitude, which lives partially in the interpolating re-

(33) . with is a real equilibrium only for The point is a real equilibrium only for while the point . The linearization of the equilibrium point at the origin is

,

(34)

such that

(35) , but with a negative determiThe trace vanishes for and . nant. The determinant is zero both for . For it is Thus, the origin is a stable node for the origin is a a nonhyperbolic equilibrium. For it is again a nonhyperbolic equilibsaddle point. For . rium, being always an unstable node for and at , the system undergoes In fact, both at a transcritical bifurcation due to the collision of the origin with and , respectively. the equilibrium

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(a)

(b)

(c)

=0

0p

0 0

Fig. 8. Phase portraits of example 3 for different values of . (a) For 6 the origin is the global attractor of the system. (b) For = 4 the system has two stable equilibria and a saddle point (at = 3 2 the system undergoes a degenerate saddle node bifurcation). (a) = 6 and (b) = 4. (c) = 2.

The linearization at the equilibrium

such that (37) (38) , .

0

Thus, the trace (37) is zero for

is

(36)

with

0

, and

(39) (note that, with positive determinant only at although it could be possible a Hopf bifurcation, the point at this value is a virtual equilibrium). On the other hand, determinant (38) is zero for several values of , namely

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(a)

(b)

(c)

(d)

(e)

(f)

=0 0

0

Fig. 9. Phase portraits of example 3 for different values of . (a) For 0:06 the system has two stable equilibria and a saddle point. (b) For = 0:03 there exist an unstable periodic orbit and a stable limit cycle around it (at = 0:05 the system undergoes a saddle-node bifurcation of periodic orbits). (a) = 0:06 and (b) = 0:03. (c) For = 0 there are one stable limit cycle and a local center around the point (2, 0) in the right operating region [the outermost periodic orbit of the center is tangent to the border of the region at the point (1, 0)]. (d) For = 0:01 the system has only one (stable) limit cycle whose amplitude grows with . (c) = 0 and (d) = 0:01. (e) For = 0:028 the limit cycle connects with the saddle point in a homoclinic connection. (f) For = 0:04 the system has only one stable equilibrium. (e) = 0:028 and (f) = 0:04.

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and (40) being the trace nonzero at these values, and the point a nonhyperbolic equilibrium. In a similar way, analyzing the trace and determinant of the , it can be concluded that linearized system at the point the trace is always negative and the determinant is zero for

being the point a nonhyperbolic equilibrium at these values. and do not exist for Note that the equilibria . The system undergoes a saddle-node bifurcation , where appear two branches of of virtual equilibria at (a stable node), equilibria, corresponding to the point (a saddle point). Furthermore, these and the point branches will become real equilibria of the whole system at for , and for . Therefore, from the point of view of the whole system, represents a bifurcation value, in fact, a nonsmooth saddle-node (NSN) bifurcation takes place just at the boundary between the interpolating (middle) and the right operating rethe only real equilibrium is the origin gions. When there appears two new (a globally stable node). At ): the equilibria starting from the point ( being a saddle point, and the equilibrium equilibrium of the right operating region, which is a stable node. the stability or instability of the origin Thus, for will be only local (see Fig. 8). It should be remarked that this bifurcation (NSN) occurs at a point where the system is not differentiable. However, such phenomenon has been detected with the methodology presented in this paper. , the system undergoes a tranOn the other hand, at scritical bifurcation corresponding to the collision of the point with the origin. Thus, for , the point is an unstable node, while for it is a saddle point. In there exists a transcritical bifurcation a similar way, at with the origin. Thus, due to the collision of the point , is a saddle point, while for it is a for stable node. Finally, there appear also two global bifurcations. First, a saddle-node bifurcation of periodic orbits takes place at , resulting in the appearance of an unstable limit cycle, together with a stable limit cycle around it (see Fig. 9). The amplitude of the stable periodic orbit grows with the value of the bifurcation parameter, while the amplitude of the unstable limit cycle decreases with . Thus, the unstable periodic orbit will with the outermost orbit of decrease until it connects at the local center (see Fig. 9). , the only periodic orbit is the As mentioned before, for stable one. This limit cycle will grow in amplitude approaching the perithe saddle point at the origin. Thus, for odic orbit becomes a loop connecting one branch of the unstable manifold of the origin with one branch of its stable manifold, in

=

Fig. 10. Bifurcation diagram of example 3 (NSN Nonsmooth Saddle Node; T Transcritical; SNPO Saddle Node of Periodic Orbits; LC Local Center; HC Homoclinic Connection).

=

=

=

=

Fig. 11. Trajectories of the quadratic system (42) for different values of K . When K ( (1=4); 0) trajectories are closed curves; these closed curves belong completely to the middle region only for K ( (1=4); (1=4e ))

2 0

2 0

0

a global homoclinic bifurcation. For the relative positions of these manifold branches change, giving as a result the disappearance of the limit cycle (see Fig. 9). The whole bifurcation analysis is summarized in Fig. 10, where it is shown the bifurcation diagram for this example. The the origin is the only main conclusions are that for , the global attractor of the system and for system exhibits two stable attractors. Also it can be concluded the origin is not an operating point for the that for system. VI. CONCLUSION In this paper, the great variety of behaviors of some simple T–S fuzzy systems has been studied with the tools supplied by bifurcation theory. Qualitative analysis and bifurcations are of great relevance in every family of nonlinear systems and so in T–S-systems. It has been shown that even very elementary T–S

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systems can display local and global bifurcations, some of them due to the structure of the regions and to the loss of differentiability. In this last case (low differentiability), we have spoken of degenerate bifurcations. It should be emphasized that these degenerate bifurcations might not persist after some smoothing of the piecewise linear membership functions, even when the perturbations were small ones. It has been also shown that small variations in the values of the parameters associated with the fuzzy rules can give rise to unexpected behaviors. Only through a bifurcation analysis it is possible to detect the richness of asymptotic dynamics a system can exhibit. The use of piecewise linear approximation of membership functions has been useful in the approach. It has been stressed that the task is nontrivial even for the simplest cases, but as seen in the examples analyzed, the gained information is of great value, for instance, in robustness studies, since only far from the bifurcation points the system displays behaviors that are structurally stable. The samples included in the paper are relevant enough to rise interest in bifurcation analysis by the fuzzy systems community. The moral of the tale is that this kind of careful analysis could help to cope with the criticism raised regarding the blind use of fuzzy systems. APPENDIX BIFURCATION ANALYSIS OF EXAMPLE 1 FOR The standard techniques of Hopf bifurcation analysis do not provide any interesting information, apart from the fact that in this case, the bifurcation is degenerate. Here, it is proposed an , system (8) is alternative approach. For

for for for

From (43) it is easily deduced that all trajectories of the quadratic system are symmetric with respect to the axis, since they are defined by

(44)

provided that there exists a range of values of , where the expression in the radical is positive. In fact, that range always . For instance, when , the radappears for and the trajectory is given by the ical is positive for . parabola , the corresponding trajectory degenerates For trajectories in a point (the origin), while for form closed curves, see (Fig. 11). Obviously, from above trajectories only the portion in the inrepresent actual trajectories for system terval (41). However, the complete system (41) is invariant under the following transformations: for for what indicates that, apart from the aforementioned symmetry axis in the middle region, of trajectories with respect to the trajectories are symmetric with respect to the origin in the outer regions. In these regions the dynamics are of focus type (one unstable and one stable) and so all the trajectories are closed curves. The conclusion follows. Lemma A-2: For and small, the system (8) has no periodic orbits. Proof: For sake of brevity, technical details will be omitted. After an elemental scaling of time, (8) can be written in Liénard form, that is

(41)

Lemma A-1: System (41) has a global nonlinear center. Proof: First, consider the quadratic system that coincides with system (41) in the middle region. Such quadratic system is not difficult to integrate by rewriting it as follows:

(42)

(45) small has only one equilibrium at the origin. Then and for a necessary condition for existence of periodic orbits, which is based upon Filippov transformations, is given in [1, Theorem 5] (see also the related remark therein). Basically, the condition such that the system needed is that there exist equation

is an integrating factor. Thus, trajectoand observing that ries are implicitly given by (46) (43)

for each value of the constant

.

. Computations show is fulfilled, where that the above system cannot be compatible and the conclusion follows.

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