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DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMSSERIES B Volume 6, Numb er 5, September 2006

Website: http://AIMsciences.org
pp. 11571174

PARAMETRIC PERTURBATIONS AND NONFEEDBACK CONTROLLING CHAOTIC MOTION

Alexander Loskutov
Physics Faculty, Moscow State University 119992 Moscow, Russia

(Communicated by Miguel Sanjuan)
Abstract. In this paper we generalize analytic studies the problems related to suppression of chaos and nonfeedback controlling chaotic motion. We develop an analytic metho d of the investigation of qualitative changes in chaotic dynamical systems under certain external periodic perturbations. It is proven that for polymodal maps one can stabilize chosen in advance periodic orbits. As an example, the quadratic family of maps is considered. Also we demonstrate that for a piecewise linear family of maps and for a two-dimensional map having a hyperbolic attractor there are feedback-free perturbations which lead to the suppression of chaos and stabilization of certain periodic orbits.

1. Intro duction. In the last few years an unexpectedly interesting property of chaotic dynamical systems has come to light: weak external perturbations can qualitative change their behavior. It was found that chaotic dynamics is surprisingly pliable to certain actions. Such problems as controlling chaotic motion [1] and suppression of chaos [3, 4, 2] are related to this phenomenon. Due to important and profitable applications (see [5] and refs. cited therein) they have attracted a considerable interest. It is sufficient to mention the following: information processing and communications (see, e.g., [6, 7, 8, 10]), an artificial pattern formation in chaotic spatiotemporal systems [9], a selforganization problem [11, 12], controlling cardiac chaos and suppression of complex cardiac rhythms (see, e.g., [13, 14, 15, 16, 17, 18, 19] and also references therein), and other significant problems of nonlinear dynamics [5]. Apparently, the control of dynamical systems by parametric perturbations has been first proposed in papers [3, 4]. Later, in a series of publications (see [2] and refs. cited therein) this phenomenon has been justified analytically and studied numerically. As is known, qualitative changes in the behavior of chaotic systems with external perturbations can be realized by two different ways. First of them provides the removing a system from the chaotic state into a regular regime by means of external actions without a feedback. In other words, this method does not take into account the current value of the dynamical variables of the system. The qualitative different method ensures the correction in dynamical variables according to their values and
2000 Mathematics Subject Classification. Primary: 58F13, 58F03; Secondary: 53F12, 58F30. Key words and phrases. Chaos, suppression of chaos, perturbed maps, periodic orbits.

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ALEXANDER LOSKUTOV

thus, this method involves the feedback as a component of the system. By the established convention, the first way is called suppression of chaos (or sometimes nonfeedback controlling chaotic motion) and the second one is called (feedback) controlling chaos. In turn, each of these ways can be subdivided into a parametric (multiplicative) method and a forcing method. Although there are several papers devoted to a theoretical substantiation of the questions of the control (as reviews see [5, 2]), it would be interesting to develop a sequential theory and establish a rigorous foundation of the possibility of the chaos suppression and its control. Apparently, it is very difficult to make this in general. But we may resolve such a problem for certain families of dynamical systems. These questions are a ma jor focus of interests for the present paper. We describe rigorous methods of the investigation of qualitative changes in the behavior of chaotic maps under external periodic perturbations and propose an analytical key which allows to find such perturbations. The article is organized as follows. First, n-dimensional maps in a general form with external periodic parametric perturbations are considered. Then it is shown that for polymodal maps there are -periodic parametric perturbations which lead to stabilization of prescribed periodic orbits. As an example, a quadratic family of maps is considered in detail. In the next sections a one-dimensional piecewise linear family and a twodimensional map having a hyperbolic type of attractor are studied. It is shown that chaotic behavior of these maps may be suppressed by the stabilization of certain periodic orbits. 2. External p erturbations of n-dimensional maps. In general, the control (feedback or feedbackfree) of dynamical systems involves a certain additive or/and multiplicative variable(s) which take(s) into account additive or/and multiplicative perturbations, respectively. Therefore, first we study the properties of dynamical systems with external perturbations. Let us consider a n-dimensional oneparametric family of maps in a general form Ta : x - f (x, a) , (1) where x = {x1 , . . . , xn } M , f = {f1 , . . . , fn } is a certain nonlinear function, a is a control parameter, and M is a compact invariant set. Let A be a set of the admissible values of the parameter a. Introduce a parametric perturbation for the map (1), G : A A, G : a - g (a), a A. (2) In this case the perturbed map (1) has the form T : y - h(y) , (3) where y M в A, y = (x, a), h(y) = (f (x, a), g (a)). We may involve parametric perturbations into the system by two different ways. If an external source is associated with a multiplicative action with respect to dynamical variables then (multiplicative) parameters of the system are modified. If, however, external sources are included into the model as additive terms with respect to dynamical variables then the force perturbation takes place. Thus, depending on the type of external actions, a certain component of the perturbed dynamical systems is modified. Below only periodic (cyclic) perturbations are considered. Then for period we have from (2): ai+1 = g (ai ), i = 1, 2, . . . , - 1, a1 = g (a ), ai = aj for i = j, ai A, 1 i . Now, any type of periodic perturbations with period

NONFEEDBACK CONTROLLING CHAOTIC MOTION

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can be associated with the following vectors: a = (a1 , . . . , a ), a IR . So, the ^ ^ ^ set A = {a A A · · · A : a = (a1 , . . . , a ), ai = aj , 1 i, j , i = j , ^ ^ ^ a1 , . . . , a A}, A IR , corresponds to all possible sets of -periodic perturbations which operate on A. Note that the explicit form of the function g (a) in (2) is not essential. For our aims it is sufficient to know that g transforms the parameter values into an infinite series {ai } consisting of the repeated subsequences (a1 , a2 , . . . , a ). It is quite clear that period t of any obtained periodic orbit in the perturbed map (3) is multiple to the period of perturbation, t = k , where k is a positive integer. Really, if the perturbed map (3) has a t-periodic orbit then the coordinate sequences which form this periodic orbit are also periodic with period t. But the sequence {ai } is already periodic with period . Therefore, t = k .
times

Figure 1. A periodic orbit of the periodically perturbed map (3) in the space (x, a) (a), and in the pro jection onto the coordinate axis (b).

Figure 2. A periodic orbit with coincident coordinates of the periodically perturbed map (3) in the space (x, a) (a), and in the pro jection onto the coordinate axis (b). However, there is an important remark to be made. If we pro ject an obtained periodic orbit onto the initial space M (i.e. when the perturbed system (3) as a nonautonomous one is considered) then we may get in M an orbit which can not be a periodic orbit in the usual sense. The matter is that the points of this periodic orbit, which differ from each other only in the coordinate a, are pro jected onto one and the same point in the space M . Thus, in non-autonomous systems such a behavior can not correspond to periodic orbits because the representative point of the map hits several times in some points forming the given periodic orbit.

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ALEXANDER LOSKUTOV

For example, for onedimensional (n = 1) maps, in a general case, in the projection onto the initial space M = I we get a periodic orbit of the period k (Fig.1a,b). But in I we may obtain a orbit with the coincided xcoordinates when xi = xm , ai = am , i = m, where (xi , ai ) and (xm , am ) are points of the periodic orbit of the perturbed map (3) (Fig.2a,b). However, the described situation can be considered as a degenerated one, and it may occur only in special cases. Introduce a quite simple construction which is necessary for the further analysis. Any -cyclic transformation for the map (1) means that the perturbed system (3) can be written as follows: Ta1 : x - f (x, a1 ) f1 , Ta2 : x - f (x, a2 ) f2 , T= (4) ................., Ta : x - f (x, a ) f . Consider functions of the following form: F1 = f (f -1 (...f2 (f1 (x))...)) , F2 = f1 (f (f -1 (...f3 (f2 (x))...))) , ...................... , F = f -1 (f -2 (...f1 (f (x))...) ,
(1) (n) (1) (n)

(5)

where x = {x1 , . . . , xn }, and fi = {fi , . . . , fi }, Fi = {Fi , . . . , Fi }, i = 1, 2, . . . , , are the n-component functions. Thus, the perturbed map (3) one can rewrite in the form T1 T2 .. T : : . : x x .. x - - ... - F1 (x, a1 , F2 (x, a1 , ....... F (x, a1 , . . . . .., .., .. .., a ) , a ) , .., a ) ,

(6)

for which initial conditions are determined as follows: x1 = f1 (x0 ), x2 = f2 (x1 ), . . ., x -1 = f -1 (x -2 ). Thus, analysis of non-autonomous -periodic perturbed maps may be reduced to the consideration of autonomous maps of the form (6). Moreover, to describe the dynamics of the initial nonautonomous system it is sufficient to carry out the analysis of only one of the functions (5) [20]. This result follows from Prop osition 1. If the map Tk , 1 k has a periodic orbit of period t and the function fk (x) is a C 0 -function then the map Tp , p = k + 1 (mod ), also has a periodic orbit of the same period t. Moreover, if i) a periodic orbit of the map Tk is stable then a periodic orbit of the map Tp is stable as wel l; ii) fk is a homeomorphism then the maps Tk and Tp are topological ly equivalent. Proof see in [20].

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3. Stabilization of the required p erio dic orbits and suppression of chaos. In this Sect. we demonstrate that for polymodal maps one can stabilize the prescribed periodic orbits by means of simple periodic parametric perturbations. It is also shown that for a certain class of the piecewise linear maps and for a so-called Belykh map, which has a hyperbolic attractor, such perturbations lead to the chaos suppression and stabilization of certain initially unstable orbits. Since we consider chaotic maps, let us introduce a subset Ac A such that if a Ac A then the map (1) possesses chaotic dynamics. A rigorous definition of chaos may be found in [2, 23, 24]. 3.1. Polymo dal maps. Consider a onedimensional map in a general form Ta : x - f (x, a), (7) where x I , f is a certain function and a is a control parameter. As before, let us introduce a -periodic parametric perturbation, G : A A, G : a - g (a), a A A, such that ai+1 = g (ai ), i = 1, 2, . . . , - 1, a1 = g (a ), ai = aj for i = j . Rewrite the perturbed map in the form: x - f (a, x) , (8) T= a - g (a) . Now we can get the following interesting result. Suppose that the map (8) satisfies the following conditions: (i) there is a subset I such that for any x1 , x2 we have a A for which f (x1 , a ) = x2 ; (ii) for an arbitrary a A there is a critical point xc , i.e. f (x, a)/ x|x=xc Dx f (xc , a) = 0. It is quite clear that in this case for any x2 , x3 , . . . , x we can find such values x1 and a1 , a2 , . . . , a that in the perturbed map T the periodic orbit (x1 , x2 , . . . , x ) is stable for a = (a1 , . . . , a ). Really, let us choose the following arbitrary values ^ x1 , x2 , . . . , x . In view of the condition (i), the system f (x1 , a1 ) = x2 , f (x2 , a2 ) = x3 , . . . , f (x , a ) = x1 (9) has the solution of the form a = (a1 , a2 , . . . , a ). This means that the sequence of ^ the points (x1 , x2 , . . . , x ) = p is a -periodic orbit of the map T at the periodic perturbation a = (a1 , a2 , . . . , a ). To stabilize this orbit p it is sufficient to choose ^ the element x1 in a quite small neighborhood of the critical value xc because the corresponding multiplier (p) =
N i=1

Dx f (xi , ai ) and Dx f (xc , a) = 0 for an arbitrary

a. Thus, the orbit (xc , x2 , . . . , x ) = p is stable for xi M , i = 1, 2, . . . , , and parameters a = (a1 , a2 , . . . , a ) satisfying the system (9). ^ 3.1.1. Example. Consider as an example the following quadratic family of maps in the form Ta : [0, 1] [0, 1], Ta : x - f (a, x) = ax(1 - x) , that is a simplest model It is well known that for behavior. However for introduce a subset Ac (10) of some nonlinear phenomena (see, e.g., [21, 22, 23, 24]). a [0, a ), a = 3.569 . . ., the map (10) has only regular a (a , 4] it can be both regular and chaotic. Let us (a , 4] such that for a Ac the map (10) has the chaotic

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ALEXANDER LOSKUTOV

behavior. It can be proven [25, 26] that the subset Ac has the positive Lebesgue measure and the point a = 4 is a density point of this subset. Let us consider the periodically perturbed map (10). In previous papers (see [27, 2] and references cited therein) it has been shown that periodic parametric ^ perturbations operating in the chaotic subset Ac can stabilize its dynamics. ^ ^ ^ Theorem 1. There exists a subset Ad Ac such that if a Ad then the perturbed ^ quadratic family possesses stable periodic orbits. Proof see in [27]. It is evident that for the quadratic family (10) the set is the interval [xb , xe ], where xb and xe are the solutions of the equation xint = f (x, 4), and xint is the intersection point of the curves y = 4x(1 - x) and y = x. Thus, [xb , xe ] = [1/4, 3/4]. Suppose that this map has an orbit of the period t = , i.e. p = (x1 , x2 , . . . , xt ). Then the points forming this periodic orbit obey the following system: x2 = a1 x1 (1 - x1 ), x3 = a2 x2 (1 - x2 ), . . . , x1 = at xt (1 - xt ). (11) To determine the parameters for which the perturbed map has the periodic orbit p = (x1 , x2 , . . . , xt ), it is necessary to express the values ai from system (11): x2 x3 x1 a1 = , a2 = , . . . , at = . (12) x1 (1 - x1 ) x2 (1 - x2 ) xt (1 - xt ) It is obvious that not for all possible xi (0, 1) the expression ai [0, 4] takes place. However, if it holds then for a given periodic orbit p = (x1 , x2 , . . . , xt ) one can find the parameter values (a1 , a2 , . . . , at ) for which the perturbed map has such an orbit. If | (p)| = get:
t i=1

ai (1 - 2xi ) < 1 then it is stable. Therefore, from (12) we
t

| (p)| =
i=1

1 - 2xi <1. 1 - xi

(13)

Owing to the equality xc = 1/2 we find that (1 - 2xc )/(1 - xc ) = 0. Thus, the expression (13) always holds. The set of values (x1 , x2 , . . . , xt ), for which there are ai [0, 4] and inequality (13) holds, forms a certain region in the coordinate space IRt . Using the system (12) we can obtain the corresponding region in the parametric space IRt . Let us consider the case of twoperiodic perturbation, = 2. In the space (x1 , x2 ) the region of stable orbits is defined by the following set of inequalities: x2 x1 1 - 2x1 1 - 2x2 4, 0 < 4, <1. (14) x1 (1 - x1 ) x2 (1 - x2 ) 1 - x1 1 - x2 Realization of the first and the second inequalities corresponds to the region of all orbits of period two. The third inequality cuts from it the existence region of the stable orbits. Let us fix the value x1 (0, 1). Taking into account singularities, we find from (14): 3x1 - 2 1 0 < x2 < , 0 < x1 < , 5x1 - 3 3 1 3 x1 , < x1 < , 0 < x2 < 3x1 - 1 3 5 3x1 - 2 x1 3 < x2 < , < x1 < 1 . 5x1 - 3 3x1 - 1 5 0<

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Thus, we get the existence region of all possible stable orbits of period two, p = (x1 , x2 ), for the perturbed map (Fig.3a). To construct a parametric region corresponding to these orbits, it is necessary to transform Fig.3a by means of (12). Let us divide the region in Fig.3a into subregions which are mapped onto (a1 , a2 ) by a oneto-one manner. Now it is necessary to transform the boundaries of the subregions. Thus, we find the existence region of the stable period-two orbits, p = (x1 , x2 ), in the parametric space (a1 , a2 ) (Fig.3b). To clarify the action of the map (11), the region in Fig.3a is divided into 4 parts which marked by different shading. The corresponding regions in Fig.3b have the same structure. Now we can easy analyze the twoperiodically perturbed quadratic family.

Figure 3. The existence region of the period two orbits for the perturbed ( = 2) quadratic map in the space (x1 , x2 ) defined by the curves x2 = 4x1 (1 - x1 ) (a), x1 = 4x2 (1 - x2 ) (b), x2 = (3x1 - 2)/(5x1 - 3) (c), x2 = x1 /(3x1 - 1) (d), and in the parametric space (a1 , a2 ) given by the curves a2 = 1/a1 (e), a2 = 8/[a1 (4 - a1 )] (f ), a1 = 8/[a2 (4 - a2 )] (g). 1) Obviously, the symmetry of the parameter space is the reason of the symmetry of the twodimensional stability diagrams. 2) Because Fig.3b has intersecting subregions, then the map (11) is a singlevalued map but it is not a one-to-one transformation. 3) Intersections mean that the perturbed map is bi-stable. At certain parameter values it may simultaneously have two stable periodic orbits. 3.1.2. Estimation of admissible noises. Numerical approach allows us to make only approximate calculations. Therefore, it is necessary to know estimation of the admissible errors in the parameter values of the perturbed map. This estimation can be done using the following result. Theorem 2. [20] Suppose that f (x, a) C 2 [M в A] and at a = (a1 , a2 , . . . , at ) the ^ perturbed map T has a stable orbit of the period t, p = (x1 , x2 , . . . , xt ). Then , if |ai | a = 1
x,a

tSa LS

t- 1 x

t

L = max |D2 f (x, a)|, Sx = max |Dx f (x, a)|, this map has also a stable orbit p = x
x,a

i=1

i Sx , where i = 1, 2, . . . , t, Sa = max |Da f (x, a)|, x,a

(xc + x1 , x2 + x2 , . . . , xt + xt ) of period t at a = (a1 + a1 , a2 + a2 , . . . , at + ^

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ALEXANDER LOSKUTOV

at ) with |xi | x = 1 . t LSx-1

Proof. Assume that all parameters ai are perturbed, ai = ai + ai . Let us find a value x1 = x1 - xc , where x1 = F1 (x1 , a1 , a2 , . . . , at ). Then xc + x1 F1 (xc , a1 , a2 . . . , at ) + Dx F1 (xc , a)x1 + ^
t t t

account the expressions xc = F1 (xc , a) and Dx F1 (xc , a) = (p) = 0 we get: x1 = ^ ^
i=1 l=i+1

i=1

Dai F1 (xc , a)ai . Therefore, taking into ^

Dx f (xl , al )Da f (xi , ai )ai . Thus,
t t t

|x1 |

a i=1 l=i+1

Dx f (xl , al · Da f (xi , ai ) a tSa
i=1

i Sx .

(15)

Let us estimate the change in the multiplier of the orbit. We have: (p ) =
t i=1

D2 f (xi , ai ) x

t
l=1 l=i

Dx f (xl , al )xi +

t i=1

D2 f (xi , ai ) ax

t
l=1 l=i

Dx f (xl , al )ai . Obviously,
t

in both of these sums only the first components are nonzero because Dx f (x1 , a1 ) = Dx f (xc , a1 ) = 0. Thus, (p ) = D2 f (xc , a1 )x1 +D2 f (xc , a1 )a1 x ax But D2 f (xc , a1 ) = D ax
a

Dx f (xl , al ).

l=2

Dx f (xc , a)

a=a

= Da (0) = 0. Therefore, the multiplier
1

of the changed periodic orbit is | (p )| = |x1 | D2 f (xc , a1 ) x

t l=2

Dx f (xl , al ) . For
t l=2

the stability of this orbit it is necessary that |x1 | D2 f (xc , a1 ) x

Dx f (xl , al )

t t |x1 |LSx-1 < 1. Hence, |x1 | x = 1/(LSx-1 ). Therefore, if the perturbation x1 is less than the value x , then the periodic orbit is stable. However, at parametric perturbations the maximum of x1 is t i t given by the inequality (15). Thus, finally we get: a tSa i=1 Sx = 1 LSx-1 or

a = 1

tSa LS

t- 1 x

t i=1

S

i x

.

For the quadratic family of maps at a [0, 4] and x [1/4, 3/4] we find: Sa = 1 2 max f (x, a) = , Sx = max f (x, a) = 2, L = max f (x, a) = 8. x,a a x,a x x,a x2 4 This gives a sufficient estimate of the admissible errors in the parameter values: t a 1 t2t i=1 2i , x 1/2t+2 . From the obtained results we may get the following important consequence. If the induced periodic dynamics in a polymodal family under a periodic perturbation is observed, then it can not be destroyed by a small enough external noise which smears the parameter values a. The maximal noise level can be estimated by the ^ above theorem 2.

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3.2. A piecewise linear family of maps. Consider the following family of maps of the interval [0, 1] into itself: q (a)x + r(a) , 0 x a, Ta : x - f (x, a) = (16) p(a)(x - 1) , a < x 1, where a (0, 1) is a control parameter and q (a) = (1 - a)/(a(2 - a)), r(a) = 1/(2 - a), p(a) = -1/(1 - a). All maps of the family (16) are constructed in such a way that the critical point xc = a hits in the unstable fixed point r(a) x after ~ three iterations (Fig.4). First, let us prove that these maps has a mixing attractor for an arbitrary a (0, 1), i.e. they are chaotic.

Figure 4. A piecewise linear map determined by the family (16). Let T be a map of a set M into itself and M be a compact invariant subset which is different from a periodic orbit. We say (see [28, 29]) that is a mixing set if for any open set U in and any finite covering = {j } of the set there exist m = m(U, ) and r 1 depending only on such that T
m r -1 i=0

T iU

j = for

all j . If, in addition, the set attracts almost all tra jectories from a neighborhood V , i.e. there exists V such that V = , T V V and T i V = , then the set is said to be a mixing attractor. The existence of the mixing attractor in maps of an interval into itself is a sufficiently strong property. For example, a map with the mixing attractor does not have stable periodic orbits, and it possesses sensitive dependence on initial conditions. Moreover, for such maps it is possible to construct absolutely continuous invariant measure. Thus, maps with the mixing attractor can be called chaotic. Moreover, the following result holds [28]: if T : x f (x), x I , and f C 0 (I , I ) where I is an interval, then the mixing attractor consists of one or several intervals which are cyclically mapped into each other, and periodic points are dense on it. Now we may get the chaoticity conditions in the family (17). Theorem 3. For any a (0, 1) the map Ta (16) has the mixing attractor = [0, 1].
i>0

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Proof. We make it by two stages. 1) To show that = [0, 1] is an attractor for Ta , it is sufficient to consider the map (16) in any extended interval, for example, in [-1/2, 3/2] (Fig.5). One can see that for any a (0, 1) there exists a neighborhood V of the set such that f V V and f 2 V = . Thus, is the attractor on the interval [0, 1].

Figure 5. An extended piecewise linear map (16). 2) To prove that the attractor = [0, 1] is a mixing, note that it consists of two subintervals, J1 and J2 , which are mapped into each other under the action of Ta , = J1 J2 , Ta J1 = J2 , Ta J2 = J1 , where J1 = [0, x], J2 = [x, 1] (Fig.6). The ~ ~ rest of proof uses the following result.

Figure 6. The intervals J1 , J2 composed the attractor = [0, 1] for the map (16). Lemma 1. For any open set U there exists a number n such that n i) Ta U = or n ii) Ta U = Ji , where i = 1, 2.

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2 2 2 Proof. Consider the map Ta = Ta Ta (Fig.7). Because Ta J1 = J1 and Ta J2 = 2 J2 , then Ta is decomposed into two maps, TJ1 and TJ2 , operating on J1 and J2 , respectively. Moreover, owing to p2 = p(a)p(a) > 1, |p(a)q (a)| = 1/(a(2 - a)) > 1, a (0, 1), the maps TJ1 , TJ2 are expanding ones. Therefore, for any open set 2 2 U J1 (U J2 ) there exists a number m such that Ta m U = J1 (Ta m U = J2 ). In other words, for n = 2m we have the condition ii). Now it is necessary to consider 2 the sets U x. For such sets (Fig.7) there exists m : Ta m U = . ~

2 Figure 7. The second iteration, Ta , of the map Ta (16).

Thus, to prove theorem 3 it is necessary, for any open set U , to choose r = 2 m and m from the above lemma. In this case Ta (Ta U U ) = because, if U J1 m m then Ta U J2 , and if Ta U = J1 then Ta (Ta U ) = J2 . By the same manner we obtain the result for U J2 and U x. ~ It is not hard to understand the structure of periodic orbits of the family (16). 2 Because Ta is decomposed into two independent maps, TJ1 and TJ2 which operate in the subintervals J1 and J2 respectively, it makes it possible to construct maps 2 having orbits of an arbitrary even period, Ta k . These orbits are dense in = [0, 1]. k k For this it is sufficient to find TJ1 and TJ2 . A completely different situation takes place for odd iterations of Ta . Really, 2 2 Ta k+1 = Ta Ta k . Therefore, except for a fixed point, Ta does not have orbits of odd periods. Thus, the family (16) has the mixing attractor with dense periodic orbits of even periods. Now let us consider the perturbed family (16). Restrict ourselves by the case of a 2-periodic transformation of the parameter a: T1 : x - F1 (x) Ta2 Ta1 , (17) T2 : x - F2 (x) Ta1 Ta2 . Without loss of generality suppose that 0 < a1 < a2 < 1. Introduce the following notations: a1 = a, a2 = a + , > 0 (Fig.8). Thus, the map T1 has three fixed points which exist for any a1 , a2 (0, 1). These fixed points correspond to three

1168

ALEXANDER LOSKUTOV

different 2-periodic orbits of the perturbed map (17). The orbit corresponding to the middle fixed point (Fig.8) arises from the fixed point of the unperturbed map (17). Two orbits of period 2 corresponding to two other fixed points arise from an orbit of period 2 (Fig.7).

Figure 8. Construction of the map T1 (17). Let us show that it is possible to find such parameter values that the latter fixed points become stable. We have: |q1 p2 | = (1 - a)/(a(2 - a)(1 - a - )) s1 (), |q2 p1 | = (1 - a - )/ (a + )(2 - a - )(1 - ) s2 (). Consider the functions s1 (), s2 () in the region 0 < < 1 - a (Fig.9). Thus, we find that for any a (0, 1) there is a range of the values ( , 1 - a) where s2 () < 1. In other words, in the interval ( , 1 - a) the perturbed map (17) has the stabilized periodic orbit arisen from the unstable orbit of the unperturbed map (16) (see Figs.7,8), and almost all phase points from the interval [0, 1] are attracted to it. This orbit becomes stable by means of continuous change in the parameters (a1 , a1 ) of the map (17) to the values (a1 , a2 ) such that s2 (a1 , a2 ) < 1 (see Fig.9). In view of the this obtained result the following question arises: Is it possible to stabilize a prescribed unstable orbit in the chaotic map (16) via the feedbackfree perturbation (2)? Our preliminary considerations show that this problem requires detailed analysis of the map (16) and application of specified perturbations. 3.3. Twodimensional map with a hyp erb olic attractor. External perturbations can crucially affect on dynamics of certain maps if their dimension is more than one. As an example of such a map let us consider so-called Belykh map which appears in the physical systems of phase synchronization [30].

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Figure 9. The functions s1 (), s2 () for a = 1/2. Let Q = {(x, y ) : |x| < 1, |y | < 1} be a square in the plane (x, y ). Then the Belykh map is given by the following construction [31]: T : (x, y ) - f (x, y ), where f (x, y ) = 1 (y + 1) - 1 , (x, y ) Q1 , 2 1 (y - 1) + 1 , (x, y ) Q2 , 3 (x - 1) + 1, 4 (18)

1 (x + 1) - 1,

(19)

and the regions Q1 , Q2 are presented as the separation of the function h(x) : [-1, 1] [-1, 1] into two parts: Q Q
1 2

= =

(x, y ) Q : y < h(x) , (x, y ) Q : y > h(x) . (20)

In addition, the constants 1 , 2 , 3 , 4 and the function h(x) should be chosen in such a way that the square Q is mapped into itself, T Q Q (Fig.10). We consider the case h(x) = ax, 1 = 3 , 1/2 = 1/4 2 , i.e. 1 (x + 1) - 1, 2 (y + 1) - 1 , y < ax, (21) T : (x, y ) - f (x, y ) = 1 (x - 1) + 1, 2 (y - 1) + 1 , y > ax, |a| < 1 (Fig.11). The Belykh map is remarkable for the fact that it has a hyperbolic attractor for certain parameter values. As is known (see, e.g., [29, 32]), an invariant set for a diffeomorphism f : Q Q of a compact manifold Q is said to be hyperbolic attractor if is an attractor and simultaneously hyperbolic set (a rigorous definition can be found in [32]). Let us clarify the sense of the notion of the hyperbolic attractor. If the set is an attractor then there is an open neighborhood which shrinks to with iterations.

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Figure 10. Construction of the Belykh map (18)-(20).

Figure 11. A special case (21) of the Belykh map (18)(20). For maps, the property of hyperbolicity means that in any point p of there are two invariant directions. Along one of them the points of the manifold Q exponentially tend to the initial point p at t , and along the other direction the points tend to p at t -. In turn, the existence of stable and unstable manifolds implies that maps with hyperbolic attractors have sensitive dependence on initial conditions. Moreover, such maps possess invariant measures which determine statistical properties of typical tra jectories. We consider the Belykh map (21) which is however not hyperbolic in a rigorous sense, because it has discontinuities. But this map is typified by hyperbolic dynamical systems with singularities. Such maps appear in many physical problems. With the proviso that a set of discontinuities has a zero measure and the other additional conditions [33], one can obtain the rigorous results concerning discontinuous hyperbolic dynamical systems. In particular, for every regular point it is possible to form the local stable and unstable manifolds. Moreover, based on the concrete type of points of discontinuity one can construct an ergodic invariant measure. It is not difficult to find conditions at which there exists the hyperbolic attractor in the Belykh map. First, let us note that for |a| < 1 the map (21) has two fixed points X = (1, 1) and Y = (-1, -1). Second, for all points of the square where the map (21) is determined, the derivative is Df = diag{1 , 2 }. For the property of hyperbolicity it is necessary that |1 | < 1, |2 | > 1 or vice versa,

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and T Q Q. It is not hard to verify that the latter condition is satisfied only if 0 < 1 < 1, 0 < 2 < 2/(1 + |a|), |a| < 1. Finally, the transformation T should be a one-to-one map. Geometrically, we may come to the conclusion that the map (21) is a homeomorphism only in the parameter range 0 < 1 < 1/2. Thus, for a hyperbolicity we get the following inequalities: 1) 0 < 1 < 1/2; 2) 1 < 2 < 2/(1 + |a|); and 3) |a| < 1.

Figure 12. The generalized Belykh map (22). Let us generalize the Belykh map (21) into the case of |a| > 1. To this end one should note that for |a| > 1 the point X falls in the range y < ax, and the point Y falls in the region y > ax (Fig.12). Therefore, to determine for |a| > 1 the existence conditions of these fixed points it is necessary to rewrite the map (21) in the form: (1 (x + 1) - 1, 2 (y + 1) - 1), y > ax, (22) T : (x, y ) - f (x, y ) = (1 (x - 1) + 1, 2 (y - 1) + 1), y < ax. Thus, we can obtain the map (22) from the map (21) by the substitution x y and a 1/a. Consequently, we get that the hyperbolicity condition for the map (22) is the validity of the following inequalities 0 < 2 < 1/2 and 1 < 1 < 2/(1 + 1/|a|). It should be however noted that in contrast to the map (21), in this case |2 | < 1 and |1 | > 1. Let us consider a construction of the Belykh map for the case of a twoperiodically perturbed parameter a. To find qualitative changes in its dynamics it is necessary to switch the parameter a near a = 1 in such a way that a1 < 1, a2 > 1. In order that for both these cases the perturbed map would be hyperbolic it is necessary to vary also the parameter 1 , 2 . Taking into account these conditions, we may write the perturbed Belykh map as follows: (x, y ) - f (a2 , 2 , 2 ) f (a1 , 1 , 1 )(x, y ) 1 2 1 2 Ї T= (23) (x, y ) - f (a1 , 1 , 1 ) f (a2 , 2 , 2 )(x, y ) 1 2 1 2 for even and odd iterations, respectively. Because both maps (with a1 < 1 and a2 > 1) have the fixed points X = (1, 1) and Y = (-1, -1), then these points remain to be the fixed ones also for the perturbed Ї map (23). Moreover, the differential DT of the perturbed map (for odd and even

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iterations) is: Ї DT = 2 1 0
1 1

2 1 0 0

0 2 2
21 2 2

·
def

1 1 0 1 0

0 1 2 0 2 .

=

=

Therefore, in view of the fact that for a1 < 1 we have 0 < 1 < 1/2, 1 < 1 < 1 2 2/(1 + |a1 |), and 1 < 2 < 1/(1 + 1/|a2 |), 0 < 2 < 1/2 for a2 > 1, the eigenvalues 1 2 Ї and of DT are changed in the following range: 0 < < 1/(1 + 1/|a2 |), 1 2 1 0 < 2 < 1/(1 + |a1 |). In other words, we find that | | < 1, | | < 1 and the fixed 1 2 points X , Y of the map become stable. This means that the hyperbolic attractor is degenerated and replaced with a simple attractor. Thus, introducing periodic perturbations of the map with the hyperbolic attractor we get a qualitative change in the dynamics: From a chaotic map it transforms into a regular one with the stable fixed points. 4. Conclusion. Thus, we have shown that external periodic perturbations can crucially effect on the behavior of the quadratic map, a piecewise linear family and a map with the hyperbolic attractor. Moreover, as shown in Section 3.1, for maps having critical points the chosen in advance periodic orbits can be stabilized. Hence, for dynamical systems the behavior of which is effectively described by polymodal onedimensional maps, the nonfeedback control is possible. Therefore, in general, the obtained results allows us to put a question concerning a rigorous validation of the existence of a feedbackfree parametric excitation needed for the stabilization of the prescribed periodic orbits embedded in a chaotic attractor. In addition, the developed technique allows, in principle, to find analytically a way for the chaos suppression phenomenon for chaotic maps with external perturbations. This fact can essentially simplify investigations of the maps under periodic perturbations. Thus, the described results permit us to find an analytic approach to the problem of the chaos suppression for dynamical systems with continuous time (i.e. for flows). Let us suppose that a system possesses a chaotic attractor. Then, if we appropriately choose external periodic perturbations then one can expect that they lead to appearance of stable periodic orbits. These orbits either have not existed in the initial (unperturbed) system or they have not been stable ones. Some investigations justify this conjecture (see [2] and references therein). However the main problem is to show the presence of an appropriate chaotic attractor in the system. At the same time, some results concerning the induced stable periodic behavior in continuous nonchaotic dynamical systems possessing only steady state and/or unstable cycles have been described [2]. On the other hand, great successes of the chaos suppression phenomena in applications led to the opinion that chaos can always be suppressed by external perturbations. However this is not the case. It has been found that periodic perturbations of certain parameters can not lead to the suppression of chaos. As a consequence, for an arbitrary system we do not know in advance, what parameter is appropriate for the stabilization of the system dynamics [34]. In this connection, it would be useful to have a basic criterion which allows us to determine in what cases the chaotic motion can be stabilized. This important question will arise every time in

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practice. But unfortunately, up to the present there is no even general analytical criteria of the existence of chaos in simple dynamical systems.
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Received November 2005, revised January 2006.
E-mail address : loskutov@chaos.phys.msu.ru