Controlling chaos for the dynamical system of coupled dynamos

Chaos, Solitons and Fractals 13 (2002) 341±352

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Tutorial

Controlling chaos for the dynamical system of coupled dynamos H.N. Agiza Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt Accepted 10 October 2000

Abstract This work is a tutorial on the di€erent methods to control chaotic behaviour of the coupled dynamos system. Feedback and nonfeedback control techniques are proposed to suppress chaos to unstable equilibrium or unstable periodic solution. The stabilization of unstable ®xed point of the chaotic behaviours is achieved also by bounded feedback method. Stability of the controlled systems are studied by Routh±Hurwitz criterion. Nonfeedback method and a derived method based on the delay feedback control are used to control chaos to periodic orbits. Numerical simulation results are included to show the control process of the di€erent methods. Ó 2001 Published by Elsevier Science Ltd.

1. Introduction In the last decade, controlling chaos has become on focus in the nonlinear problems ranging from physics and chemistry to biology and economics. Recently, there has been increasing interest in the research area of controlling chaotic dynamical systems. There are many practical reasons for controlling chaos. First of all, chaotic system response with little meaningful information content is unlikely to be useful. Secondly, chaos can lead systems to harmful or even catastrophic situations, therefore chaos should be reduced as much as possible or totally suppressed. Traditional engineering design always tries to reduce irregular behaviours of a system and, therefore completely eliminates chaos. Various control algorithms have been proposed in recent years to control chaotic systems. The existing control algorithms can be classi®ed mainly, into two categories: feedback and nonfeedback. Feedback methods [1±9,23] are primarily devised to control chaos by stabilizing a desired unstable periodic orbit embedded within a chaotic attractor. The nonfeedback methods [10±17] suppress chaotic behaviours by converting the system dynamics to a periodic orbit. In these methods, one does not perform changes in the system according to its position in phase space, but applies instead weak periodic perturbations on some control parameters or variables. The essential advantages of nonfeedback technique lie in their speed and these make them especially promising for controlling systems such as chaotic circuits, fast electro-optical systems, and so on. The nonfeedback methods include weak periodic parametric perturbation [18,24], addition of second periodic force [9], constant bias [15,16], addition of weak periodic pulses, and addition of weak noise signal [12±14]. A bounded feedback method is proposed to control chaos of the system of coupled dynamos [19]. The proposed method preserves the steady states of the original system. It vanishes after the stabilization is achieved. The simplicity of the application of this method is one of its features.

E-mail address: [email protected] (H.N. Agiza). 0960-0779/01/$ - see front matter Ó 2001 Published by Elsevier Science Ltd. PII: S 0 9 6 0 - 0 7 7 9 ( 0 0 ) 0 0 2 3 4 - 4

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The delayed feedback control (DFC) is proposed by Pyragas [3] to control chaos in continuous dynamical systems. The method deals with a chaotic system that can be simulated by a system of ordinary nonlinear di€erential equations. Ushio [20] has extended the DFC method to control chaos in discrete dynamical systems. The DFC method allows one to stabilize unstable periodic orbits of a strange attractor over a large domain of parameters. In this paper, we propose another method of control based on the DFC method. The controlled system is converged to a periodic orbit such that the period is small. The work which should be regarded as a tutorial is organized as follows. In Section 2, a brief description of the model of two coupled dynamos is introduced. In Section 3, stabilization of unstable equilibria is achieved and the stability of the corresponding controlled system is analyzed. Two feedback methods are proposed to control chaos to the unstable equilibria of the original system. In Section 4, chaos is controlled to limit cycles by using nonfeedback and a derived method from the delay feedback method. Numerical simulations are carried out to show the control processes. 2. The system of coupled dynamos The system of coupled dynamos is a nonlinear dynamical system which is capable of chaotic behaviours [25±27]. The system consists of two dynamos connected together so that the current generated by any one of them produces the magnetic ®eld for another. We denote the angular velocities of their rotors by x1 , x2 and the currents generated by x, y, respectively. Then, with appropriate normalization of variables, the mathematical model equations [19] for this system are x_ ˆ x1 y

l1 x;

y_ ˆ x2 x

l2 y;

x_ 1 ˆ q1

1 x 1

xy;

x_ 2 ˆ q2

2 x2

xy;

…1:1†

where q1 and q2 are the torques applied to the rotors, and l1 , l2 , 1 and 2 are positive constants representing dissipative e€ects. The model is described by a system of four di€erential equations. The system (1.1) is simpli®ed by setting 1 ˆ 2 ˆ 0, and q1 ˆ q2 ˆ 1. Finally, the dynamical equations of the coupled dynamos can be written in the simple form [19] x_ ˆ

lx ‡ y…z ‡ a†;

y_ ˆ

ly ‡ x…z

a†;

z_ ˆ 1

xy;

…1:2†

where l1 ˆ l2 ˆ l, x1 ˆ z ‡ a and x2 ˆ z a and a is a constant of the motion. The divergence of the ¯ow (1.2) is given by r
Fˆ

oF1 oF2 oF3 ‡ ‡ ˆ ox oy oz

2l < 0;

where F ˆ …F1 ; F2 ; F3 † ˆ … lx ‡ y…z ‡ a†; ly ‡ x…z a†; 1 xy†. Then, system (1.2) is a forced dissipative system similar to Lorenz system [28,29]. Thus the solutions of the system of Eq. (1.2) are bounded as t ! 1 for positive values of a and l. The equilibrium points of the system (1.2) are E1 ˆ …b1 ; b2 ; c†; E2 ˆ … b1 ; b2 ; c†; p p p where b1 ˆ …c ‡ a†=l, b2 ˆ …c a†=l, c ˆ a2 ‡ l2 . The Jacobian of the linearized system about the ®rst equilibrium E1 is 2 3 l c ‡ a b2 J ˆ 4c a l b1 5: b2 b1 0 The eigenvalues of the matrix J are s 2c k1 ˆ 2l; k2;3 ˆ  : l

…1:3†

…1:4†

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Fig. 1. Three-dimensional image of the coupled dynamos chaos at a ˆ 1 and l ˆ 2.

It follows that the linearized system has a transient term that decays to zero as t increases (since k1 < 0) plus an oscillatory solution. It is therefore neutrally stable in the limit. The same result is true at the other equilibrium E2 . Consequently, we cannot conclude anything about the behaviour of the dynamical system (1.2) by means of linearization. However, a numerical integration of these equations is quite revealing. It shows that the equilibria are actually unstable and that the orbits encircle one of them a number of times before switching suddenly to encircle the other equilibrium where it oscillates for awhile before again switching rapidly back about the original point. The orbits are never captured by either equilibrium point, and the limiting behaviour is apparently chaotic since the number of oscillations in the neighbourhood of each equilibrium point is unpredictable. Fig. 1 shows a three-dimensional image of the coupled dynamos chaotic trajectory at l ˆ 1 and a ˆ 2.

3. Controlling chaos to equilibrium points In this section, the chaos of the dynamical system (1.2) is controlled to one of the two equilibria of the system. Two feedback methods are applied; linear feedback and bounded feedback to achieve this goal. 3.1. Feedback control method The linear feedback control is applied to system (1.2). Our goal is to guide the chaotic trajectories of the system to one of the two unstable equilibria (E1 or E2 ). For the purpose of controlling chaos by feedback control approach, let us assume that the equations of the controlled coupled dynamos system are given by x_ ˆ

lx ‡ y…z ‡ a† ‡ u1 ;

y_ ˆ

ly ‡ x…z

a† ‡ u2 ;

z_ ˆ 1

xy ‡ u3 ;

…2:1†

where u1 , u2 and u3 are external control inputs. It will be suitably designed to drive the trajectory of the system, speci®ed by …x; y; z† to any of the two equilibrium points of the uncontrolled (i.e., u1 ˆ u2 ˆ u3 ˆ 0) system by only a single state variable feedback. For practical applications, a simple feedback controller is more desirable so the control law is 2

3 u1 4 u2 5 ˆ u3

2

k11 4 0 0

0 k22 0

32 x 0 4 5 0 y k33 z

3 x y 5; z

where …x; y; z† is the desired unstable equilibrium of the chaotic system (1.2), and k11 , k22 and k33 , positive feedback gains, are needed to be chosen such that the trajectory of the controlled system is stabilized to any of the two equilibrium points of the uncontrolled system. Hence (2.1) is rewritten in the form x_ ˆ

lx ‡ y…z ‡ a†

k11 …x

x†;

y_ ˆ

ly ‡ x…z

a†

k22 …y

y†;

z_ ˆ 1

xy

k33 …z

z†:

…2:2†

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The controlled system (2.2) has one ®xed point E ˆ …x; y; z†. The stability of the controlled system is studied by the Routh±Hurwitz criterion, which states that all real eigenvalues and all real parts of complex conjugate eigenvalues are negative if and only if the following conditions hold: …i†

c1 ; c2 ; c3 > 0;

…ii†

c1 c2

c3 > 0;

…2:3†

where c1 , c2 and c3 are the coecients of the characteristic equation of the Jacobian matrix for the linearized system at E ˆ …x; y; z† k3 ‡ c1 k2 ‡ c2 k ‡ c3 ˆ 0: In the case E ˆ E1 the controlled system (2.2) has the form x_ ˆ z_ ˆ 1

lx ‡ y…z ‡ a† xy

k33 …z

k11 …x

b1 †;

y_ ˆ

ly ‡ x…z

a†

k22 …y

b2 †;

c†:

…2:4†

3.1.1. Stabilizing the equilibrium E1 ˆ …b1 ; b2 ; c† Lemma 1. The equilibrium solution E1 ˆ …b1 ; b2 ; c† of the controlled system (2.4) is asymptotically stable such that k11 > 0; k22 ˆ 0 and k33 ˆ 0. Proof. The linearized system of (2.4) about the positive equilibrium E1 ˆ …b1 ; b2 ; c† is given by x_ ˆ

…l ‡ k11 †x ‡ …c ‡ a†y ‡ b2 z;

y_ ˆ …c

a†x

…l ‡ k22 †y ‡ b1 z;

z_ ˆ

xb2

b1 y

k33 z:

The Jacobian matrix of the linearized system is 2 3 l k11 c ‡ a b2 J ˆ4 c a l b1 5; b2 b1 0 where k22 ˆ 0 and k33 ˆ 0. The characteristic equation of the Jacobian J has the form k3 ‡ c1 k2 ‡ c2 k ‡ c3 ˆ 0; where c1 ˆ 2l ‡ k11 > 0, c2 ˆ lk11 ‡ b21 ‡ b22 > 0, c3 ˆ 2cb1 b2 ‡ l…b21 ‡ b22 † ‡ k11 b21 † > 0 and c1 c2 c3 ˆ 2 lk11 ‡ …2l2 ‡ b22 †k11 > 0. Then the Routh±Hurwitz conditions (i) and (ii) of (2.3) are satis®ed and the asymptotic stability of E1 ˆ …b1 ; b2 ; c† of the controlled system (2.4) is established.  Remark 1. The gain constants k11 , k22 and k33 can be chosen in di€erent choices which give the same stability results for example (k11 ˆ 0, k22 > 0 and k33 ˆ 0) or (k11 ˆ 0, k22 ˆ 0 and k33 > 0). 3.1.2. Stabilizing the equilibrium E2 ˆ … b1 ; b2 ; c† In order to control chaos of system (1.2) to the unstable equilibrium E2 ˆ … b1 ; b2 ; c†, linear feedback control is proposed to obtain the controlled system x_ ˆ z_ ˆ 1

lx ‡ y…z ‡ a† xy

k33 …z

k11 …x ‡ b1 †; c†:

y_ ˆ

ly ‡ x…z

a†

k22 …y ‡ b2 †; …2:5†

Lemma 2. The equilibrium E2 ˆ … b1 ; b2 ; c† of (2.5) is asymptotically stable such that k11 ˆ 0; k22 > 0 and k33 ˆ 0.

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Proof. The Jacobian matrix of the linearized system of (2.5) at the equilibrium E2 ˆ … b1 ; b2 ; c† is given by 2 3 l c‡a b2 J ˆ 4c a l k22 b1 5: b2 b1 0 The eigenvalues of the Jacobian matrix J are the roots of the characteristic polynomial k3 ‡ c1 k2 ‡ c2 k ‡ c3 ˆ 0;
where c1 ˆ 2l ‡ k22 > 0, c2 ˆ lk22 ‡ b21 ‡ b22 > 0 and c3 ˆ 2cb1 b2 ‡ l b21 ‡ b22 ‡ k22 b22 > 0. 2 The condition c1 c2 c3 ˆ lk22 ‡ …2l2 ‡ b21 †k22 > 0, holds. Then according to Routh±Hurwitz conditions (2.3), the real eigenvalues and all the real parts of complex conjugate eigenvalues are negative. Hence the asymptotic stability of E2 ˆ … b1 ; b2 ; c† is established and the proof is completed.  3.1.3. Numerical simulations The chaotic and controlled systems of the ordinary di€erential equations (1.2) and (2.1), respectively, are integrated by using fourth-order Runge±Kutta method with time step size 0.01. The parameters a and l are taken a ˆ 2 and l ˆ 1 to ensure chaotic behaviour in the absence of the control. The initial conditions x ˆ 0:3, y ˆ 0:4 and z ˆ 0:5 are chosen in all simulations. The gain matrix K is chosen to be k11 ˆ 0:2, k22 ˆ 0 and k33 ˆ 0. Fig. 2 shows the convergence of the trajectory of the controlled system (2.4) to the equilibrium point E1 ˆ …b1 ; b2 ; c† ˆ …2:05817; 0:48586; 2:33606† of system (1.2) in three-dimensional image. Fig. 3 shows in three-dimensional image the stabilization of the equilibrium point E2 ˆ … 2:05817; 0:48586; 2:33606), where k11 ˆ 0, k22 ˆ 0:5 and k33 ˆ 0. 3.2. Bounded feedback control In controlling chaotic systems of practical problems, we need bounded control. In order to ensure small values of the control perturbations, the controller u…t† is restricted in the following manner: 8 < u0 ; u…t† < u0 ; u…t† ˆ u…t†; u0 < u…t† < u0 ; : u0 < u…t†; u0 ; where u0 is a small saturating positive value of the control.

Fig. 2. Three-dimensional image of stabilizing the positive equilibrium E1 ˆ …2:05817; 0:48586; 2:33606† of the controlled system (2.2) by using linear feedback at a ˆ 1, l ˆ 2, k11 ˆ 0:2 and k22 ˆ k33 ˆ 0. Fig. 3. Three-dimensional image of the stabilization of the equilibrium point E2 ˆ … 2:05817; 0:48586; 2:33606†, of the controlled system (2.5), where k11 ˆ k33 ˆ 0, k22 ˆ 0:5, l ˆ 1 and a ˆ 2.

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3.2.1. Stabilizing the equilibrium E1 ˆ …b1 ; b2 ; c† In order to stabilize the unstable equilibrium E1 ˆ …b1 ; b2 ; c† by bounded feedback control, the proposed control is designed for system (1.2) as follows: x_ ˆ

lx ‡ y…z ‡ a†;

y_ ˆ

ly ‡ x…z

a†;

z_ ˆ 1

xy ‡ u…t†;

…2:6†

where u…t† ˆ k…y…z ‡ a† lx†, k > 0. The controller u…t† acting as a damping to the system. This damper u…t† will be positive as x…t† decreasing and negative as x…t† increasing, thus chaos is suppressed to stable attractor in a short settling time. The controller u…t† tends to zero if the solutions converge to one of the two unstable equilibria. The original and controlled systems have the same equilibrium points E1 and E2 . Lemma 3. The equilibrium solution E1 ˆ …b1 ; b2 ; c† of the controlled system (2.6) is asymptotically stable if the constant k > 0. Proof. Let the controller u…t† ˆ k…y…z ‡ a† lx†, k > 0 be designed to the chaotic system (1.2). The Jacobian matrix J of the linearized system of (2.6) at E1 ˆ …b1 ; b2 ; c† is 2 3 l c‡a b2 …2:7† l b1 5: J ˆ4 c a b2 ‡ kl b1 k…c ‡ a† kb2 The characteristic equation of the Jacobian matrix (2.7) has the form k3 ‡ c1 k2 ‡ c2 k ‡ c3 ˆ 0; where the coecients of the polynomials are satisfying (i) c1 ˆ 2l ‡ kb2 > 0, (ii) c2 ˆ ‰k…lb2 ‡ cb1 ‡ ab1 † ‡ b21 ‡ b22 Š > 0 since c > a, (iii) c3 ˆ l…b21 ‡ b22 † ‡ 2cb1 b2 > 0 (iv) and   
 c1 c2 c3 ˆ lb22 ‡ b2 b1 …c ‡ a† k 2 ‡ 2l2 b2 ‡ 2b1 …c ‡ a† ‡ b2 b21 ‡ b22 k 
 ˆ 2ck 2 ‡ 2l2 b2 ‡ 2b1 …c ‡ a† ‡ b2 b21 ‡ b22 k > 0: Then the conditions of Routh±Hurwitz theorem are satis®ed. Hence the equilibrium E1 of the controlled system (2.6) is asymptotically stable.  3.2.2. Stabilizing the equilibrium E2 ˆ … b1 ; b2 ; c† For stabilizing the unstable equilibrium E1 ˆ … b1 ; b2 ; c† of (1.2) by bounded feedback control, the proposed control is designed for system (1.2) as follows: x_ ˆ

lx ‡ y…z ‡ a† ‡ u…t†;

y_ ˆ

ly ‡ x…z

a†;

z_ ˆ 1

xy;

…2:8†

where u…t† ˆ k…1 xy†, k > 0. The controller u…t† will be positive as z…t† decreasing and negative as z…t† increasing, and the controller u…t† converges to zero as chaos suppressed to the equilibrium E2 . In order to study the stability of the controlled system (2.8) in the neighbourhood of E2 ˆ … b1 ; b2 ; c†, linearization about the equilibrium E2 ˆ … b1 ; b2 ; c† gives the Jacobian matrix 2 3 l kb2 c ‡ a kb1 b2 J ˆ4 c a …2:9† l b1 5 : b2 b1 0 The eigenvalues of the Jacobian matrix satisfy the following characteristic equation: k3 ‡ c1 k2 ‡ c2 k ‡ c3 ˆ 0;

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where the coecients of the polynomials are satisfying (i) c1 ˆ 2l ‡ kb2 > 0, (ii) c2 ˆ ‰k…lb2 ‡ cb1 ab1 † ‡ b21 ‡ b22 Š > 0 since c > a, (iii) c3 ˆ l…b21 ‡ b22 † ‡ 2cb1 b2 > 0 (iv) and c1 c2

c3 ˆ ‰lb22 ‡ b2 b1 …c

a†Šk 2 ‡ ‰2l…lb2 ‡ b1 …c

a†† ‡ b2 …b21 ‡ b22 †Šk > 0

since …c a† > 0 from the de®nition of c; a and l. Applying Routh±Hurwitz theorem of classical stability theory, we get the necessary and sucient conditions (2.3) are satis®ed and asymptotic stability for the equilibrium point E2 ˆ … b1 ; b2 ; c† is established and this completes the proof of the following lemma. Lemma 4. The equilibrium solution E2 ˆ … b1 ; b2 ; c† of the controlled system (2.8) is asymptotically stable for k > 0. 3.2.3. Numerical simulations Numerical experiments are carried out to integrate the controlled systems (2.6) and (2.8). System (2.6) is numerically integrated by using fourth-order Runge±Kutta method with time step 0.01. The parameters a and l are chosen a ˆ 2 and l ˆ 1 to ensure the existence of chaos in the absence of the control. The initial conditions x ˆ 0:3, y ˆ 0:4 and z ˆ 0:5 are chosen in all simulations. The equilibrium point E1 ˆ …b1 ; b2 ; c† ˆ …2:0581; 0:4857; 2:236† is stabilized for k ˆ 0:1 and u0 ˆ 0:5. Fig. 4(a)±(c) shows the behaviour of the states x, y and z of system (2.6) and the controller u…t† with time. The control is started at t ˆ 100. The equilibrium point E2 ˆ … b1 ; b2 ; c† ˆ … 2:0581; 0:4857; 2:236† of system (2.8) is stabilized for k ˆ 0:1 and u0 ˆ 0:5. Fig. 5(a)±(c) shows the behaviour of the states x, y and z of system (2.8) and the controller u…t† with time. The control is activated at t ˆ 100. 4. Controlling chaos to limit cycles The strange attractor of system (1.2) of coupled dynamos contains in®nite number of unstable periodic orbits (UPOs) embedded within the attractor. In order to suppress the chaotic behaviour of system (1.2) to one of these UPOs, we apply two methods of control in this section. The nonfeedback and a derived method based on the delay feedback control are applied to system (1.2) to guide the chaotic trajectory to a limit cycle. 4.1. Nonfeedback control method The essential advantages of nonfeedback technique lie in their speed ¯exibility, which are suited for practical applications. The nonfeedback methods include weak periodic force used to controlling the chaotic behaviour of system (1.2). The proposed nonfeedback control of system (1.2) consists of a constant and an external periodic force function in the form u…t† ˆ f1 ‡ f2 sin…x; t†;

…3:1†

where f1 is a constant bias, f2 the amplitude and x is the frequency of the external periodic force signal. The nonfeedback method with the controller (3.1) is applied in two cases. Case 1. In this case, the controller (3.1) is added to the second equation of (1.2). The controlled system has the following equations: x_ ˆ

lx ‡ y…z ‡ a†;

y_ ˆ

ly ‡ x…z

a† ‡ f1 ‡ f2 sin…xt†;

z_ ˆ 1

xy:

…3:2†

The controlled system (3.2) is dissipative and has bounded solutions with chaotic behaviour for some ranges of its parameter l and a. Chaos is suppressed to a periodic solution within the chaotic attractor for certain values of f1 , f2 and x.

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Fig. 4. The states of the controlled system (2.6) and the control u…t† response with time before and after control activation. (a) The response of the state x, (b) the response of the state y, and (c) the response of the state z. The control is activated at t ˆ 100, a ˆ 1, l ˆ 2, k ˆ 0:1 and u0 ˆ 0:5.

Case 2. In this case, the controller (3.1) is added to the third equation of (1.2). The resulting controlled system has the form x_ ˆ

lx ‡ y…z ‡ a†;

y_ ˆ

ly ‡ x…z

a†;

z_ ˆ 1

xy ‡ f1 ‡ f2 sin…xt†:

…3:3†

Also chaos is suppressed to another periodic solution within the chaotic attractor for certain values of f1 , f2 and x. 4.1.1. Numerical simulations In this part, a number of numerical simulations is carried out by using fourth-order Runge±Kutta method with time step size 0.01. The initial conditions are chosen in all these simulations to be x ˆ 0:3, y ˆ 0:4 and z ˆ 0:5. The parameters a and l are chosen a ˆ 2 and l ˆ 1 so that system (1.2) will exhibit chaotic behaviour if no controls are applied. Fig. 6 shows the projection of the chaotic attractor in x±y phase plane. Many numerical experiments are carried out to guide chaos to a periodic solution around the positive equilibrium E1 . The controlled system (3.2), for the values f1 ˆ 0:452, f2 ˆ 1:499 and x ˆ 2:712, chaos is

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Fig. 5. The states of the controlled system (2.8) and the control u…t† response with time before and after control activation. (a) The response of the state x, (b) the response of the state y, and (c) the response of the state z. The control is activated at t ˆ 100, a ˆ 1, l ˆ 2, k ˆ 0:1 and u0 ˆ 0:5.

suppressed to a limit cycle as shown in Fig. 7(a). The orbits of the second controlled system (3.3), with the same values f1 ˆ 0:452, f2 ˆ 1:499 and x ˆ 2:712 are converged to another limit cycle that surrounds the two equilibria Fig. 7(b). Fig. 8 shows the time response of the state x of system (3.2) where the controller force (3.1) is activated at t ˆ 100. 4.2. Delayed feedback control The DFC is proposed by Pyragas [3] to control chaos in continuous dynamical systems. The DFC method [21,22] is based on a feedback of the di€erence between the current state and the delayed state. The delay time is set to correspond to the desired period of the periodic orbit, therefore the feedback vanishes after stabilizing UPO. This method does not require ®nding UPO, but it needs its period only. The DFC method is applied to system (1.2) to stabilize a UPO of period s. The controlled system is obtained by adding a delayed feedback term to (1.2) as follows: x_ ˆ lx ‡ y…z ‡ a† k11 …x…t† x…t z_ ˆ 1 xy k33 …z…t† z…t s††;

s††;

y_ ˆ

ly ‡ x…z

a†

k22 …y…t†

y…t

s††; …4:1†

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Fig. 6. The chaotic attractor of system (1.2) in the x±y phase plane at l ˆ 1 and a ˆ 2.

Fig. 7. (a) The limit cycle of the controlled system (3.2) in x±y plane for the values f1 ˆ 0:452, f2 ˆ 1:499 and x ˆ 2:712. (b) The limit cycle of the controlled system (3.3), for the values f1 ˆ 0:452, f2 ˆ 1:499 and x ˆ 2:712, is plotted in the x±y phase plane.

where K is 3  3 feedback gain matrix and s is the delay time which is the period of the target UPO. The controller in system (4.1) converges to zero when the trajectories converge to periodic orbit with period s. The controlled and uncontrolled systems have the same ®xed points E1 , E2 . 4.2.1. Approximated delay feedback method Our aim is to stabilize the chaotic system (1.2) to a limit cycle. Instead of applying DFC method, we approximate the controlled system (4.1) by applying Taylor theorem, for small s we have x…t

s† ˆ x…t†

s_x…t† ‡ O…s2 †:

Hence, we get x…t†

x…t

s† ˆ s_x…t† ˆ s…y…z ‡ a† ˆ s…x…z

a†

ly†;

lx†; z…t†

y…t† z…t

y…t

s† ˆ s_y …t†

s† ˆ s_z…t† ˆ s…1

xy†:

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Fig. 8. The time responses of the controlled system (3.2) for tracking a limit cycle with a ˆ 1, l ˆ 2, f1 ˆ 0:452, f2 ˆ 1:499 and x ˆ 2:712. The control is activated at t ˆ 100.

This method can be applied for small value of s and the chaotic orbit …x…t†; y…t†; z…t†† of (1.2) is very close to the periodic orbit with the period s. Thus, system (4.1) can be approximated to the following controlled system: x_ ˆ …1

k11 s†…y…z ‡ a†

lx†;

y_ ˆ …1

k22 s†…x…z

a†

ly†;

z_ ˆ …1

k33 s†…1

xy†:

…4:2†

The controlled system (4.2) has the same equilibrium points E1 , E2 of the original system (1.2). In order to keep (4.2) as a dissipative system the conditions, 0 < k11 s < 1, 0 < k22 s < 1 and 0 < k33 s < 1 should be satis®ed. Another study in the future to analyze and test this control method will be presented. 4.2.2. Numerical simulations The controlled system of ordinary di€erential equations (4.2) is integrated by using fourth-order Runge± Kutta method with time step 0.01 , a ˆ 2 and l ˆ 1. The initial values are taken at x ˆ 0:3, y ˆ 0:4 and

Fig. 9. (a) The periodic trajectory of period three of the controlled system (4.2), for the values k11 ˆ k22 ˆ 6:1, k33 ˆ 0 and s ˆ 0:1 is plotted in the x±y phase plane. (b) Periodic orbit is stabilized in the x±y phase plane of the controlled system (4.2) at the values k11 ˆ k22 ˆ 7:1, k33 ˆ 0 and s ˆ 0:1.

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z ˆ 0:5. The time delay s is chosen to be s ˆ 0:1. When the gain matrix has the values k11 ˆ k22 ˆ 6:1 and k33 ˆ 0, the chaotic orbit converges to a periodic orbit with three period, see Fig. 9(a). Fig. 9(b) shows the stabilization of the limit cycle in the xy-phase plane when the gain matrix has the values k11 ˆ k22 ˆ 7:1, k33 ˆ 0 and s ˆ 0:1. 5. Conclusions In this tutorial, we have presented four methods for controlling chaos of the nonlinear dynamical system (1.2). The mathematical controllability conditions are derived from Routh±Hurwitz theorem. Both the methods of linear feedback control and bounded feedback control suppress the chaotic behaviour of the coupled dynamos system to one of the two unstable equilibrium points E1 or E2 . The third method, nonfeedback control suppresses chaotic behaviour of system (1.2) to unstable periodic orbit. The fourth method which is proposed as an approximation to the DFC method suppresses chaotic behaviour to a limit cycle. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

Ott E, Grebogi C, Yorke J. Controlling chaos. Phys Rev Lett 1990;64:1196±9. Singer J, Wang YZ, Bau HH. Controlling a chaotic system. Phys Rev Lett 1991;66:1123±5. Pyragas K. Continuous control of chaos by self-controlling feedback. Phys Lett A 1992;170:421±8. Chen G, Dong X. Controlling Chua's circuit. J Circuits Syst Comput 1993;3:139±49. Chen G, Dong X. On feedback control of chaotic dynamic systems. Int J Bifurc Chaos 1992;2:407±11. Chen G. On some controllability conditions for chaotic dynamics control. Chaos, Solitons & Fractals 1997;8(9):1461±70. Pyragas K. Control of chaos via extended delay feedback. Phys Lett A 1995;206:323±30. Agiza HN. On the analysis of stability, bifurcation, chaos and chaos control of Kopel map. Chaos, Solitons & Fractals 1999;10(11):1909±16. Kapitaniak T. Controlling chaos. New York: Academic Press; 1996. Rajasekar S, Lakshmanan M. Algorithms for controlling chaotic motion: application for the BVP oscillator. Phys D 1993;67: 282±300. Murali K, Lakshmanan M. Controlling of chaos in driven Chua's circuit. J Circuits Syst Comput 1993;3:125±37. Rajasekar S. Controlling of chaotic motion by chaos and noise signals in a logistic map and a Bonhoe€er±van der Pol oscillator. Phys Rev E 1995;51:775±8. Rajasekar S, Murraali K, Lakshmanan M. Control of chaos by nonfeedback methods in a simple electronic circuit system and the Fitz Hugh±Nagumo equation. Chaos, Solitons & Fractals 1997;8(9):1554±8. Ramesh M, Narayanan S. Chaos control by nonfeedback methods in the presence of noise. Chaos, Solitons & Fractals 1999;10(9):1473±89. Ramirez J. Nonlinear feedback for control of chaos from a piecewise linear hysteresis circuit. IEEE Trans Circuits Syst 1995;42:168±72. Franz M, Zhang Minghua. Supression and creation of chaos in a periodically forced Lorenz system. Phys Rev E 1994;52(4): 3558±65. Hegazi AS, Agiza HN, El-Dessoky MM. Controlling chaotic behaviour for spin generator and Rossler dynamical systems with feedback control. Forthcoming in Chaos, Solitons & Fractals. Gills Z, Iwata C, Roy R. Tracking unstable steady states: extending the stability regime of a multimode laser system. Phys Rev Lett 1992;69(22):3169±72. Cook PA. Nonlinear dynamical systems. London: Prentice-Hall; 1986. Ushio T. IEEE Trans CAS-I 1996;43:815. Dressier U, Nitsche G. Controlling chaos using time delay coordinates. Phys Rev Lett 1992;68(1). So P, Ott E. Controlling chaos using time delay coordinates via stabilization of periodic orbits. Phys Rev ES 1995;1(4):2955±62. Ramirez JA. Nonlinear feedback for controlling the Lorenz equation. Phys Rev F 1994;50:2239±342. Braiman Y, Goldhirsch I. Taming chaotic dynamics with weak periodic perturbations. Phys Rev Lett 1991;66:2545±8. Jackson EA. Controls of dynamic ¯ows with attractors. Phys Rev A 1991;44:4839±53. Lorenz HW. Nonlinear dynamical economics and chaotic motion. Berlin: Springer; 1993. Farmer JD. Chaotic attractors of an in®nite-dimensional dynamical system. Phys D 1982;4:366±93. Lornez EN. Deterministic non-periodic ¯ows. J Atmos Sci 1963;20:130±41. Sparrow C. The Lorenz equations, bifurcation, chaos and strange attractors. New York; 1982.