Anti-control of continuous-time dynamical systems

Commun Nonlinear Sci Numer Simulat 17 (2012) 2617–2627

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Anti-control of continuous-time dynamical systems Simin Yu a, Guanrong Chen b,⇑ a b

College of Automation, Guangdong University of Technology, Guangzhou 510006, China Department of Electronic Engineering, City University of Hong Kong, Hong Kong SAR, China

a r t i c l e

i n f o

Article history: Received 3 July 2011 Received in revised form 2 October 2011 Accepted 2 October 2011 Available online 19 October 2011 Keywords: Chaos Continuous-time system Anti-control Global boundedness Lyapunov exponent placement

a b s t r a c t Based on two basic characteristics of continuous-time autonomous chaotic systems, namely being globally bounded while having a positive Lyapunov exponent, this paper develops a universal and practical anti-control approach to design a general continuoustime autonomous chaotic system via Lyapunov exponent placement. This self-unified approach is verified by mathematical analysis and validated by several typical systems designs with simulations. Compared to the common trial-and-error methods, this approach is semi-analytical with feasible guidelines for design and implementation. Finally, using the Shilnikov criteria, it is proved that the new approach yields a heteroclinic orbit in a three-dimensional autonomous system, therefore the resulting system is indeed chaotic in the sense of Shilnikov. Ó 2011 Elsevier B.V. All rights reserved.

1. Introduction Anti-control of chaos, or chaotification, refers to the desire of generating chaos from a non-chaotic system by a simple control input. For continuous-time dynamical systems, although several successful techniques have been developed for the task, such as time-delay feedback, topological conjugate mapping, and impulsive control [1–10], there are no very effective and universal methodologies available in the literature today. Most reports in the existing literature took a trialand-error approach to anti-controlling continuous-time autonomous systems, through parameter tuning, numerical simulation and Lyapunov exponent calculation, which by no means provide unified theoretical guidelines for designers to follow [11–13]. For discrete-time systems, namely for mappings, the situation is much more promising. Anti-control of discrete-time systems has developed several relatively complete theories and relatively mature techniques with analytic guidelines for the users supported by rigorous mathematical chaos theory [14–25]. As a result, the traditional numerical approach of trial-and-error with computer simulation has literally become only a means of verification in general. The first milestone of a mathematically-rigorous anti-control theory and method was attributed to the Chen–Lai anti-control algorithm initiated in 1996 [14–21], followed by the Wang–Chen chaotification scheme and the Shi–Chen theory of coupled-expanding maps developed in the 2000s [22–25]. By comparison, it is very natural to ask whether or not the discrete-time anti-control methods can be directly modified and applied to the continuous-time setting. The answer is generally no, because they are described by difference and differential equations respectively, which have many essential distinctions. One prominent difference in point is the Lyapunov exponent placement: a discrete chaotic system can have all positive Lyapunov exponents but a continuous counterpart typically needs to have positive, zero and negative Lyapunov exponents so as to stretch and fold the orbit flows in the phase space. Nevertheless, they still share many similarities and analogies in both system structure and dynamics. ⇑ Corresponding author. E-mail address: [email protected] (G. Chen). 1007-5704/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2011.10.001

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In this paper, based on the aforementioned two basic characteristics of continuous-time autonomous chaotic systems, namely being global bounded while having a positive Lyapunov exponent, a universal feedback controller design criterion is derived for anti-controlling continuous-time autonomous dynamical systems to become chaotic. First, general forms of uncontrolled and controlled systems and the feedback controller to be used are established such that the controlled system outputs are globally bounded while the controlled system has positive, zero and negative Lyapunov exponents. These generic forms allow a systematic design and parameter determination of the anti-controlled system, overcoming the timeconsuming and uncertain trial-and-error parameter tuning disadvantages. To this end, the Shilnikov criteria are applied to a three-dimensional autonomous system as an example to show that the resulting anti-control system possesses a heteroclinic orbit therefore is chaotic in the sense of Shilnikov, which means the existence of Smale horseshoes. The rest of the paper is organized as follows. Section 2 describes the two questions to be investigated. Section 3 proposes the general criterion for anti-controlling continuous-time autonomous dynamical systems. Section 4 shows several design examples in general forms. Section 5 demonstrates by the Shilnikov criteria the existence of a heteroclinic orbit in the designed anti-controlled three-dimensional autonomous system, thereby proving the chaoticity of the resulting system. Section 6 concludes the investigation. 2. Problem statements Consider an n-dimensional continuous-time linear autonomous system

x_ ¼ Ax

ð1Þ T

where x = [x1, x2, . . . , xn] with a real system matrix

0

1

a11

a12

   a1n

Ba B 21 A¼B B .. @ .

a22 .. .

   a2n C C .. .. C C . . A

an1

an2

   ann

ð2Þ

In modern control theory, a basic technical problem is: assuming that the origin of the uncontrolled system (1) is an unstable equilibrium, design a linear feedback controller for the system such that the origin of the controlled system becomes asymptotically stable. On the contrary, a basic problem of anti-control theory is: assuming that the origin of the uncontrolled system (1) is an asymptotically stable equilibrium, design a simple nonlinear feedback controller f(rx, e) such that the controlled system

x_ ¼ Ax þ Bf ðrx; eÞ

ð3Þ

becomes chaotic, where B is a control matrix to be designed:

0

1

b11

b12

   b1n

Bb B 21 B¼B B .. @ .

b22 .. .

   b2n C C .. .. C C . . A

bn1

bn2

   bnn

ð4Þ

and the nonlinear feedback controller

1 f1 ðr1 x1 ; e1 Þ B f ðr x ; e Þ C B 2 2 2 2 C C f ðrx; eÞ ¼ B .. C B A @ . 0

ð5Þ

fn ðrn xn ; en Þ where



0

1

B 0 B

r2   

C C C C A

0

r1

0

@ .

.. .

..

.

0 .. .

0

0



rn

r¼B B ..

ð6Þ

is the gain matrix, and e is an upper bound for the controller (5), which is also to be designed:

e ¼ ½e1 ; e2 ; . . . ; en T

ð7Þ

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In this paper, a universal approach to anti-control of continuous-time autonomous systems is proposed in a general form. Two tasks will be accomplished: (i) Design A, B, f(rx, e), r, and e such that the controlled system is dissipative and globally bounded, with positive, zero and negative Lyapunov exponents. (ii) Apply the Shilnikov criteria to a three-dimensional autonomous system to show that the resulting controlled system has a heteroclinic orbit which is chaotic in the sense of Shilnikov. 3. Anti-control principles and design criteria Consider a given system as shown in (1) and its controlled setting as (3). The anti-control principles and design criteria are developed in this section. 3.1. General principle of anti-control In this subsection, some general anti-control principles for system (3) are discussed. First, observe that, unlike discrete-time systems, for continuous-time systems one cannot simply set all of their Lyapunov exponents to be positive. Instead, all n-dimensional (n P 3) autonomous chaotic systems have positive, zero and negative Lyapunov exponents. The following gives a necessary condition for the controlled system (3) to be chaotic. Theorem 1. If the real parts of all eigenvalues of matrix A are negative, and if

sup kf ðrx; eÞk 6 kek < 1

ð8Þ

06t<1

then the orbits of the controlled system (3) are globally bounded, where k  k is the Euclidean norm. Proof. It can be easily verified that the solution of (3) is given by

xðtÞ ¼ expðAtÞ  xð0Þ þ

Z

t

exp½Aðt  sÞ  Bf ðrxðsÞ; eÞds

ð9Þ

0

where x(t) = (x1(t),x2(t), . . . , xn(t))T. Since the real parts of all eigenvalues of A are negative, there exist constants a, b > 0 such that

sup k expðAtÞk 6 aebt

ð10Þ

06t<1

Furthermore, since f(rx, e) is uniformly bounded, by combining (3) and (9) one has

sup kxðtÞk 6 sup aebt  kxð0Þk þ sup a  kBk  kek 06t<1

06t<1

06t<1

¼ sup aebt  kxð0Þk þ sup 06t<1

06t<1

Z

t

ebðtsÞ ds

0

a  kBk  kek b

ð1  ebt Þ 6 a  kxð0Þk þ

a  kBk  kek b

<1

ð11Þ

Namely, the orbits of system (3) are globally bounded, completing the proof of the theorem. h 3.2. Design criteria In this subsection, based on the global boundedness discussed above, it is to design A, B, f(rx, e), r and e, such that the controlled system (3) is dissipative and possesses positive, zero and negative Lyapunov exponents, thereby becoming chaotic. The following gives a sufficient condition for the controlled system (3) to be chaotic. Theorem 2. Consider the controlled system (3). If the following five conditions are all satisfied, then the system (3) is dissipative and chaotic. (i) Controller f(rx, e) is uniformly bounded. (ii) Real parts of all eigenvalues of matrix A are negative. (iii) For the diagonal entries of the Jacobian matrix of the controlled system evaluated at the origin x = 0, namely, for

0

J 11 BJ B 21 J0 ¼ B B .. @ . J n1

J12 J22 .. . J n2

1 1 0    J 1n a11 þ b11 @f1 ðr@x1 x11 ;e1 Þ    a1n þ b1n @fn ðr@xn xnn ;en Þ C    J 2n C C B .. .. .. C ¼B .. .. C C A @ . . . . . A @f1 ðr1 x1 ;e1 Þ @fn ðrn xn ;en Þ    ann þ bnn an1 þ bn1 @x1 @xn x¼0    J nn x¼0

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the following inequality holds: n X

J ii ¼

i¼1

n  X

aii þ bii

i¼1

@fi ðri xi ; ei Þ @xi



<0

ð12Þ

xi ¼0

(iv) Choose B such that all its diagonal entries are zero and there is at most one nonzero entry in each of its rows, namely,

0

0



   b1i





1

B 0 B B B    b2j B B¼B .. B .. . B . B . .. B . @ . .



   b2j

0 .. . .. .

 .. . .. .

 .. . .. .

C C C C C .. C C . C .. C C . A



bnl

...

...

0

...

ð13Þ

where main diagonal elements bpp = 0 (p = 1, 2, . . . , n) are all zeros. On each line, except b1i, b2j, b3k, . . . , bnl, other elements are zeros. b1i = b2j = b3k =    = bnl = 0, ±1 (but not all 0), i, j, k, . . . , l are not equal to each other, i = 2, 3, . . . , n, j = 1, 3, . . . , n, k = 1, 2, 4, . . . , n, . . . , l = 1,2, . . . , n  1. (v) Choose f(rx, e), r and e such that the n roots of the algebraic polynomial jJ0  kIj = 0 satisfy the following four conditions: (1) There are n  2 real roots ci (i = 1, 2, . . . , n  2) and one pair of complex conjugate roots k± = g ± jx, satisfying the Shilnikov inequalities jcij > jgj. (2) The origin is a saddle-node of index 2 or 3. (3) They enable the placement of positive, zero and negative Lyapunov exponents into the controlled system (3) by modifying both real and imaginary parts of the Jacobian eigenvalues. Proof. By Theorem 1, it follows from conditions (i) and (ii) that the orbits of the controlled system (3) are globally bounded. Condition (iii) implies that the controlled system is dissipative. Thus, condition (iv) furthermore guarantees that the Jacobian matrix of the controlled system will never be diagonally dominant, therefore it is possible to enable the placement of positive, zero and negative Lyapunov exponents simultaneously. Finally, condition (v) guarantees that the origin is a saddle-node of index 2 or 3. So, on the one hand, the controlled system has an expanding manifold therefore one can place positive Lyapunov exponents; on the other hand, there exist eigenvalues with negative real parts and the summation of all the negative real parts is large than that of all the positive real parts, therefore one can also place zero and negative Lyapunov exponents. Moreover, all these can be done by suitably choosing the parameters in A, B, f(rx, e), r and e, through modifying both real and imaginary parts of the Jacobian eigenvalues of the controlled system (3). This completes the proof of the theorem. h

Remark 1. Theorem 2 actually only provides some sufficient conditions for the design of a working anti-controller, but it does not tell how to choose those parameters. In this regard, the following three aspects need to be taken into consideration in the anti-controller design: (1) If the eigenvalues of the system matrix A do not satisfy the condition (ii) in the theorem, then one may consider adopte þ Bf ðrx; eÞ, and choose A e such that A þ A e together satisfies the condition (ii). ing an anti-controller of the form A (2) The nonlinear feedback controller f(rx, e) can have the simple form of (5), namely, its every component fi(rixi, ei) is only a function of one state variable xi, i = 1, 2, . . . , n. (3) The state equations of the anti-controlled system can be expressed as follows:

8 x_ 1 ¼ a11 x1 þ a12 x2 þ    þ a1n xn þ b1i fi ðri xi ; ei Þ > > > > > > < x_ 2 ¼ a21 x1 þ a22 x2 þ    þ a2n xn þ b2j fj ðrj xj ; ej Þ x_ 3 ¼ a31 x1 þ a32 x2 þ    þ a3n xn þ b3k fk ðrk xk ; ek Þ > > >  > > > : x_ n ¼ an1 x1 þ an2 x2 þ    þ ann xn þ bnl fl ðrl xl ; el Þ

ð14Þ

It is noted that in order to ensure be able to place positive, zero and negative Lyapunov exponents, it is required to avoid possible diagonal dominant of the controlled system Jacobian matrix, i.e., each component fi(rixi, ei) should not appear in the corresponding state equation. For this purpose, system (14) should satisfy condition i – j – k –    – l, i = 2, 3, . . . , n, j = 1, 3, . . . , n, k = 1, 2, 4, . . . , n, . . . , l = 1, 2, . . . , n  1. On the contrary, for the discrete-time setting, such as for the Chen–Lai and Wang–Chen algorithms, the diagonal dominance property is acceptable, system (14) can only satisfy condition i = 1, j = 2, k = 3, . . . , l = n.

S. Yu, G. Chen / Commun Nonlinear Sci Numer Simulat 17 (2012) 2617–2627

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4. Design examples Based on the general guidelines developed above, this section presents detailed design examples of typical three-, fourand five-dimensional continuous-time autonomous systems for anti-control of chaos, which all serve for different purposes of illustration. In these examples, the nonlinear feedback controller uses a simple piecewise linear saw-tooth function, yet it should be noted that the method developed here is universal in the sense that it works for all uniformly bounded nonlinear functions such as the modulo and sine functions, and it works for any finite-dimensional system although the design will become more complex as the dimension increases. 4.1. Three-dimensional systems Start from the general form of a three-dimensional autonomous system,

0

1 0 a11 x_ 1 B_ C B @ x2 A ¼ @ a21 x_ 3

a31

a12

a13

10

x1

1

0

b11

a22

CB C B a23 A@ x2 A þ @ b21

a32

a33

x3

b31

b12

b13

10

f1 ðr1 x1 ; e1 Þ

1

b22

C CB b23 A@ f2 ðr2 x2 ; e2 Þ A

b32

b33

ð15Þ

f3 ðr3 x3 ; e3 Þ

Take x1 as the anti-control state variable for feedback, and use a saw-tooth controller of the form



f1 ðr1 x1 ; e1 Þ ¼ e1  sawtoothðpr1 ðx1  e1 =r1 Þ=e1 ; pÞ

ð16Þ

f2 ðr2 x2 ; e2 Þ ¼ f3 ðr3 x3 ; e3 Þ ¼ 0 where p = 1, r1 = 6.943, e1 = 3.4715, and sup06t<1kf1(r1x1, e1)k 6 e1 = 3.4715 < 1, with f1(r1x1, e1) as shown in Fig. 1. Giving

0

1 0:4 1:95 2:05 B C A ¼ @ 1:95 2:3 0:05 A; 2:05 0:05 1:7

0

1 0 0 0 B C B ¼ @ 1 0 0 A 0 0 0

and substituting them into (15) yields

8 > < x_ 1 ¼ 0:4x1  1:95x2 þ 2:05x3 x_ 2 ¼ 1:95x1  2:3x2 þ 0:05x3  f1 ðr1 x1 ; e1 Þ > :_ x3 ¼ 2:05x1 þ 0:05x2 þ 1:7x3

ð17Þ

It can be easily verified that the uncontrolled system matrix A has eigenvalues k1 = 0.3 and k2,3 = 0.35 ± j2.0, therefore the origin of the uncontrolled system is asymptotic stable. Furthermore, according to (17), the eigenvalues of the controlled system Jacobian at the origin are k1 = 4.3912 and k2,3 = 1.6956 ± j1.6763, where the origin is a saddle-node of index 2. Thus, all conditions stated in Theorem 2 are satisfied. Indeed, chaos can be generated as shown in Fig. 2. As another example, in (14), use x1, x2, x3 as feedback control states, along with the saw-tooth feedback anti-controller

fi ðri xi ; ei Þ ¼ ei  sawtoothðpri ðxi  ei =ri Þ=ei ; pÞ

ð18Þ

where p = 1,r1 = 2.1, r2 = 0.35, r3 = 1.0, e1 = 0.75, e2 = 0.175, e3 = 0.5, and it can be verified that sup06t<1kfi(rixi, ei)k 6 0.75 < 1, i = 1, 2, 3. Also, choose

0

0:7 2:9 3:1

B A ¼ @ 2:9

3:1

1

C 3:5 0:1 A; 0:1

2:5

0

0

0

0

1 0

B B ¼ @ 1 0

1

1

C 0A

f1 (σ 1 x1 ,ε 1 )

σ1

ε1

0 ε 1 /σ 1

Fig. 1. The saw-tooth function f1(r1x1, e1).

x1

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Fig. 2. The chaotic attractor generated by anti-controller using state variable x1.

Fig. 3. The chaotic attractor generated by anti-controller using state variables x1, x2, x3.

Substituting them into (15) gives

8 > < x_ 1 ¼ 0:7x1  2:9x2 þ 3:1x3 þ f3 ðr3 x3 ; e3 Þ x_ 2 ¼ 2:9x1  3:5x2 þ 0:1x3  f1 ðr1 x1 ; e1 Þ > :_ x3 ¼ 3:1x1 þ 0:1x2 þ 2:5x3 þ f2 ðr2 x2 ; e2 Þ

ð19Þ

It can be easily verified that the uncontrolled system matrix A has eigenvalues k1 = 0.5 and k2,3 = 0.6 ± j3.0, therefore the origin of the uncontrolled system is asymptotic stable. Furthermore, according to (19), the eigenvalues of the controlled system Jacobian at the origin are k1 = 2.8978 and k2,3 = 0.5989 ± j3.1697, where the origin is a saddle-node of index 2. Thus, all conditions stated in Theorem 2 are satisfied. Indeed, chaos can be generated as shown by Fig. 3. 4.2. Four-dimensional systems Consider a general form of a four-dimensional autonomous system,

1 0 a11 x_ 1 B x_ C B a B 2 C B 21 B C¼B @ x_ 3 A @ a31 0

a41

x_ 4

a12 a22

a13 a23

a32

a33

10 1 0 x1 a14 b11 Bx C Bb a24 C CB 2 C B 21 CB C þ B a34 A@ x3 A @ b31

a42

a43

a44

x4

b41

b12

b13

b22

b23

b32 b42

b33 b43

1 f1 ðr1 x1 ; e1 Þ C B b24 CB f2 ðr2 x2 ; e2 Þ C C C CB b34 A@ f3 ðr3 x3 ; e3 Þ A b44 f4 ðr4 x4 ; e4 Þ b14

10

ð20Þ

Give x1, x2, x3, x4 to be feedback anti-control state variables, with sine function controller

fi ðri xi ; ei Þ ¼ ei  sinðri xi Þ

ð21Þ

where r1 = 1, r2 = 2, r3 = 2, r4 = 2, e1 = 0.5, e2 = 1, e3 = 1, e4 = 1, and it satisfies sup06t<1kfi(rixi, ei)k 6 1 < 1, i = 1, . . . , 4. Moreover, choose

0

1

0:5 4:9

5:1

1:0

B 4:9 5:3 B A¼B @ 5:1 0:1

0:1

1:0 C C C; 1:0 A

1:0

2:0

4:7

3:0 1:0

0

1

0 1

0

0

B1 0 B B¼B @0 0

0

0C C C 1A

0

0

0 1 0

S. Yu, G. Chen / Commun Nonlinear Sci Numer Simulat 17 (2012) 2617–2627

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Substituting them into (20) leads to

8 x_ 1 > > > < x_ 2 > _3 x > > : x_ 4

¼ 0:5x1  4:9x2 þ 5:1x3 þ 1:0x4 þ f2 ðr2 x2 ; e2 Þ ¼ 4:9x1  5:3x2 þ 0:1x3 þ 1:0x4 þ f1 ðr1 x1 ; e1 Þ

ð22Þ

¼ 5:1x1 þ 0:1x2 þ 4:7x3 þ 1:0x4 þ f4 ðr4 x4 ; e4 Þ ¼ 1:0x1 þ 2:0x2  3:0x3  1:0x4  f3 ðr3 x3 ; e3 Þ

The uncontrolled system matrix A has four eigenvalues k1 = 1, k2 = 0.3, and k3,4 = 0.4 ± j5.0, so the origin of the uncontrolled system is asymptotically stable. It follows from (22) that the eigenvalues of the controller system Jacibian at the origin are k1 = 15.2615, k2 = 9.8142, k3,4 = 1.6737 ± j22.3207, so the origin is a saddle-node of index 3. Thus, all conditions in Theorem 2 are satisfied, with chaotic attractor generated as shown in Fig. 4. 4.3. Five-dimensional systems Again, start from a general five-dimensional autonomous system

1 0 a11 x_ 1 B_ C B B x2 C B a21 B C B B x_ 3 C ¼ B a31 B C B B_ C B @ x4 A @ a41 a51 x_ 5 0

a12

a13

a14

a22

a23

a24

a32

a33

a34

a42

a43

a44

a52

a53

a54

a15

10

x1

1

0

b11 CB C B a25 CB x2 C B b21 CB C B B C B a35 C CB x3 C þ B b31 CB C B a45 A@ x4 A @ b41 a55 x5 b51

b12 b22

b13 b23

b14 b24

b32

b33

b34

b42

b43

b44

b52

b53

b54

1 10 f1 ðr1 x1 ; e1 Þ b15 C CB b25 CB f2 ðr2 x2 ; e2 Þ C C CB C B b35 C CB f3 ðr3 x3 ; e3 Þ C C CB b45 A@ f4 ðr4 x4 ; e4 Þ A f5 ðr5 x5 ; e5 Þ b55

ð23Þ

Give x1, x2, x3, x4, x5 to be anti-control state variables, and use hyperbolic tangent feedback controller

fi ðri xi ; ei Þ ¼ ei  tanhðri xi Þ

ð24Þ

where r1 = 1, r2 = 3, r3 = 4, r4 = 2, r5 = 2, e1 = 5, e2 = 15, e3 = 10, e4 = 10, e5 = 10, with sup06t<1kfi(rixi, ei)k 6 15 < 1, i = 1, . . . , 5. Take

0

0:5 4:9

5:1

1:0

1:0

1

C B 1:0 1:0 C B 4:9 5:3 0:1 C B B A ¼ B 5:1 0:1 4:7 1:0 1:0 C C; C B 2:0 3:0 1:0 1:0 A @ 1:0 1:0 1:0 1:0 1:0 1:0

0

0

1 0

0

0

1

C B B1 0 0 0 0C C B B B ¼ B0 0 0 1 0C C C B @0 0 0 0 1A 0

0

1 0

0

It then follows that (23) becomes

8 x_ 1 > > > > > > < x_ 2 x_ 3 > > > _4 x > > > : x_ 5

¼ 0:5x1  4:9x2 þ 5:1x3 þ 1:0x4 þ 1:0x5 þ f2 ðr2 x2 ; e2 Þ ¼ 4:9x1  5:3x2 þ 0:1x3 þ 1:0x4 þ 1:0x5 þ f1 ðr1 x1 ; e1 Þ ¼ 5:1x1 þ 0:1x2 þ 4:7x3 þ 1:0x4  1:0x5 þ f4 ðr4 x4 ; e4 Þ

ð25Þ

¼ 1:0x1 þ 2:0x2  3:0x3  1:0x4  1:0x5 þ f5 ðr5 x5 ; e5 Þ ¼ 1:0x1 þ 1:0x2 þ 1:0x3 þ 1:0x4  1:0x5 þ f3 ðr3 x3 ; e3 Þ

The eigenvalues of the uncontrolled system matrix A are k1 = 1.0782, k2,3 = 0.5255 ± j5.0828, k4,5 = 0.4854 ± j1.0264, so the origin of the uncontrolled system is asymptotically stable. By (25), the eigenvalues of the controlled system Jacobian at the origin are k1 = 22.7700, k2 = 23.6662, k3 = 18.5388, k4,5 = 11.2675 ± j22.8665, therefore the origin is new a saddle-node of

Fig. 4. The chaotic attractor generated by anti-controller using state variables x1, x2, x3, x4.

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Fig. 5. The chaotic attractor generated by anti-controller using state variables x1, x2, x3, x4, x5.

index 2. It can be easily seen that all conditions of Theorem 2 are satisfied, so a chaotic attractor can be generated as shown in Fig. 5. 5. Existence of a heteroclinic orbit In this section, it is furthermore proved that in the three-dimensional setting, any anti-controlled system designed above, satisfying the Shilnikov criteria, indeed has a heteroclinic orbit, therefore is chaotic in the sense of Shilnikov [26,27]. For a higher-dimensional system, as long as it has a three-dimensional subsystem that satisfies the Shilnikov criteria, it is considered to be chaotic in the sense of Shilnikov. It follows from (15)–(17) that

8 > < x_ 1 ¼ a11 x1 þ a12 x2 þ a13 x3 x_ 2 ¼ a21 x1 þ a22 x2 þ a23 x3  f1 ðr1 x1 ; e1 Þ > :_ x3 ¼ a31 x1 þ a32 x2 þ a33 x3

ð26Þ

where f1(r1x1, e1) = e1  sawtooth (pr1x1/e1, p) is a saw-tooth function. By setting x_ ¼ y_ ¼ z_ ¼ 0 in system (26), one obtains the equilibrium equation

8    > < a11 x1 þ a12 x2 þ a13 x3 ¼ 0 a21 x1 þ a22 x2 þ a23 x3 ¼ f1 ðr1 x1 ; e1 Þ > : a31 x1 þ a32 x2 þ a33 x3 ¼ 0

ð27Þ

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Let

          a11 a12 a13    a11  a11 a12 0 a13  0 0 a12 a13                     D ¼  a21 a22 a23 ; D1 ¼  f1 ðr1 x1 ; e1 Þ a22 a23 ; D2 ¼  a21 f1 ðr1 x1 ; e1 Þ a23 ; and D3 ¼  a21 a22 f1 r1 x1 ; e1 :           a31 a32 a33    a31  a31 a32 0 a32 a33  0 a33  0 Then, the above equilibrium equation has a solution

8   D1 a13 a32 a12 a33  > f1 r1 x1 ; e1 > D < x1 ¼ D ¼   13 a31 f1 r1 x1 ; e1 x2 ¼ DD2 ¼ a11 a33 a D > > : x ¼ D3 ¼ a12 a31 a11 a32 f r x ; e  1 1 1 1 3 D D

ð28Þ

Denote k1 = D/(a13a32  a12a33), k2 = D/(a11a33  a13a31) and k3 = D/(a12a31  a11a32), and then rewrite (28) as

8     > < k1 x1 ¼ f1 r1 x1 ; e1  k2 x2 ¼ f1 r1 x1 ; e1 >   : k3 x3 ¼ f1 r1 x1 ; e1

ð29Þ

    By the odd-symmetry of (29), one obtains the two equilibria, P 1 x1 ; x2 ; x3 and P2 x1 ; x2 ; x3 , which are closest to the origin, as well as the distributions of the others, as shown in Fig. 6. Now, computing P1 and P2 one can get

8  > < x1 ¼ 2e1 =ðr1  k1 Þ x2 ¼ 2k1 e1 =k2 ðr1  k1 Þ > :  x3 ¼ 2k1 e1 =k3 ðr1  k1 Þ

ð30Þ

Since system (26) is piecewise linear, a straight line in the stable manifold ES(P1) in the subspace corresponding to the   equilibrium P 1 x1 ; x2 ; x3 is given by

ES ðP1 Þ :

x1  x1 x2  x2 x3  x3 ¼ ¼ l m n

ð31Þ

where l, m, n are constants; while a plane in the unstable manifold EU(P1) is determined by

      EU ðP 1 Þ : a x1  x1 þ b x2  x2 þ c x3  x3 ¼ 0

ð32Þ

where a, b, c are constants.   Similarly, for another equilibrium P 2 x1 ; x2 ; x3 , a straight line in the corresponding stable manifold ES(P2) is given by

ES ðP2 Þ :

x1 þ x1 x2 þ x2 x3 þ x3 ¼ ¼ l m n

ð33Þ

and a plane in the unstable manifold EU(P2) is described by

      EU ðP 2 Þ : a x1 þ x1 þ b x2 þ x2 þ c x3 þ x3 ¼ 0

ð34Þ

Thus, according to (31)–(34), the eigen-subspace corresponding to P1 and P2 can be specified, as shown in Fig. 7 As can be seen from Fig. 7, the cross point of ES(P1) and S : x1 = 0 is given by

 m n Q 1 ¼ ES ðP1 Þ \ S ! Q 1 0; x2  x1 ; x3  x1 l l

ð35Þ

while the cross line between EU(P1) and S : x1 = 0 is U

L1 ¼ E ðP1 Þ \ S !

(       a x1  x1 þ b x2  x2 þ c x3  x3 ¼ 0

ð36Þ

x1 ¼ 0

f1 (σ 1 x1 ,ε 1 )

k1 x1 P2 − x1∗

σ1 0 ε 1 /σ 1

ε1

x1∗ P1

x1

Fig. 6. The two equilibria P1 and P2 which are closest to the origin.

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S. Yu, G. Chen / Commun Nonlinear Sci Numer Simulat 17 (2012) 2617–2627

x3 x1

x2 L1 Q2

P1

0 Q1

P2

E U ( P1 ) E S ( P1 )

L2 V2

S

V1

E ( P2 ) E ( P2 ) S

U

Fig. 7. The eigen-subspace corresponding to P1 and P2.

Fig. 8. The dependence of d on parameter r1.

Similarly, ES(P2) and S : x1 = 0 intersect at a point,

 m n Q 2 ¼ ES ðP2 Þ \ S ! Q 2 0; x2 þ x1 ; x3 þ x1 l l

ð37Þ

and EU(P2) and S : x1 = 0 intersect along a line, U

L2 ¼ E ðP2 Þ \ S !

(       a x1 þ x1 þ b x2 þ x2 þ c x3 þ x3 ¼ 0 x1 ¼ 0

ð38Þ

It can be seen from Fig. 7 that if Q1 is located above L2, then by (35) and (38), one has

  m n ax1 þ b 2x2  x1 þ c 2x3  x1 ¼ 0 l l

ð39Þ

Similarly, if Q2 is located above L1, then by (36) and (37), the same equality (39) will hold automatically, which is clear from the symmetry about the origin as shown in Fig. 7. Hence, if Q2 is located above L1, then Q1 must be above L2. If Q1 is located above L2 and also Q2 is located above L1, then it means the controlled system (26) has a heteroclinic orbit     connecting the two equilibria P 1 x1 ; x2 ; x3 and P 2 x1 ; x2 ; x3 .

S. Yu, G. Chen / Commun Nonlinear Sci Numer Simulat 17 (2012) 2617–2627

2627

Now, substituting (30) into (39), one has

 m n k1 k1 þ 2b þ 2c ¼ 0 d¼ ab c l l k2 k3

ð40Þ

This is precisely the condition that the involving parameters should satisfy for the heteroclinic orbit to exist. Observe that it is not necessary to analytically find parameter values to satisfy the equality (40) to prove the existence of the heteroclinic orbit, which clearly will be very difficult, so numerical approach is adopted. To do so, one may vary r1 = 5–10 and let the parameters of matrices A and B be fixed as in (17). Since a, b, c, l, m, n, k1, k2, k3 are all dependent on r1, the compound parameter d is also dependent on r1, as shown in Fig. 8. As can be seen, when r1 = 6.943, one has d = 0, satisfying the condition for the existence of a heteroclinic orbit, as expected. Consequently, the anti-controlled system (26) is chaotic in the sense of Shilnikov. 6. Conclusions This paper has developed a universal and practical anti-control approach to designing an anti-controller for chaotifying a general continuous-time autonomous linear system. The method used is Lyapunov exponent placement while keeping the system orbits globally bounded. This self-unified approach has been verified both mathematically and numerically, with several typical systems designed and simulated, which demonstrates the feasibility of the new method. Compared to the common trial-and-error methods, this approach is semi-analytical, providing guidelines for analysis, design and implementation. More importantly, for the three-dimensional setting, it has been proved that the new approach yields a heteroclinic orbit therefore the resulting system is chaotic in the sense of Shilnikov. Acknowledgments This work was supported by the National Natural Science Foundation of China under Grants 61172023 and 60871025, the Natural Science Foundation of Guangdong Province under Grants 8151009001000060 and S2011010001018, and the Science and Technology Program of Guangdong Province under Grant 2009B010800037; and by the Hong Kong Research Grants Council under the GRF Grant CityU1114/11E. References [1] Wang XF. Generating chaos in continuous-time systems via feedback control. In: Chen G, Yu X, editors. Chaos control: theory and applications. Berlin: Springer-Verlag; 2003. p. 179–204. [2] Wang XF, Chen G. Chaotifying a stable LTI system by tiny feedback control. IEEE Trans Circuits Syst I 2000;47:410–5. [3] Wang XF, Chen G, Yu X. Anticontrol of chaos in continuous-time systems via time-delay feedback. Chaos 2000;10(4):771–9. [4] Zhou T, Chen G, Yang Q. A simple time-delay feedback anti-control method made rigorous. Chaos 2004;14(3):662–8. [5] Wang XF, Chen G. 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