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Copyright <0 IFAC Nonlinear Control Systems, St. Petersburg, Russia, 2001

COl> Publications www.elsevier.comllocatelifac

ON CONTROL OF NONLINEAR CHAOTIC DYNAMICAL SYSTEMS N.A. Magnitskii· S.V. Sidorov··

• Institute for Systems Analysis of Russian Academy of Sci ., 9, Prospect 60-1et Oktyabrya, Moscow 117312, Russia E-mail: [email protected] •• Russian Correspondence Institute of Textile and Light Industry, 38-2, Narodnogo Opolcheniya Str., Moscow 123298, Russia E-mail: [email protected] TU

Abstract: In the present paper the problem of localization and stabilization of an unstable periodic trajectory of a non linear ordinary and delay equations with chaotic behavior is considered. For example unstable periodic orbits of Rossler system and Mackey-Glass equation are exstracted from their chaotic attractors. Copyright ~ 2001IFAC Keywords: nonlinear dynamical system, chaos, periodic orbit, stablilzation

1. INTRODUCTION

a qualitative change in the system dynamics (by passing from a neighborhood of one cycle into the neighborhood of another cycle) using small perturbations of the system parameters.

The presence of chaos is an essential property of most dynamical systems describing complicated physical, chemical, biological, and social processes and phenomena. Chaotic systems are characterized by their high sensitivity to small perturbations of the system parameters and initial conditions, and for a long time the behavior of such systems has been considered unpredictable. There has been the opinion that a desired behavior of a system can be achieved only by suppressing the chaotic behavior with the help of changes, possibly extensive and expensive, in the system itself, which leads to changes in the system dynamics as a whole. However, the special role of chaos in the self-organization of various natural phenomena has recently been recognized (Ott, et al., 1990, Shinbotr, el al., 1993, Zao and Wang, 1994, Loskutov and Shishmarev, 1994). In fact , chaos is a necessary condition for rather than to the efficiency of complicated systems like the human brain (Sepulchre and Babloyanz, 1994). It is only owing to the presence of a chaotic attractor containing, as a rule, infinitely many unstable periodic trajectories (cycles) that one can achieve

In this connection, the problem of chaos control naturally splits two parts: a) how to get into a neighborhood of a given unstable periodic trajectory by small perturbations of the system parameters and moving along trajectories of a chaotic attractor; b) how to approach an unstable periodic trajectory from a neighborhood of it arbitrarily closely and remain near it for an arbitrarily long time. Some approaches to solving the first problem were suggested (Shinbort , et al., 1993) . The present paper deals with the second problem, which is reduced to the localization (detection) and stabilization of unstable periodic trajectories (in particular, steady states) of chaotic dynamical systems (including chaotics mappings). The idea of the localization and stabilization method, whose foundations were given by Magnitskii (Magnitskii, 1996a, 1997b, 1997c, 1997d), is to construct a higher-dimensional dynamical system such that a given unstable periodic trajectory of the original chaotic system is the projection 783

of some asymptotically orbitally stable periodic trajectory of the new system. The scope of the method is rather wide and includes chaotic mappings, chaotic dynamical systems described by ordinary differential equations, distributed chaotic dynamical systems, and delay dynamical systems.

Q(y, s, t,p)

x E jRm, pE JR,

VCr) = A(x(r,p))V(r), V(O) = J,

(1)

Obviously, Q(x*(t,p),T(p),t,p) = 0 for all t and p. Therefore, for any p ( in particular, for p > pO) the vector u*(t,p) = (x*(t,p),p, T(p))T is a periodic solution (a cycle) of the extended system (2). Now in (2) it suffices to choose the matrices E and D of control parameters, the vector C, and the scalar (3 such that the cycle u * (t, p) is an asymptotically orbitally stable limit cycle of system (2) in some neighborhood of the parameter value p, p* :S p :S pi. In this case, the vector yet) which cosists of the first m coordinates of the solution u*(t,p) of system (2) with the initial conditions, say, u(O,p) = (y(O,p),q(O,p),s(O,p))T = (x*(O,p*),p, T(p*)f, tends to the unstable limit cycle x*(t,p) of system (1) for all p E [P*,pil . Theorem. 1. There is constant matrices E and D of control parameters, a vector C, scalar (3, and a value pi > p* of system parameter such that the cycle u* (t, p) is an asymptotically orbitally stable limit cycle of system (2) for all values of the parameter p E (p*, pil .

Proof. Let linearize system (2) in a neighborhood of the solution u*(t,p) and denote the linearization matrix K(t,p). Since

Let the cycle x*(t,p) of system (1) has period T = T(P). Note that if p > p* , then the period of the cycle x* (t, p) is unknown and cannot be

I

8Q(y, s, t,p)

8s

JRm ,

JR is scalar function, yet)

= x*(T,p),

u=u"(t,p)

K(t,p) =

(2)

s = CTQ(y,s,t,p), E

I

then

iJ = F(y,p) + Z(y, t,p)E(q -

where set)

= [Z(x*(T,p))-IlZ- 1 (x*(t,p)),

8Q(y, s,t,p) 8y u=u"(t,p)

found by analyzing solutions of system (1). Let us consider the (m + k + I)-dimensional system

{

(4)

where A(x(r,p)) = DxF(x(r,p)). Therefore, for any point (y, t) E jRm+l, the matrix Z(y, t,p) occurring in (2) can be obtained as the solution of the matrix nonautonomous equation (4) at time t taken along the trajectory x(r,p) of system (1) such that x(t,p) = y.

given by a family F of smooth mappings. Let x* (t, p) be a closed periodic trajectory (a limit cylcle) of system (1) depending on the system parameter p. Without loss of generality, let assume that there exists a critical value of the system parameter p* such that the trajectory x*(t,p) is an asymptotically orbitally stable cycle of system (1) for p :S p* and is unstable cycle for p > p* . In the latter case, system can have attractors in the form of other stable limit cycles of various periods or stable tori, and for larger values of the parameter p, the appearance of strange attractors is possible, which indicates the chaotic dynamics of system in this case. The problem is to localize and stabilize the unstable cycle x* (t, p) of system (1) with the help of small perturbations of system parameter p in domain p > p* of chaotic behavior of trajectories of the system for the case in which there is almost no information about the cycle x*(t,p) itself. The main idea of the method for solving this problem remains the same: in a space of higer dimension, construct a dynamical system such that the unstable cycle x*(t,p) of system (1) is the projection of some limit cycle of the new system; the latter cycle must be asymptotically orbitally stable in the domain p > p*.

pe), q = DQ(y, s, t,p) + (3(q - pe),

(3b)

where x(r,p) is a solution of system (1) at time r under the condition x(t,p) = y. Since the matrix Z(y, t, p) occuring (3) is the derivative of solution of system (1) with respect to initial conditions, we have Z(y, t,p) = Z(x(t,p)), where Z(x(r,p)) is the solution of nonautonomous matrix ordinary differential equation

This problem is one of the most important parts of the chaotic control problem. Let consider a non linear dynamical system

= F(x,p),

(3a)

x(s,p) - x(O,p),

Z(y, t,p) = 8y/8xlx=x(o,p),

2. LOCALIZATION AND STABILIZATION OF UNSTABLE CYCLES OF CHAOTIC DYNAMICAL SYSTEMS.

x

=

=

A(x*(t, p)) 8Q D (u;;t,p))

Z(x*(t,p))E

0

)

(3J

Dx*(T,p).

0

C T x*(T,p) .

E

q(t) E JRk is a vector function, pe E jRk, e =

( C T8Q (U;;t,p))

(l, ... ,I)T, 1 :S k < m; D kxm and Emxk are constant matrices, C E jRm is a constant vector,

The first-approximation linear system acquires the form wet) = C(t,p)w(t) (5)

and (3 E R Let define mappings Q(y, s, t,p) and Z(y , t,p) as follows

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p- 1 S has a unique nonzero coordinate in the (m - k)th position. Let denote it by r . Therefore , eAtDx*(T,p) = (0, eA(HT»)p-ls = 0, and Cx*(T,p) = (0, c, O)p-lS = cr.

in neighborhood of solution u"(t,p) of system (2), where wet) = u(t,p) - u"(t,p). Since Z(x"(t,p)) is a principial solution matrix of the periodic linear system (4), it follows from the Floquet theory that it can be represented as Z(x"(t ,p)) = R(t)exp(Bt) , where R(t) is a periodic matrix with period T = T(P) and R(O) = R(T) = 1 . By A = A(P) let denote the Jordan form of the matrix B ; thus, B = PAP-I . The matrix A has one zero for any value of system parameter p lying in a neighborhood of the point p*. In addition, since the point p* lies on the boundary of stability domain of the cycle x* (t, p) of system (1), it follows that the matrix A has at least one more .A eigenvalue crossing the imaginary axis from the left to the right for p = p*. Without loss of generality, let assume that this condition is satisfied for last k, 1:S; k < m, eigenvalues of the matrix A. Therefore, if p = p" then Re.Ai < 0, i = 1, ... ,m-k-l, .Am-k = 0, and Re.Ai = 0, i = mk + 1, . . . ,m. The number k determines the order (m + k + 1) of system (2) and linearized system (5).

Now set c = -sgn(r) . Let F = diag(ai) have equal diagonal entries in each block corresponding to a Jordan block of the matrix X. Represent the matrix A = diag(A, X), where matrix A coincides with the left top (m - k) x (m - k) block of the matrix A. Then Eq. (7) acquires the form

wet)

(~

o

= N(P)w(t) =

}_ I 0

pL, ~) 0

(8)

wet)

-Irl

Let us show that the diagonal entries ai of the matrix F can allways be chosen so that for p = p' the matrix N(P) has a unique zero eigevalue and the remaining eigenvalues of N(P) have negative real parts. To this end, it sufficies to chose ai such that all 2k eigenvalues of matrix

In addition, suppose that the matrix exp(AT) has a unique eigenvalue equal to unity for p = p" that is multiplicator of the cycle x* (t, pO) of system (1) , so that .Ai "1= 0, i = m- k+ 1, ... ,m. Let represent the solution wet) of the linear system (5) in the form

u_(_X F) - eAT - 1 (31 + X have negative real parts. Let us consider the Jordan block Aof the matrix X with multiplicity j corresponding to some eigenvalue .Ai. Then the eigenvalues J1. of the matrix U satisfy the equation

wet) = G(t)w(t) = = diag[R(t)P, exp( -Xt) , l]w(t),

(6) where X is the matrix concinding with the rigt bottom k x k block of the matrix A. The matrix exp(XT) has no eigenvalues equal to + 1. By constuction, the matrix G(t) is periodic for p = p" and is bounded as t -+ 00 in some neighborhood p > pO. Substituting (6) to (5), one can obtain

det(U - J1.1)

=

- IT det (~eAT- -J1.11 (31 +ailA- J1.1 ) -

i

- 0 - .

(9)

In turn , each determinant occurring in (9) vanishes if and only if

wet) =

[(.Ai - J1.)((3 +.Ai - J1.) - ai(e AiT - 1)]1 = O. The last equation has two j-multiple roots satisfying the relation

J1.2 - J1.((3 + 2.Ai) + ((3 + .Ai).Ai - ai(e A- iT - 1) = O. where J(t)

(10)

= P(e At -1)e- At .

One can choose the parameters (3 = -2d, and = -~ /(e AiT - 1) , d > 0. In this case, each equation in (10) has two equal roots J1.i = -d + .Ai with negative real part in some neighborhood p E [P", pi] of the value p". Therefore, for all pE [P",pi] the principial solution matrix W(t ,p) of the first-approximation linear system (5) can be represented in the form W(t,p) = G(t,p)eN(p)t , where matrix G(t,p) is bounded as t -+ 00, and matrix N(P) has exactly one zero eigenvalue corresponding to the unit multiplicator of the periodic solution u*(t, p) of system (2) . All remaining m+k eigenvalues of the matrix N(P) have negative real parts for any p E [P*, pi], whence the periodic solution u"(t ,p) of system (2) is asymptotically

Let choose the matrices E and D and C in following way: E = P(O, F)T , D = (0, H)P-l, and CT = (0, C, O)P-l, where H k x k is the identity matrix, F is a matrix commuting with matrix X, and vector C has a unque nonzero entry c in the (m - k)th column. In addition we have

ai

eAt p- 1 Ee- At = (0, F)T, eAt DJ(t) = (0, eAT -1) , C T J(t) = O. Next, note that the vector x* (t, p) is a solution of the linear system (4). Therefore, there exists a vector s = x"(O, p) = x"(T,p) such that 'x"(t,p) = V(t)s = R(t)PeAtp-1s. Consequently, p- 1s = eAT p- 1s , hence the vector

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orbitally stable for p E [p' , pi]. The proof of the theorem is complete.

with operator right-hand side, x(t + 19) , -T ~ 19 ~ 0, and

To illustrate the method, let consider the Rossler chaotic dynamical system and the Mackey-Glass nonlinear chaotic delay equation. The Rossler dynamical system

S(t)xt(19)

={

f(x(O) , x( -T), p) , if 19 = 0, dx if -

d19 '

for a = 0.5 and b = 0.75 has stable limit cycle in the domain 1.375 < p < 1.88, then the limit cycle of system (11) loses stability, and a strange attractor appears in the system for p = 2.35. This attractor is well know as Rossler attractor. In the numerical experiment it was localized and stabilized the limit cycle of system (11) with the help of system (2) . Fig. 1 shows the attractor of the Rossler system: 1 is the chaotic trajectory of the Rossler system for p = 2.4 , 2 is the limit cycle of the system for p = 1.85, and 3 is the trajectoty of system (2) stabilizing the limit cycle of system (ll) for p = 2.4.

i

= 1, . .. ,rn -

Yo

fo(Yo , Ym, p) , Yi

=

i

= m (Yi-l T

= 1, . . . ,rn.

Yi) , (14)

Thus the equation (12) is red used to the system of ordinary differential equations

yE ]Rm+l .

(15)

In this case solution x(t , p) of Eq. (12) corresponds to the values of the coordinate Yo(t , p) of system (15), and trajectory of Eq. (12) in the extended phase space]R x C[ - T; OJ corresponds to the trajectory of system (15) in the phase space ]Rm+l. Consequently, the problem of stabilization of an unstable periodic solution x(t,p) of the delay equation (12) can be reduced to the stabilization of the corresponding unstable periodic trajectory y(t , p) of system (15) .

1 ~

Fig. 1 By way of second example, let consider the problem of localization and stabilization of an unstable trajectory in the nonlinear equations with delay argument, which have a chaotic behavior

By way of example, let cosider the Mackey-Glass equation (Mackey and Glass, 1997)

. X

f3 0Tl x(t -

~

0,

(12)

T)

= -ax(t) + () 0+ x (t - T ) , TI

f(x(t) , x(t, -T),p), t

1.

Then, using a finite-difference approximation of the derivative, one can obtain

Y = F(y , p) ,

=

19 < O.

Let represent system (13) as a finite-dimensional system of ordinary differential equations. To this end, we divide the interval [-T ; OJ into rn equal parts and set

x=-(y+ z ), y=x+ay, z=b+z(x-p), (ll)

x(t)

T ~

Tl

(16)

where a, f3o, () , and n are positive constans such that f30 > a > 0 , nB > 2, and 6aB > f3o ,; here B = (f3o - a) / f3o . Equation (16) has the unique

where x(t), f(-) are scalar functions and T > 0 is a constant delay. Let x'(t,p) is a periodic trajectory of Eq. (12) with period T(p) . As a rule there exists a critical value of the parameter p' such that the limit cycle x' (t, p) is asymptotically orbit ally stable for p ~ p' and unstable for p > p' . The problem is to localize and stabilize this cycle for p > p' , including the values of p for which Eq. (12) has chaotic behavior.

steady state x

= ()

V

f30 : a, which loses stability

arccos( -a/ b) ~ , b = a( nB - 1), as a result yb 2 - a 2 of a Hopf bifurcation. For a = 1, f30 = () = 2, and n = 10 equation (16) has steady stationary solution x = 2 in domain 0 < T < 0.4708. For T = 0.4708 there is a steady periodic solution (or a limit cycle in the extended phase space). For further growth of T , a sequence of period doubling bifurcations take place, and for T > 1.68 chaotic dynamics is observed. for

Let us show that the approach suggested above can be used for this problem. Let C[ -T ; 0] be the space of continuous real functions 'P(') defining initial conditions for Eq. (12) on the interval [-T ;O] . Let equip this space with the norm IIx(19) 11 = sup( lx(19)I, - T ~ 19 ~ 0) . To Eq. (12) with the initial condition x(19) = cp(19), -T ~ 19 ~ 0, one can assign the system of equations

T

>

To solve the problem of localization and stabilization of an unstable periodic soluton for the values of T corresponding to chaotic oscillations, let reduce Ed. (16) to the finite dimensional system (15). Then in the numerical experiment let localize

(13)

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Loskutov, A. and A. Shishmarev (1994). Control of dynamical system behavior by parametric perturbations: An analytic approach. Chaos, 4, N 2, 391-395.

and stabilize the limit cycle of the system (15) with the help of (m + 3)-dimensional system (2).

Sepulchre, J. and A. Babloyantz (1994) . Controlling chaos in a network of oscillators. Physical Rev., E48(2), 119-125. Magnitskii, N.A. (1997a) On stabilization of the fixed points of chaotic dymamical systems. Dokl. RAN, 352, No 5, 61()"'{)12. Fig. 2.

Magnitskii, N.A. (1997b). On stabilization of unstable cycles of chaotic mappings Dokl. RAN 355, No 6, 747-749.

The results concerning the stabilization of an unstable solution of Eq. (16), using system (15), for the parameter value r = 1.8 corresponding to chaotic oscillations are shown in Fig.2:

Magnitskii, N.A . (1995c) . Hopf bifurcation in the Rossler system. Differents. Uravn., 31, No 3, 538-541. Magnitskii, N.A. (1997d) Stabilization of unstable periodic orbits of chaotic maps. Computers Math . Applic., 34, N 2-4, 369-372.

a) the projection of the phase portrait of the chaotic attractor on plane (Yo, Ym) for finitedimensional system (15) or on plane (x(O), x( -r)) for Eq. (16) with r = 1.8;

Mackey, M. and L. Glass (1997) Oscillations and chaos in physiological control systems. Science, 197, 287-289.

b) a similar projection of the phase portrait of the limit cycle of the finite-dimensinal system (15) [and, respectively, Eq. (16)] for r = 1.2; c) the projection of the phase portrait of the unstable limit cycle stabilized by system (2) for r = 1.8 on plane (Yo, Ym) for system (15) or on plane (x(O),x(-r)) for original equation (16) .

3. CONCLUSION In numerous cases, the presence of chaos can be treated as an advantage of a dynamical system, which allows one to qualitatively change its dynamics by slight perturbations of the control parameters. This requires finding and stabilizing unstable periodic trajectories of the system braided into the web of its chaotic attractor . In the present paper a method is suggested that allows one to successively perform this procedure both for finite-dimensional and infinite-dimensinal chaotic dynamical systems and for chaotic mappings.

REFERENCES Ott, E ., C. Grebogi and J .A. Yorke (1990). Controlling chaos. Phys. Rev. Lett., 4 , 1196-1199. Shinbort, T ., C. Grebogi and J .A. Yorke (1993). Using small perturbations to control chaos. Nature , 363, N 3, 411-417. Zhao, H. and Y. Wang (1994) . Controlling timedependent chaotic dynamical systems. Dynamical Systems and Chaos, 2 , 363- 366.

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