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Mechanisms of non feedback controlling chaos and suppression of chaotic motion Alexander Loskutov Moscow State University Loskutov@moldyn.phys.msu.ru

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Historical remarks Suppression and non-feedback control of chaotic motion: V.Alekseev & A.Loskutov, 1985, 1987 Synchronization and force resonance actions: Yu.Kuznetsov, P.Landa et al., 1983, 1985; E.Lűsher, A. Hűbler et al., 1987, 1988, 1989 Analytical methods: R. Lima & M. Pettini, 1989, 1990; R. Chacŏn, 1995; A.Loskutov, 1993, 1994 Controlling chaos: E.Ott, C.Grebogi, & J.A.Yorke, 1990 Mathematical foundation: A.Loskutov et al.,1993, 1994, 1995 Reviews: E.Ott & M.L.Spano, 1995; S.Boccaletti, C. Grebogi, et al., 2000; Loskutov, 2001

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1. Introduction We describe a rigorous approach to the investigation of qualitative changes in the behaviour of chaotic dynamical systems under external periodic perturbations and propose an analytical key to find such perturbations. 1

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2. External perturbations of n-dimensional maps 2 In general, controlling dynamical systems involves a certain additive or/and multiplicative variable(s) which take(s) into account additive or/and multiplicative perturbations, respectively. External parametric perturbation: Dynamical system: where

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In this case the perturbed map has the form: 3 Perturbations We consider only periodic (cyclic) perturbations: multiplicative force

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4 Perturbations with period

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Proposition [Loskutov, 1991]. Period t of any obtained periodic orbit in the perturbed map is multiple to the period of perturbation,, where is period of the perturbation and k is a positive integer. 5

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-cyclic transformation: 6

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Thus, 7 with initial conditions:

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Lemma [Loskutov & Rybalko, 1993]. If the map T k has a periodic orbit of period t and the function f k (x) is a C 0 -function, then the map T p, p=k+1 (mod ), also has a periodic orbit of the same period t. Moreover, if i) periodic orbit of the map T k is stable then periodic orbit of the map T p is stable as well; ii) if f k is a homeomorphism then the maps T k and T p are topologically equivalent. 8 Analysis of the non-autonomous perturbed maps are reduced to the consideration of only one of autonomous maps T p.

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3. Stabilization of the required periodic orbits and suppression of chaos 3. Stabilization of the required periodic orbits and suppression of chaos 9 Introduce a subset of the parameter values a corresponding to chaotic dynamics of the map. Thus, we suppose that if then this map has the chaotic behaviour.

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3.1 Polymodal 1D maps 10 where Consider a one-dimensional transformation: Introduce -periodic perturbation:

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11 Perturbed map: Theorem 1 [Loskutov & Rybalko, 1998]. Suppose that map satisfies the following conditions: (i) there is a subset such that for any we have for which ; (ii) for an arbitrary there is a critical point, i.e.. Then for any x 2,x 3,...,x we can find such values x 1 and a 1,a 2,...,a that the periodic orbit (x 1,x 2,...,x ) is stable for the perturbed map T at = (a 1,a 2,...,a ).

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Example. 12 Theorem 2 [Loskutov 1992, 1994]. There exists a subset such that if then the perturbed quadratic family of maps possesses stable periodic orbits. Logistic map Introduce. Periodic parametric perturbations operating only in the chaotic subset can stabilize its dynamics:

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For the quadratic family, where are the solution of is the intersection point of, i.e. 13 Suppose that this map possesses an orbit with period. Thus, STABILITY:

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= 2: The existence regions of period two stable orbits for the 2-perturbed quadratic map family in the coordinate and parametric spaces. 14

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Theorem 3 [Loskutov & Rybalko, 1998]. Suppose that and the perturbed map T at has a stable orbit of the period t,. Then, if where i=1,2,...,t, then this map also has stable t-periodic orbit at with 15 Estimation of the admissible errors:

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Quadratic family for : 16 Thus, the admissible errors:

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3.2 Piecewise linear family where 17

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Note: the quadratic map on the interval is conjugated with this family at. The set is said to be a mixing attractor, if 1) for any open set U in and any finite covering there exist and : for all j. 2) : and 18

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Consider the perturbed family with period 2: Introduce and suppose that. Theorem 4 [Loskutov & Rybalko, 1994]. For any the map has the mixing attractor. 19

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Thus, 20

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Now we can find parameters such that these orbits becomes stable: 21

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3.3 2D map with hyperbolic attractor Let be a square in : 22

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We consider : 23

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Thus, where. 24

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This map (Belykh map) has a hyperbilic attractor. An invariant set for a diffeomorphism of a compact manifold is said to be a hyperbolic attractor if is an attractor and simultaneously a hyperbolic set. A f-invariant subset is said to be hyperbolic if there is a splitting of the tangent bundle of restricted to into a (Whitney) sum of -invariant subbundles, and such that the restriction of is a contraction and is an expansion. Fully developed chaos 25

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For the hyperbolicity of the Belylh map: Let us construct the map with 2-cyclic perturbation of parameter a:. For the hyperbolicity we should also change. for even and odd iterations, respectively. 26

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The fixed points: For : 27 The differential:

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Eigenvalues of are changed in: The fixed points X, Y become stable. 28 The hyperbolic attractor is degenerated and replaced with a simple attractor.

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4. Dynamical systems with continuous time This fact gives us a method for the chaos suppression by an additive excitation of the system with a homoclinic structure. 29 The basis: the Melnikov method that gives a sufficient criterion of chaoticity in the separatrix neighbourhood. Main result: an explicit form of external perturbations leading to the stabilisation of chaotic dynamics can be analytically found.

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4.1 The Melnikov method Consider a two-dimensional autonomous system having a unique saddle point: Chaotic motion appears only in the last case d. 30 Suppose that the unperturbed system has a separatrix loop (Fig.a), so that in the presence of perturbations the separatrix loop is destroyed. Thus, here we have three different cases: Figs. b, c and d.

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The Melnikov method consists in the evaluating by the perturbation theory the distance between stable and unstable manifolds at time measured along the homoclinic trajectory: If changes its sign then separatrices intersect each other and dynamics in this area is chaotic. 31

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4.2 Function of stabilisation Consider the following equations: where is a periodic perturbation and are some smooth functions. We analyse the chaos suppression phenomenon in systems with separatrix loops. Such an approach allows us to find an analytical form of the perturbations at which the Melnikov distance does not change sign. where For this equation the Melnikov distance is: 32

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a function of stabilisation To suppress chaos we should get a function of stabilisation:. Thus, consider the inverse problem. For systems for which it is possible to make an additional shift from the critical value of the Melnikov function, the external stabilising perturbation has the explicit form (Loskutov and Janoev, 2002): where is a Dirac delta-function. Suppose that. In this case, omitting intermediate expressions, we finally get (Loskutov and Janoev, 2002): 33

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Example: Duffing oscillator The Duffing equation with damping and parametric perturbation of the cubic term can be written as follows: where and are the amplitude and the frequency of the perturbation, respectively, and. Chaos in a neighbourhood of the separatrix does not take place ( i.e. does not change sign, in our case ) if ( where p is an integer ) : 34

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In our analysis, we use the left-hand side of this inequality. Consideration of the second case can be carried out by the same manner. Expression for the external stabilising perturbation has the following form (Loskutov and Janoev, 2002): where is a solution on the separatrix loop. 35

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Numerical analysis Parameters:,,,,. It is known that these values corresponds to the chaotic motion in the Duffing system. The phase portrait and the relevant time evolution are shown in the following figures. 36

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Introduce the stabilising perturbation. The corresponding phase portrait and the solution of the Duffing equation are shown here: 37

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4. Conclusion Thus, we have shown that external periodic perturbations can crucially effect on the behaviour of the quadratic map family, a piecewise linear map family and the map with the hyperbolic attractor. Moreover, for maps having critical points the chosen in advance periodic orbits can be extracted and stabilized. Thus, for dynamical systems the behaviour of which can be effectively described by unimodal one-dimensional maps, the non-feedback control is possible. Therefore, in general, the obtained results allows us to rise questions concerning a rigorous substantiation of the existence of a feedback-free cyclic parametric excitation needed for the stabilization of the prescribed unstable periodic orbits embedded in a chaotic attractor, i.e. to establish the realistic (in the sense of Ott-Grebogi-Yorke) non-feedback controlling chaos. 38

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Analysis of the separatrix splitting is a quite convenience method for investigations of dynamical systems. This is due to the fact that it allows us to get conditions of the integrability and non-integrability in applied problems. On the basis of the Melnikov method we analytically considered the effect of perturbations on a two- dimensional non-autonomous system. In general, we have got an explicit analytical form of the external stabilized perturbations which allows us to suppress chaos. By this reason the obtained results can be applied to the systems and models of various nature for which the separatrix splitting phenomenon is inherent. 39

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In addition, the described results allow us to approach analytically the problem of non-feedback controlling chaotic behaviour for dynamical systems with continuous time (i.e. for the flows). Let us suppose that a system under consideration 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 (Loskutov, 2001). However the main problem here is to show the presence of an appropriate chaotic attractor in the system. 40

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