Analysis of a type of nonsmooth dynamical systems

Chaos, Solitons and Fractals 30 (2006) 1153–1164 www.elsevier.com/locate/chaos

Analysis of a type of nonsmooth dynamical systems Guofeng Zhang a, Guanrong Chen b, Tongwen Chen

c,*

, Yanping Lin

a

a

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta, Canada T6G 2G1 b Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China c Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, Canada T6G 2V4 Accepted 30 August 2005

Abstract In this paper, a class of nonsmooth dynamical systems is analyzed. Extensive simulations reveal the chaotic behavior in these systems. By introducing a parameter, a chain of systems with one end being a linear stable system and the other being a chaotic system is constructed. Then the phase transition process through the chain is investigated as the parameter varies. Difficulties involved in analyzing this class of systems are also discussed.  2005 Elsevier Ltd. All rights reserved.

1. Introduction Various chaotic theories for smooth dynamical systems have been established in light of symbolic dynamics, fractal dimensions, entropies or ergodicity, and so on [5,7]. It is fair to say that theories for chaos in smooth systems have reached a certain maturity. However, in the real world, discontinuous dynamics are prevalent. And it seems that they are able to generate chaos more easily as demonstrated by many numerical simulations. Unfortunately, to prove that a discontinuous dynamical system especially a switched system is chaotic turns out to be much more difficult. On the other hand, during the shift from smooth to nonsmooth chaotic systems, many properties will become invalid; nevertheless, there are still some that remain persistent. Probably these remaining properties constitute the essence of chaos. In this sense, the study of nonsmooth dynamical systems is of great importance both practically and theoretically. In our earlier papers [13,14], the following system has been studied in detail: 

8   a þ b 0 x1 ðkÞ > > ; > < x1 ðk þ 1Þ 1 0 x2 ðkÞ ¼    > a b x1 ðkÞ x2 ðk þ 1Þ > > ; : 0 1 x2 ðkÞ 

if jx1 ðkÞ  x2 ðkÞj > d; ð1Þ otherwise;

*

Corresponding author. Tel.: +1 780 492 3940; fax: +1 780 492 1811. E-mail addresses: [email protected] (G. Zhang), [email protected] (G. Chen), [email protected] (T. Chen), [email protected] (Y. Lin). 0960-0779/$ - see front matter  2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2005.08.181

1154

G. Zhang et al. / Chaos, Solitons and Fractals 30 (2006) 1153–1164

where d > 0, a, b are real constants satisfying ja + bj < 1. It has been shown that • • • • •

Depending on the parameters a and b, the system may have converging, periodic and (or) aperiodic orbits. The system behaves chaotically. The topological entropy of the system is 0. When the system has periodic orbits, it has a dense set of periodic orbits of the same period. The system is not structurally stable.

System (1) is governed by a simple switching law. As is known, switching occurs in many industrial control systems such as multi-body systems with unilateral constraints, cooperative and autonomous systems, intelligent systems, robots and social systems (see [8] for examples), thus making the analysis of switching fundamental and extremely important. System (1) is obvious a very simple type among switched systems, so we hope this study can shed some lights on the nature of switching. Bifurcation phenomena in nonsmooth dynamical systems are reviewed in great details in [1] for three classes of nonsmooth dynamical systems. However, nonsmooth transitions considered there only involve either an equilibrium point approaching (or leaving) a single boundary or a periodic orbit, namely, either grazing with a boundary or approaching the intersection point between two boundaries (see [3] for one practical example). The family of systems studied here clearly does not belong to those three classes. This paper is organized as follows: Topological transitivity and sensitive dependence on initial conditions are discussed in Section 2. Problem is precisely formulated in Section 3. A family of nonsmooth dynamical systems is analyzed in Section 4. A higher-dimensional example is discussed in Section 5. Some difficulties involved in analyzing this class of dynamical systems are pointed out in Section 6. Some open problems are posed in Section 7.

2. More evidence of chaos In this section, we illustrate the chaotic behavior of system (1). Though there is no universally agreed definition of chaos, a chaotic map typically has an invariant set on which it is topologically transitive and has sensitive dependence on initial conditions. So, we first demonstrate the topological transitivity of the map defined by system (1) via numerical examples. Fix parameters to be a¼

9 ; 10

b¼

3 ; 10

d¼

1 . 100

Choose a trajectory x1(i), i = 1, . . . , N, starting from x1 ð0Þ ¼

d 1:0 þ 2 108

with N = 5 · 106. Here, the numerical precision is set to be 32. Then, choose several points along this trajectory and check the points within each interval of length  centered at each of these chosen points (except themselves). The computational result is summarized in Table 1. We clearly see the recurrence of the trajectory to itself. Next, we choose another orbit, x2(k), starting from x2 ð0Þ ¼

2  d 1:0 þ 8. 3 10

Table 1 Recurrent behavior of x1 Val x1(90,000) x1(1,000,006) x1(1,000,006) x1(N  11) x1(N  11) x1(N  11)

Number

e 10

10 1010 109 1010 1012 1011

1 0 14 135 0 1

G. Zhang et al. / Chaos, Solitons and Fractals 30 (2006) 1153–1164

1155

Accordingly, the following points are obtained: x2 ð1; 000; 006Þ ¼ 0:005478247409168660668943840211631603889702611599158266702; x2 ðN  11Þ ¼ 0:005507833414434384965183963494955909879041106961617575674; x2 ð300; 006Þ ¼ 0:005411439540349742303466366471586072486903229411453756781. We then check the number of points of x1(k) inside a small neighborhood of each point above. Table 2 summarizes the result. We found that within each small neighborhood of each chosen point in x2, there are always points of x1. (Note that the occurrence of 0 in these two tables is due to the limited amount of data we had. By choosing N bigger, the neighborhoods given by , can be smaller.) Since x1 and x2 are chosen arbitrarily, it is plausible to infer that there must be an invariant set within which almost all the orbits are dense. Such a set, say K, can be approximated as K ¼ limfx1 ðkÞg;

ð2Þ

where x1 is an oscillating orbit of the map, the outer overline denotes the closure, and lim means the upper limit. It is worth pointing out that here the uniqueness of an invariant set in the oscillating region is implicitly assumed. Our extensive simulations strongly support this conjecture. Clearly K is closed and compact. Moreover, Table 1 tells us that each point in K is an accumulating point. As is known, dissipative chaos is always closely related to some Cantor set. To make K a Cantor set, we need to show that K is nowhere dense, which can be guaranteed if the map involved is dissipative. For a continuously differentiable dynamical system, its dissipativity can be verified quite easily. However, for a nonsmooth dynamical system, there still lack effective tools to do so. We have to admit that it is extremely hard to find the invariant set K analytically. This is best demonstrated by Fig. 3, where it is hard to find any explicit pattern. Next, we study the sensitive dependence on initial conditions, a defining nature of chaos. Fix a¼

11 ; 10

b¼

3 ; 10

d¼

1 . 100

Table 2 x1 around x2 Val

Number

e 9

x2(1,000,006) x2(1,000,006) x2(N  11) x2(300,006) x2(300,006)

10 1010 1010 1010 1013

13 0 2 2 0

Table 3 Sensitive dependence on initial conditions traj1

traj2

Maximal distance

  d 2d b  1 þ a þ 6  2 10 1  ða þ bÞ

  d 3d b  1 þ a þ 6  2 10 1  ða þ bÞ

0.004388827490502198

  d 2d b  1 þ a þ 7  2 10 1  ða þ bÞ

  d 3d b  1 þ a þ 7  2 10 1  ða þ bÞ

0.004388810010553412

  d 2d b  1 þ a þ 8  2 10 1  ða þ bÞ

  d 3d b  1 þ a þ 8  2 10 1  ða þ bÞ

0.004388807107462013

  d 2d b  1 þ a þ 10  2 10 1  ða þ bÞ

  d 3d b  1 þ a þ 10  2 10 1  ða þ bÞ

0.004388806797374203

1156

G. Zhang et al. / Chaos, Solitons and Fractals 30 (2006) 1153–1164

We then check the maximal distance between two initially nearby trajectories. In Table 3, there are four rows of data. In each row, the first and second columns are the initial points of two trajectories, denoted ‘‘traj1’’ and ‘‘traj2’’, respectively. The third column contains the maximal distance between these two trajectories. The initial differences are tiny, at levels of 106, 107, 108, 1010, respectively. However, the maximal distances are all around 0.0043888, so the sensitive dependence on initial conditions is clear. As commented before, topological transitivity on an invariant set and sensitive dependence on initial conditions are two characteristic features of a chaotic map. Based on our forgoing simulations, the system discussed here does show chaotic behavior. It is our strong belief that a map is chaotic if it has a Cantor set as an invariant set and on which the map is topologically transitive.

3. Problem formulation Define a family of systems 8   x1 ðkÞ > A >   > < 1 x ðkÞ ; x1 ðk þ 1Þ 2 ¼   > x x2 ðk þ 1Þ 1 ðkÞ > > ; : A2 x2 ðkÞ where

 A1 ¼

 aþb 0 ; 1 0

 A2 ¼

if jx1 ðkÞ  x2 ðkÞj > d; ð3Þ otherwise;

a þ kb k

ð1  kÞb ð1  kÞ

 ð4Þ

in which ja + bj < 1. Note that when k = 1, A2 = A1, the resulting system is a stable linear system. When k = 0,   a b A2 ¼ , we have system (1). Hence, by introducing k 2 [0, 1], we get a family of systems. The advantage of this 0 1 reformulation is obvious: If we know clearly the transition as k moves from 1 to 0, we can have some better understanding of the original system (1). This is carried next.

4. System analysis In this section, we study the transition process in the family of systems defined by Eqs. (3) and (4) as k moves from 1 to 0. We illustrate this process by examples. In the following two subsections, we only discuss the case with a ¼ 9=10;

b ¼ 3=10;

d ¼ 1=100.

ð5Þ

4.1. The region of converging orbits The following result is given in [14]: Proposition 1. For each k 2 (0, 1] and under Eq. (5), the system defined by Eqs. (3) and (4) has a unique fixed point (0, 0), which is (locally) asymptotically stable. According to Proposition 1, there is a neighborhood around the origin (0, 0), each orbit starting within which will converge to the origin. One question is: Is it possible to determine this local stability region analytically? This problem turns out to be difficult. For example, assume k = 16/100. Then starting from an initial point (10, 10), the orbit is oscillating (see Fig. 1). Based on this, one may guess that each orbit starting from an initial point further away from (0, 0) than (10, 10) will be oscillating. However, this is not true. In fact, the trajectory starting from (100, 870) converges to the origin (see Fig. 2). This example tells us that the set of converging orbits is hard to find, even if it is possible. Clearly, this problem is closely related to the problem of computational complexity (see [2] and some references therein). 4.2. Phase transition In this subsection, we study the transition process as k moves from 1 to 0. Here, we fix an initial point (10, 10) and study the evolution of its orbits as k varies. To be precise, we use Mathematica to find these analytic orbits instead of

G. Zhang et al. / Chaos, Solitons and Fractals 30 (2006) 1153–1164

1157

–3

8

x 10

6 4

x

2

1

0 –2 –4 –6 –6

–4

–2

0

2

x2

4

6

8 –3

x 10

Fig. 1. An oscillating orbit starting from (10, 10).

–28

0.5

x 10

0 –0.5 –1

x1 –1.5 –2 –2.5 –3 –3.5 –2

0

2

x

2

4

6

8 –29

x 10

Fig. 2. A converging orbit starting from (100, 870).

numerical ones, for the reason to be given at the end of this subsection. We start from k = 165/104. It is found that the orbit is converging. So, we reduce k (we expect that the smaller the k is, the more unstable the system will be) by choosing k = 162/104, and the orbit oscillates. Then, we choose a bigger k, k = 163/104, but in this case the orbit converges to the origin. We then reduce k again by choosing k = 1625/105, and the trajectory is oscillating. However, for k = 1626/ 105, it converges. At k = 1, we have a stable linear system. When k is slightly less than 1, the system is nonlinear, but is still globally asymptotically stable. As k goes further toward 0, the nonlinearity becomes more and more prevalent and ends up with chaos at k = 0. We are now in a position to pose two questions: 1. As far as a particular orbit is concerned, say, the one starting from (10, 10), is there a k0 2 (0, 1) such that the orbit converges to the origin for all 1 P k > k0, whereas it oscillates for all 0 6 k < k0? If so, how will the orbit starting from (10, 10) behave at this k0: converging to the origin or to a periodic orbit, oscillating randomly or being periodic? 2. Is there a k1 2 (0, 1) such that all orbits will converge for all 1 P k > k1, whereas there is at least one oscillating orbit for each 0 6 k < k1? Again, if so, how will trajectories behave when k = k1?

1158

G. Zhang et al. / Chaos, Solitons and Fractals 30 (2006) 1153–1164

To answer the above questions, one necessary step is to study some particular systems. For convenience, we denote the system with k = 1 by Rs, and that with k = 0 by Ru. The following cases are worth discussing (Rs is always a linear stable systems for all these cases). Case 1. a = 9/10 and b = 3/10: Ru has a local stability region and an oscillating region. Case 2. a = 1 and b = 3/10: Ru has a local stability region, at least two sets of dense periodic orbits of periods 15 and 24 respectively, and an oscillating region. Case 3. a = 3/10 and b = 9/10: For Ru, the set of fixed points are globally attracting. Case 4. a = 1 and b = 1/2: Each orbit of Ru is an eventually periodic orbit of period 2. Apparently, the transition process as k moves should explain the above cases. Another interesting question is: Is there any continuous function of k as it goes from 1 to 0? The first candidate coming to mind is, of course, the system trajectories. However, they are unlikely. For example, as studied in [13,14], Rs is a linear stable system, and systems with k very close to 1 also have the origin as the unique global attractor, whereas Ru has an oscillating region. Hence, as k decreases, aperiodic orbits come into being, which disqualify the continuation of trajectories as a function of k. The same argument asserts that neither the system attractors nor structural stability are continuous with respect to k. Then, what about dissipativity? (Here, a system is called dissipative if the volume of its attractors has Lebesgue measure 0.) One may disprove this by saying that there are dense periodic orbits in the case of a = 1 and b = 3/10 (see [13,14] for details). However, our argument here is that systems with periodic orbits are not typical in this type of systems. As proved in [13], an extremely slight perturbation will perturb the parameter b from being rational to being irrational, and as a result there are no periodic orbits at all. Our analysis and extensive simulations strongly support that dissipativity is invariant (hence continuous) as a function of k. If this is true, following Section 2 the system Ru with a = 9/10 and b = 3/10 has a Cantor set as an invariant set on which the system is topologically transitive. Up to this point, some comments of Feigenbaum [6] come to mind, ‘‘. . ., while a vague impression of what one wants to know is sensibly clear, a precise delineation of many of these quests is not so readily available. In a state of ignorance, the most poignantly insightful questions are not yet ripe for formulation. Of course, this comment remains true despite the fact that, for technical exigencies, there are definite questions that one desperately wants the answers to.’’ In the above numerical analysis, we used the software Mathematica to get the exact orbits for each case. The reason to do this is to avoid numerical errors. Next, we discuss the problem of numerical precision by an example. Suppose a¼

9 ; 10

b¼

3  1:4142135623730950 ; 10

d¼

1 100

in Eq. (1), with an initial condition x1 ð0Þ ¼

ða þ bÞ  ðdÞ ; 2

d . 2

x2 ð0Þ ¼

–3

8

x 10

6 4 2

x1 0 –2 –4 –6 –8 –8

–6

–4

–2

0

x2

2

4

6

8 –3

x 10

Fig. 3. One trajectory generated via Mathematica.

G. Zhang et al. / Chaos, Solitons and Fractals 30 (2006) 1153–1164

1159

1

0.5

0

x1 –0.5

–1

–1.5 –1

–0.5

0

x2

0.5

1

1.5

Fig. 4. Another trajectory generated via Mathematica.

Using Mathematica, we get the result shown in Fig. 3. Now, modify b to be b ¼  314;142;135;623;730;950 , and fix the numer1017 ical precision to be 200 decimal points (the original one was 17), and we get Fig. 4. Which plot is a faithful representation of the true orbit? By using the exact value of b ¼  314;142;135;623;730;950 to produce the true trajectory, we found it 1017 corresponds to Fig. 3. Now, we see the difficulty: by increasing numerical precision, the result becomes less precise! So, we conclude that one has to be very careful when dealing with this type of systems because of its bewildering complexity. 4.3. Analysis of difficulties involved in the study the complex systems In this section, we analyze the complexity of this type of systems from the viewpoint of approximation and computational precision. Define     aþb 0 a b ; A2 ¼ ; C ¼ ½ 1 1 . A1 ¼ 1 0 0 1 Then, system (1) can be rewritten as 1 xðk þ 1Þ ¼ ððA1 þ A2 Þ þ ðA1  A2 Þð1  uðkÞÞÞxðkÞ; 2 yðkÞ ¼ CxðkÞ; uðkÞ ¼ 1  sgnðyðkÞ sgnðyðkÞÞ  dÞ.

ð6Þ

Fig. 5 is the plot of u as a function of y. Clearly, there are two discontinuous points: one is at d and the other is at d (here d = 1). Next, we use two different forms to approximate this function u. The first coming to mind is polynomials. We attempt to approximate the discontinuity at the point y = 1 up to degree 2. Hence, the polynomial to be used is P ðyÞ :¼ a3 y 3 þ a2 y 2 þ a1 y þ a0

ð7Þ 3

with ai, i = 0, . . . , 3 to be determined. Choose a small interval, [1  c, 1 + c], where c = 1/10 , and set P ð1 þ cÞ ¼ 0; P ð1  cÞ ¼ 1;

P 0 ð1 þ cÞ ¼ 0; P 0 ð1  cÞ ¼ 0.

Then, we can get the polynomial as P ðyÞ ¼ 500; 000; 000y 3  1; 500; 000; 000y 2 þ 1; 499; 998; 500y  499; 998; 499;

ð8Þ

whose plot at [1  c, 1 + c] is Fig. 6 and at [1  c  1/1000, 1 + c + 1/1000] is Fig. 7. It is evident that the polynomial P(y) defined in Eq. (8) is a bad approximation because its parameters are very big, thus very prone to perturbations (that is why Fig. 7 is quite different from that of the original system). Notice that by using a polynomial approximation we get a continuous function approximating the original discontinuous function.

1160

G. Zhang et al. / Chaos, Solitons and Fractals 30 (2006) 1153–1164 u 2

1.5

1

0.5

0 –2

–1

0

1

2

Fig. 5. u has two discontinuous points: one is at d = 1 and the other is at d = 1.

py 2

1.5

1

0.5

0 0.998

y 0.999

1

1.001

1.002

Fig. 6. One plot of polynomial equation (8).

py 2

1.5

1

0.5

0

y

0.999

0.9995

1

1.0005

1.001

Fig. 7. Another plot of polynomial equation (8).

Another way to approximate u is to use hyperbolic tangent functions. In fact, u can be approximated by 8 2; d þ q < y < d  q; > > > > > tanhðsyÞ dq6y 6dþq > < y > d þ q; uðyÞ ¼ 0; > > > > tanhðsyÞ; d  q 6 y 6 d þ q; > > : 0; y < d  q;

ð9Þ

G. Zhang et al. / Chaos, Solitons and Fractals 30 (2006) 1153–1164

1161

–3

5.5069

x 10

5.5069 5.5069 5.5069

x1

5.5069 5.5069 5.5069 5.5069 5.5069 5.5069 5.5069 –1.5

–1

–0.5

x2

0

0.5

1

Fig. 8. A trajectory with s = 2 · 106.

–3

8

x 10

6 4 2

x1 0 –2 –4 –6 –8 –8

–6

–4

–2

0

2

4

x

2

6

8 –3

x 10

Fig. 9. A trajectory with s = 3 · 106.

where q should be sufficiently small. Now, we test the effectiveness of this approximation. First, choose s = 2 · 106, use the function (9) to replace the original u, and then plot a trajectory as shown in Fig. 8. Next choose s = 3 · 106, and follow the same procedure. Then we get another trajectory, as shown in Fig. 9. The second trajectory is consistent with that of the original system. So, to get a good approximation, s will have to be very big. Remark 1. When q is small enough, the system composed of Eqs. (6) and (9) behaves like the original system. However, no matter how small q is, the modified system still has discontinuities, indicating that this approximation is not of much help. Even worse, to get a faithful approximation, s must be very large, which will inevitably leads to large numerical errors.

5. A glance at higher-dimensional cases In this section, we cast a glance at a higher-dimensional system governed by the switching law specified in Eq. (1). Here, the discussion is merely descriptive, for which we just present an example.

1162

G. Zhang et al. / Chaos, Solitons and Fractals 30 (2006) 1153–1164

Example 1. Consider the following system:  A1 xðkÞ; if jx1 ðkÞ  x3 ðkÞj > 1; xðk þ 1Þ ¼ A2 xðkÞ; otherwise; where

2

b1 a2 0

a1 6 A1 ¼ 4 b2 1

3 0 7 0 5; 0

2

a1 6 A2 ¼ 4 kb2 k

b1 a2 0

ð10Þ

3 0 7 ð1  kÞb2 5. 1k

When k = 1, A2 = A1. When k = 0, 2 3 a1 b1 0 6 7 A2 ¼ 4 0 a2 b2 5. 0 0 1 Fix a1 ¼

8 ; 10

b1 ¼ 1;

a2 ¼ 1;

b2 ¼ 

1 10

we plot the variation of the trajectory starting from ð1; 1=105 ; 1Þ as k varies (see Figs. 10 and 11: One can view the phase transition process vividly from these figures).

6. Difficulties in proving chaos The most rigorous way in proving that an autonomous dynamical system is chaotic is to show that there are Smale horseshoes [12,7]. Clearly there will be infinitely many periodic orbits if there are Smale horseshoes. Unfortunately, the systems that we are studying here generally do not possess periodic orbits. As a result, this way to prove chaos does not promise much, if any.

200

x

3

20

x

0

3 0

–200 20

50

0

x

2

0 –20 –50

–20 5 2

1

λ = –100

5

10

0

x

x

0 –5 –10 λ = –10

x

1

1

x3 0

x3 0

–5 2

x

5

0

2

0 –2 –5

x

1

–1 0.5

x

2

0

2

λ = –1

Fig. 10. Attractors in 3-d:I.

0 –0.5 –2 λ=0

x

1

G. Zhang et al. / Chaos, Solitons and Fractals 30 (2006) 1153–1164

1163

–215

x 10 4

5

x 2

x

3

3

0

0 –2 1 –215

x 10

0

x

2

–5 1

2 0 –1 –2

x

1

–215

x

x 10

2

λ=1

10

2

0

0 –1 –2 λ=5

x

1

200

x3 0

x3

–10 – 1

2

0

x2

0 –1 –2

x1

0

–200 20

x

50

0

2

λ = 10

0 –20 –50 λ = 100

x

1

Fig. 11. Attractors in 3-d:II.

A chaotic system normally is ergodic. For a given map, if it is expensive, then it is possible to prove ergodicity with resort to Perron–Frobenius operators [9]. For example, ergodicity of an unstable quantized scalar system is intensively studied in [4]. In essence, results obtained there depend heavily on the affine representation of the system by which it is piecewise expanding, i.e., the absolute value of the derivative of the piecewise affine map in each interval is greater than 1. Based on this crucial property, the main theorem (Theorem 1) in [10] and then that of [11] are employed to show that there exists a unique invariant measure on which the affine map is ergodic. Therefore, ergodicity is established for scalar unstable quantized systems. However, this is not the case for the systems we are studying. Though theses maps are piecewise linear, they are singular with respect to the Lebesgue measure and, furthermore, the derivative of the system in a certain region is (a + b), whose absolute value is strictly less than 1. Consequently, the results in [10,11] are not applicable here. Another approach to proving ergodicity is by means of the Markov transition. For a piecewise linear map, say C, suppose one subinterval, T1, contains a subset of the image of another interval, T2, that is, T1 \ C(T2) is not empty, then it is required that T1  C(T2). If this relation holds for all subintervals, then the Markov transition can be used to study the ergodicity of the map. Unfortunately, systems studied here do not satisfy this condition. Therefore, this method is not applicable either. We comment that a critical point in the existing approaches of proving chaos is that the system or map involved has some kind of hyperbolic structure [5]. Apparently, systems discussed here lack this key property. All in all, to prove chaos in the system under investigation seems to be extremely difficult, leaving an interesting and yet challenging problem for future research.

7. Conclusions In this paper, a type of nonsmooth dynamical systems induced by some switching mechanism has been studied. The following problems remain for our future research: • How to verify if the nonsmooth dynamical system (1) is dissipative? • Is it possible to determine the exact local stability region studied in Section 4.1? • What is the phase transition process for system (3) and (4) and as k varies? For example, what kind of changes will the attractors undergo? We believe that these problems, among others, are common in nonsmooth dynamical systems induced by state-dependent switchings. Therefore, we hope that the family of systems discussed here can serve as a good model in this field of research.

1164

G. Zhang et al. / Chaos, Solitons and Fractals 30 (2006) 1153–1164

Acknowledgment The first author is very grateful to the discussions with Dr. Marı´a DÕAmico, Prof. Michael Li, Prof. Feng Ding, Dr. Shujun Li and Dr. Cailin Xu.

References [1] Bernardo M, Budd C, Champneys A, Kowalczyk P, Nordmark A, Olivar G, et al. Bifurcations in nonsmooth dynamical systems. BCANM Preprint, 2005. [2] Blum L, Cucher F, Shub M, Smale S. Complexity and real computation: a manifesto. Int J Bifurc Chaos 1996;6:3–26. [3] Banerjee S, Yorke J, Grebogi C. Robust chaos. Phys Rev Lett 1998;80:3049–52. [4] Delchamps D. Stabilizing a linear system with quantized state feedback. IEEE Trans Automat Contr 1990;35:916–24. [5] Devancy R. An introduction to chaotic dynamical systems. WestView; 2003. [6] Feigenbaum M. The transition to chaos. In Chaos: a new sciences, The 26th nobel conference, 1990. p. 45–53. [7] Guckenheimer J, Holmes P. Nonlinear oscillations, dynamical systems, and bifurcations of vector fields. Springer-Verlag; 1983. [8] Liberzon D. Switching in systems and control. Boston: Birkha¨user; 2003. [9] Lasota A, Mackey M. Chaos, fractals, and noise: stochastic aspects of dynamics. 2nd ed. Springer-Verlag; 1994. [10] Lasota A, Yorke J. On the existence of invariant measures for piecewise monotonic transformations. Trans Amer Math Soc 1973;186:481–8. [11] Li T, Yorke J. Ergodic transformations from an interval to itself. Trans Amer Math Soc 1978;235:183–92. [12] Smale S. Differentiable dynamical systems. Bull Amer Math Soc 1967;73:747–817. [13] Zhang G, Chen T. Networked control systems: a perspective from chaos. Int J Bifurc Chaos 2005;15:1–27. [14] Zhang G, Chen G, Chen T, DÕAmico M. Analysis of a networked control system. Int J Bifurc Chaos, accepted for publication.