Control chaos to different stable states for a piecewise linear circuit system by a simple linear control

Chaos, Solitons and Fractals 130 (2020) 109431

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Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Control chaos to different stable states for a piecewise linear circuit system by a simple linear controlR Shihui Fu a,∗, Yuan Liu a, Huizhen Ma a, Ying Du b a b

School of Mathematics and Statistics, Zhengzhou University, Zhengzhou, Henan 450001, China School of Science, East China University of Science and Technology, Shanghai 200237, China

a r t i c l e

i n f o

Article history: Received 27 May 2019 Revised 14 July 2019 Accepted 3 September 2019

a b s t r a c t In this paper, we mainly investigate chaos control of a piecewise linear circuit system. According to the characteristic of this system, we modify Hwang’s linear continuous controller and obtain a more simple controller consisting of two parts, by which we find from theory the extent of control parameter when chaotic motion is controlled to equilibrium manifold, equilibrium point, periodic orbit or limit cycle. Numerical simulation also verifies the method is effective.

Keywords: Circuit system Chaos control Equilibrium manifold Equilibrium point Periodic orbit Limit cycle

1. Introduction Recently, non-smooth systems have been extensively investigated from theory, application and experiment [1–3]. Circuit systems, as typical non-smooth systems, are generally piecewise linear. Their nonlinear dynamics, such as bifurcation and chaos, always attracts some authors’ attention. Especially in order to obtain more complex dynamics, all kinds of circuit systems are produced [4–8], such as Chua’s circuit, n-scroll circuit system, n × m-scroll circuit system, modified Chua’s circuit and so on. The Rössler attractor [9] is a famous chaotic system with single quadratic nonlinearity, which is relatively complicated. After the quadratic nonlinearity was replaced by a piecewise linear function [10], a new non-smooth circuit system was found. Its chaotic motion and new dynamic phenomena which could not occur in smooth systems have been investigated [11]. With the development of bifurcation theory of non-smooth systems, its non-smooth bifurcation has also been studied [12]. Compared with other circuit systems, it is investigated by few authors. So in this paper, we will focus on this circuit system. A chaotic system is a deterministic system with great sensitivity to initial conditions. It can lead to the collapse of a dynamical

R This paper was supported by National Natural Science Foundation of China (Grant Nos. 11602224, 61873245, and 11672107). ∗ Corresponding author. E-mail address: [email protected] (S. Fu).

https://doi.org/10.1016/j.chaos.2019.109431 0960-0779/© 2019 Elsevier Ltd. All rights reserved.

© 2019 Elsevier Ltd. All rights reserved.

system, such as vibrations, irregular operation, fatigue failure, for example for high current proton beams, beam halo-chaos can cause excessive radioactivity of an accelerator, and significantly limits the applications of the new accelerators, which needs to control chaos. Stabilizing chaos can avoid fatal voltage collapse in power networks and deadly heart arrhythmia in humans, guide disordered circuit arrays to reach a certain level of desirable pattern formation, and regulate dynamical responses of mechanical and electronic devices. In almost circuit systems, there are chaotic motions [4–8,10,11]. So some interests are now directed to control a chaotic circuit system: A certain kind of intermittent scheme was used to control the chaos in a single chaotic Chua’s circuit to reach an arbitrary orbit [13]; For modified Chua’s circuit system, chaotic oscillation was converted into a stable equilibrium or a stable periodic orbit by a delayed feedback method [14]; After a new reaching rule in the sliding mode control (SMC) was applied on the nonlinear Chua’s circuit to control chaos [15]; Controlling chaos in the n-scroll Chua’s circuit was studied by the control method based on two low pass filters [16]; The ramp compensation method was used to control bifurcation and chaos in a discrete-time iterative nonlinear mapping model [17]; Chaos of Chua’s circuit was controlled to equilibrium manifold or point by a piecewise linear control [18]; A new feedback control was applied to a modified Chua’s circuit system in order to control chaos [19], and also designed to suppress and eliminate the chaotic behavior of Chua’s circuit [20]. It is noted that synchronization of chaos has potential application in chaos generator design, secure

2

S. Fu, Y. Liu and H. Ma et al. / Chaos, Solitons and Fractals 130 (2020) 109431

Fig. 1. The block diagram of the controlled system (2), where X = [x, y, z]T .

communication for circuit systems, which also bring active research about chaos control to some authors. From above analysis, there are some control ways applied to circuit system, such as linear feedback control [19], nonlinear control [18], delayed feedback control [14], sliding mode control [15,21], and so on. In this paper, we pay more attention to the linear control [19]. This controller consists of two portions: one is the feedback part which constructs an equilibrium manifold by modifying the dynamics of the system and the other is the proportional feedback part which will control the system to desired states on the equilibrium manifold. The advantage of this method is that with this closed-loop controller, the system attains faster settling time than the systems using previous controllers. Though the first feedback part constructs an equilibrium manifold, its structure is more complicated because it includes the second and third equations of the original system. On the other hand, there are different aims to control chaos, such as equilibrium manifold [18–20], equilibrium point [9], periodic solution [16] or limit cycle [22]. But now, we can’t also find chaos is stabilized to them only by one control. In the following, we will solve these two problems that after we design a simple controller by modifying the linear control [19], we will control chaotic motion to all kinds of stable states. The rest of the paper is organized as follows. In Section 2, according to the character of the piecewise linear circuit system, we will design a simple controller. After that, we will present the conditions from theory which control parameters satisfy when chaotic motion is respectively stabilized to equilibrium manifold, equilibrium point, periodic solution or limit cycle. In Section 3, numerical simulations verify that the theory of Section 2 is effective. Conclusions are given in Section 4. 2. Control chaos to all kinds of stable states



,



x˙ = y y˙ = z z˙ = −B|x| − y − Az + C + U

,

(2)

U = U1 + U2 = kz + k p (xre f − x ).

(3)

where

The controller U has also two feedbacks, where the first part kz is designed so as to mimic the dynamics of the second equation of system (2), then a proportional feedback k p (xre f − x ) is added to the second feedback. It is note that the third equation of system (1) isn’t included in the first part, which is more simple than that of [19] because its first part includes the second and third equation. When we desire different stable states, we can take different xref which can sometimes be tracked. After the controller is designed, the controlled system (2) is still piecewise linear. For different parameters, its equilibrium state is also different. Hence in the following theorem, according to system parameters, we mainly select appropriate xref and design control parameters k and kp to stabilize chaos to equilibrium manifold, equilibrium point, periodic orbit or limit cycle. Theorem 1. (1) Take xre f =

In this paper, the model is described as:

x˙ = y y˙ = z z˙ = −B|x| − y − Az + C

a boundary equilibrium point E0 = (0, 0, 0 ) ∈  for C = 0, B = 0 and a equilibrium manifold E = {(x, y, z )|y = z = 0} for B = C = 0.   Let E = {(x, y, z )|x > 0, y = z = 0} and E = {(x, y, z )|x < 0, y = z =      0}. It is noted that E+ ∈ E , E− ∈ E , and E = E E0 E . According to the characteristic of that the circuit system has different equilibrium states for different parameters, we will design a simple controller by modifying the linear control in [19] to control the chaotic motion to these states. A linear continuous controller U is applied to system (1) shown in Fig. 1, and the whole system is written as

(1)

where A, B, C are parameters. Due to |x|, the system is piecewisesmooth. The switching boundary function h(x ) = x = 0 separates the space R3 into three regions:

v− = {(x, y, z )|h(x ) < 0},  = {(x, y, z )|h(x ) = 0}, v+ = {(x, y, z )|h(x ) > 0}, where  is the switching boundary. It is easy to know that the system (1) has 2 admissible equilibrium points E+ = ( CB , 0, 0 ) ∈ v+ and E− = (− CB , 0, 0 ) ∈ v− for BC > 0, no equilibrium point for BC < 0,

C B.

If k < A and k p = −B, the controlled 

system (2) has a stable equilibrium manifold E located in v+ , i.e., chaos of system (1) can be controlled to the equilibrium manifold. (2) Take xre f = − CB . If k < A and k p = B, the controlled system 

(2) has a stable equilibrium manifold E located in v− , i.e., chaos of system (1) can also be controlled to the equilibrium manifold. Proof. Take xre f = CB . When k p + B = 0, i.e. k p = −B, the controlled system (2) is rewritten as



x˙ = y y˙ = z z˙ = −y + (k − A )z + f (x )

where

f (x ) =



0 x>0 . 2Bx x < 0

,

(4)

S. Fu, Y. Liu and H. Ma et al. / Chaos, Solitons and Fractals 130 (2020) 109431 

The controlled system (4) has an equilibrium manifold E ∈ v+ and a boundary equilibrium point E0 ∈  . In the following, we in vestigate E . If x > 0, the controlled system (4) is a third order differential equation:

x + (A − k )x + x = 0.

(5)

− k )2

Let  = (A − 4. If  > 0, i.e., A − k > 2 or A − k < −2, the controlled system (4) has analytical solution:

⎧ c1 λ t c2 λ t ⎨x(t ) = λ2 e 2 + λ3 e 3 + c3 , y(t ) = c1 eλ2 t + c2 eλ3 t ⎩ z(t ) = c1 λ2 eλ2 t + c2 λ3 eλ3 t −(A−k )±

λ2,3 =

where

λ y −z

√

(A−k )2 −4 , 2 λ2 y0 −z0 (λ2 −λ3 )λ3 = x0

(6)

λ y −z

λ y −z

0 0 0 0 c1 = λ3 − , c2 = λ2 − , 3 λ2 2 λ3 + ( A − k )y0 + z0 , and (x0 ,

c3 = x0 − (λ 3−0λ )λ0 − 3 2 2 y0 , z0 ) is the initial condition. If A − k > 2, i.e., k < A − 2 (λ2,3 < 0), some orbits in v+ with x0 > 0 and x0 + (A − k )y0 + z0 > 0 can  converge to (c3 , 0, 0 ) ∈ E according to (6). Similarly, we can also prove that when A − k ≤ 2, i.e., A − 2 ≤ k < A, some orbits in v+ can still converge to (c3 , 0, 0). Hence k < A, k p = −B presents that some orbits in v+ can converge to some point of the equilibrium  manifold E . i.e., chaos of system (1) can be controlled to the equilibrium manifold. Take xre f = − CB . By the similar method, we can find for the 

controlled system (2), a stable equilibrium manifold E lies in v− for k < A and k p = B, which shows that the chaos of system (1) can be stabilized to some point of the equilibrium manifold.  Theorem 2. Suppose BC > 0. (1) After we take xre f = CB and design k < A, k p > −B, k + k p < A − B, the controlled system (2) has a stable equilibrium point E+ = (xre f , 0, 0 ), i.e., chaos of system (1) can be controlled to E+ . (2) When we take xre f = − CB and design k < A, kp > B, k p + k < A + B, the controlled system (2) has a stable equilibrium point E− = (xre f , 0, 0 ), i.e., chaos of system (1) can also be controlled to E− . Proof. If BC > 0, system (1) has two equilibrium points E ± . After take xre f = CB , the controlled system (2) still has the equilibrium point E+ , whose Jacobian matrix is



J+ =

0 0 − (B + k p )



1 0 −1

0 1 . k−A

(7)

The characteristic equation is

p+ (λ ) = −λ − (A − k )λ − λ − (B + k p ) = 0. 3

2

(8)

According to Hurwitz Theorem, E+ is asymptotically stable, if and only if A − k > 0, B + k p > 0, A − k > B + k p , i.e., the control parameters satisfy k < A, k p > −B, k + k p < A − B, which implies that the chaos of system (1) can be controlled to E+ . If we take xre f = − CB , the controlled system (2) has the equilibrium point E− . Its Jacobian matrix is



J− =

0 0 B − kp

1 0 −1



0 1 , k−A

(9)

p− (λ ) = −λ − (A − k )λ − λ − (k p − B ) = 0. 2

obtain the condition under which equilibrium points E ± aren’t stable: E+ is unstable for other parameters expect k < A, k p ≥ −B, k + k p < A − B while E− is unstable for other parameters expect k < A, kp ≥ B, k p + k < A + B. In [19,20], the authors plot a borderline dividing the regions of stability about control parameter by numerical simulation. In our paper, we give the borderline dividing the regions of stability about control parameter from theory, i.e.,k = A, k p = −B, and k + k p = A − B for E+ , and k = A, k p = B and k p + k = A + B for E− .  Theorem 3. Suppose BC > 0. (1) After we take xre f = CB and design k p > −B, k + k p = A − B, the controlled system (2) has a family of stable periodic orbits around E+ , i.e., chaos of system (1) can be controlled to any of these orbits, furthermore design k p > −B, A − B < k + k p < A + B (sufficiently small k + k p − (A − B )), the controlled system (2) has a stable limit cycle, i.e., chaos of system (1) can be controlled to the limit cycle. (2) If we take xre f = − CB and design kp > B, k p + k = A + B, the controlled system (2) has also a family of stable periodic orbits around E− , i.e., chaos of system (1) can also be controlled to any of these orbits, furthermore design k p > B, A + B < k + k p < A − B (sufficiently small k + k p − (A + B )), the controlled system (2) has also a stable limit cycle, i.e., chaos of system (1) can be controlled to the limit cycle. Proof. In this theorem, we mainly prove (2). Certainly we can also apply the similar method to (1). If BC > 0, we take xre f = − CB , and we find E− is also an equilibrium point of controlled system (2). Its Jacobian matrix and characteristic equation have been given in (9) and (10). If A − k = k p − B, i.e., k p + k = A + B, Eq. (10) has eigenvalues λ1 = B − k p , λ2,3 = ±i, hence in v− , the analytical solution of controlled system (2) is

⎧ C (B−k )t ⎨x(t ) = c1 e p + c2 cost + c3 sint − B , y(t ) = c1 (B − k p )e(B−k p )t − c2 sint + c3 cost ⎩ z(t ) = c1 (B − k p )2 e(B−k p )t − c2 cost − c3 sint x +z + C

B2 x −z +BC

(11)

−B (x −z )−C

0 0 0 0 where c1 = 01+0B2 B , c2 = , c3 = y0 + , and x0 , 1+B2 1+B2 y0 , z0 are initial conditions. Let c1 = 0, i.e., the initial conditions satisfy

C x0 + z0 = − , B

(12)

the controlled system (2) has periodic orbits

⎧ C ⎨x(t ) = c2 cost + c3 sint − B y(t ) = −c2 sint + c3 cost ⎩ z(t ) = −c2 cost − c3 sint

.

(13)

Furthermore if B − k p < 0, i.e., kp > B, these periodic orbits (12) are stable. Especially, some periodic orbit grazes the switching boundary  if



c2 cost + c3 sint −

C B

=0

−c2 sint + c3 cost = 0 i.e.,



B2 x0 − z0 + BC 1 + B2

2

,

+

y0 +

(14)

−B(x0 − z0 ) − C 1 + B2

2 =

C2 , B2

(B2 x0 − z0 + BC )2 + (y0 (1 + B2 ) − B2 (x0 + z0 ) − C )2

and its characteristic equation is 3

3

(10)

Similarly, according to Hurwitz Theorem, if k < A, kp > B, k p + k < A + B, E− is stable, i.e., chaos of system (1) can be stabilized E− . It is note that in Theorem 2, we mainly show the condition under equilibrium points E ± are stable. Certainly we can also

C2 C2 (1 + B2 )2 , (1 + B2 )2 y20 + (1 + B2 )2 z02 = 2 (1 + B2 )2 , 2 B B C2 2 2 y0 + z0 = 2 . B =

(15)

According to the above analysis, if A − k = k p − B and B − k p < 0, a family of periodic orbits with initial conditions

4

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Fig. 2. Bifurcation diagram of system (1), where B = 1, C = 2, A ∈ [0.6, 0.99].

x

x y y

z

z

(a)

(b)

(c)

(d)

Fig. 3. Time history diagram of controlled system (2) for B = 1, C = 2, A = 0.63, xre f = CB = 2, k = 0.1, k p = −B = −1, (a) and (b) x, y, z output respectively for initial value (1, 0, 3) and (−1, 0, 3 ); (c) and (d) the control input respectively for initial value (1, 0, 3) and (−1, 0, 3 ).

satisfying (12) are stable, among of which some orbit with initial conditions satisfying (15) grazes the switching boundary  , which implies that when we take the control parameters kp > B, k p + k = A + B, chaos of system (1) can be stabilized to some periodic orbit including one grazing  . In the following, we will prove there is one limit cycle in controlled system (2). Let x = CB (x1 − 1 ), and controlled system (2)

can be rewritten as



X˙ 1 = J− X1 X˙ 1 = J+ X1 + b1

x1 < 1 , x1 ≥ 1

(16)

where X1 = [x1 , y1 , z1 ]T , y1 = x˙ 1 , z1 = y˙ 1 , b1 = [0, 0, 2B]T . Although, equilibrium points between system (2) and system (16) are different, they have same Jacobian matrixes J− and J+ . At this

S. Fu, Y. Liu and H. Ma et al. / Chaos, Solitons and Fractals 130 (2020) 109431

5

x

y

z

(a)

(b) C B

Fig. 4. Time history diagram of controlled system (2) for B = 1, C = 2, A = 0.63, xre f =

= 2, k = −0.5, k p = −0.5, (a) x, y, z output; (b) the control input.

2

1.5 v+

v-

1

1.5

v+

v-

1 0.5

0.5

y

y

0

0 -0.5

-0.5

-1 -1

-1.5

-1.5 -1

-0.5

0

0.5

1

1.5

2

2.5

3

-2 -1

3.5

x

x

(a)

(b) 0.8

0.6

0.6 0.4 0.4 0.2

u

0.2

u

0

0 -0.2

-0.2 -0.4 -0.4

-0.6 -0.8

-0.6 0

20

40

60

80

0

100

20

40

60

80

100

t

t

(c)

(d)

Fig. 5. Periodic orbits of controlled system (2) for B = 1, C = 2, A = 0.63, xre f = CB = 2, k = 0.13, k p = −0.5, (a) and (b) x, y, z output respectively for initial value (1, 1, 1) and (4, 0, −2 ); (c) and (d) the control input respectively for initial value (1, 1, 1) and (4, 0, −2 ).

time, we still investigate J− . When A − k lies in sufficiently small neighbourhood of k p − B, the characteristic Eq. (10) has roots λ1 = λ, λ2,3 = σ ± ωi(ω > 0 ), and

A − k = − ( λ + 2σ ) B − k p = λ (σ + ω ) 2



X˙ =

AR X + b x1 ≥ 1



(17)

X =

x1 y1 , AL = z1



Hence let

= t = k − A, m = 1, d = B − k p , T = k − A, M = 1, D = −(k p + B )

x1 < 1

AL X

where

1 = 2σ λ + σ 2 + ω 2 2

and equation (16) is rewritten as:

(18)

T M D

−1 0 0



t m d



,

(19)

−1 0 0

0 −1 , b = 0





0 −1 , AR 0



t −T m−M . d−D

6

S. Fu, Y. Liu and H. Ma et al. / Chaos, Solitons and Fractals 130 (2020) 109431

3 v+

v-

2

0.8 0.6 0.4

1

0.2

u

y

0

0 -1

-0.2 -0.4

-2 -0.6 -3 -1

0

1

2

3

4

-0.8

5

0

100

t

(a)

(b)

Fig. 6. Limit cycle of controlled system (2) for B = 1, C = 2, A = 0.63, xre f =

C B

4

150

= 2, k = 0.15, k p = −0.5, (a) x, y, z output; (b) the control input.

4 y

2

y

2 0

x/y/z

0

x/y/z

50

x

z

-2

x

-4

z

-2

x

-4

-6

-6

-8

-8 0

20

40

60

80

100

0

20

40

t

60

80

100

t

(a)

(b) 6

5

4

0

u

u

2

0

-2 -4 -6

-5 0

20

40

60

80

0

100

20

40

60

80

100

t

t

(c)

(d) − CB

Fig. 7. Time history diagram of controlled system (2) for B = 1, C = 2, A = 0.63, xre f = = −2, k = 0.1, k p = B = 1, (a) and (b) x, y, z output respectively for initial value (1, 0, 3) and (−1, 0, 3 ); (c) and (d) the control input respectively for initial value (1, 0, 3) and (−1, 0, 3 ). d Suppose tc = m = B − k p , and δ = d (m − M ) + m(D − mT ) = A − B − k p − k. According to [23], if m = 1 > 0, d = B − k p < 0 = 0, δ = A − B − k p − k = 0, system (19) undergoes a focus-center-limit cycle bifurcation for t = tc , i.e., k p + k = B + A, which means that there exists one limit cycle which appears when δ (t − tc ) > 0 ((A − B − k p − k )(k − A − (B − k p )) > 0) and t − tc is sufficiently small. In particular, if δ = A − B − k p − k > 0 (k p + k < A − B) and d < 0 (kp > B), then the limit cycle bifurcation for t > tc (k p + k > A + B) is orbitally asymptotically stable. In a word, if design kp > B and A + B < k p + k < A − B, the controlled system (2) has a stable limit cycle, i.e., the chaos of system (1) can be controlled to it. 

3. Numerical simulation In order to verify the above theory, in this section, we will give the numerical simulation. Now fix B = 1, C = 2. Actually, in [11], the authors have found chaotic motion in system (1) when system parameter A = 0.63. Now we select A ∈ [0.6, 0.99], and we give the bifurcation diagram shown in Fig. 2, where the system begins to have period-1 motion, then becomes chaos by double-period bifurcation from A = 0.99 to A = 0.6. In the following, we will design xref , k and kp to stabilize the chaotic motion at A = 0.63 to different stable states.

S. Fu, Y. Liu and H. Ma et al. / Chaos, Solitons and Fractals 130 (2020) 109431

7

8

4 y

2

6 4

0 2

u

x/y/z

z -2

0 x

-4

-2

-6

-4 -6

-8 0

20

40

60

80

0

100

20

40

60

t

t

(a)

(b)

80

100

Fig. 8. Time history diagram of controlled system (2) for B = 1, C = 2, A = 0.63, xre f = − CB = −2, k = 0.1, k p = 1.2, (a) x, y, z output; (b) the control input.

2.5

1

v-

2

v+

0.5

v+

1.5

v-

1 0.5

y

y

0

0 -0.5

-0.5

-1 -1.5

-1 -3

-2.5

-2

-1.5

-1

-0.5

0

0.5

-2 -4

1

-3

-2

(a)

(b)

2

4

1.5

3

1

2

0.5

1

0

0

u

u

-1

x

x

-0.5

-1

-1

-2

-1.5

-3 -4

-2 0

20

40

60

80

100

0

20

40

60

t

t

(c)

(d)

80

100

Fig. 9. Periodic orbits of controlled system (2) for B = 1, C = 2, A = 0.63, xre f = − CB = −2, k = 0.43, k p = 1.2, (a) and (b) x, y, z output respectively for initial value (−3, 0, 1 ) and (−4, 0, 2 ); (c) and (d) the control input respectively for initial value (−3, 0, 1 ) and (−4, 0, 2 ).

First we take xre f = CB = 2. According to Theorem 1, we design k = 0.1 < A = 0.63 and k p = −B = −1 in controlled system (2). The orbit with the initial value (1, 0, 3) quickly converges point  (4, 0, 0 ) ∈ E , the orbit with the initial value (−1, 0, 3 ) quickly  converges point (0.7, 0, 0 ) ∈ E and the controller is quickly activated at the same time shown in Fig. 3, which shows chaos of  system (1) can be stabilized to equilibrium manifold E as long as we select appropriate control parameters. According to Theorem 2,

when we design k = −0.5 < A = 0.63, k p = −0.5 > −B = −1 and k p + k = −1 < A − B = −0.37 in controlled system (2), we find the orbits quickly converges equilibrium point ( CB , 0, 0 ) = (2, 0, 0 ) and the controller is also quickly activated at the same time shown in Fig. 4, hence when we design appropriate control parameters, chaos of system (1) can also be stabilized to equilibrium point E+ . According to Theorem 3, we design k p = −0.5 > −B = −1, k = 0.13 satisfying k p + k = −0.37 = A − B = −0.37 in controlled

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S. Fu, Y. Liu and H. Ma et al. / Chaos, Solitons and Fractals 130 (2020) 109431

system (2). When we take the initial value (1, 1, 1), the orbit converges a stable periodic solution in v+ (Fig. 5(a)), while the orbit with initial value (4, 0, −2 ) converges the stable periodic solution grazing the switching boundary  in v+ (Fig. 5(b)). Hence chaos of system (1) can also be controlled to the periodic orbit. It is noted that because we select the initial values lying in the periodic orbit, hence the controller u is activated almost t = 0. Furthermore we design k p = −0.5 > −B = −1, k = 0.15 satisfying A − B = −0.37 < k p + k = −0.35 < A + B = 1.63, and there is a stable limit cycle in controlled system (2) shown in Fig. 6, which shows that chaos of system (1) can also be stabilized the limit cycle. At this time, we find the activated time has relation with k − 0.13: k − 0.13 is bigger, and the activated time is smaller. Secondly we take xre f = − CB = −2. Similarly by Theorem 1, we design k = 0.1 < A = 0.63 and k p = B = 1 in controlled system (2), and we find that the orbit with the initial value (1, 0, 3) quickly  converges point (−4, 0, 0 ) ∈ E , while the orbit with the initial  value (−1, 0, 3 ) quickly converges point (−4.518, 0, 0 ) ∈ E shown in Fig. 7, which shows chaos of system (1) can be stabilized to  equilibrium manifold E . When we design k = 0.1 < A = 0.63, k p = 1.2 > B = 1 and k p + k = 1.3 < A + B = 1.63 in controlled system (2) by Theorem 2, we find the orbits of system (2) quickly converges equilibrium point (− CB , 0, 0 ) = (−2, 0, 0 ) shown in Fig. 8, hence chaos of system (1) can also be stabilized to equilibrium point E− . By Theorem 3, we design k p = 1.2 > B = 1, k = 0.43 satisfying k p + k = 1.63 = A + B in controlled system (2). When we take the initial value (−3, 0, 1 ), there is a stable periodic solution in v− (Fig. 9(a)), while the orbit with initial value (−4, 0, 2 ) grazes the switching boundary  in v− (Fig. 9(b)). Hence chaos of system (1) can also be controlled to these periodic orbits. It is noted the corresponding activated time for the controller has also been given. 4. Conclusions After a piecewise linear function is replaced the quadratic nonlinearity of Rössler attractor, a non-smooth circuit system was produced. By now, few papers pay more attention to it, so in this paper we mainly study its chaos control, which is as follows: (1) Hwang puts up a linear feedback control consisting of two parts whose first part includes the second and third equations of the original system, which has more complicated structure. So we modify it and design a simple controller whose first part only includes the second equation of the original system. When two different controllers achieve the same control aim, the cost of the controller hardware with more simple structure is low and it is realized conveniently. On the surface, the method applied in our paper is same to that given by Hwang. Actually, there is difference because the aim to apply the controller is different. In [19], the first part of the controller includes the second and third equation of their system in order to produce the equilibrium manifold. But in my paper, the first part of the controller only includes second equation of system (1) in order to change the coefficient in front of the state variable. (2) We show from the theory the extent of control parameters when chaotic motion is controlled to different states. Numerical simulation also presents that as long as we select the appropriate control parameters, the chaotic motion can be controlled to the different states, so the method is easier to be applied.

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