Mechanismsof nonfeedback

controlling chaos and suppression of chaotic

motion

Mechanismsof nonfeedback

controlling chaos and suppression of chaotic

motionAlexander Loskutov

Moscow State UniversityLoskutov@moldyn.phys.msu.ru

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

1. Introduction1. 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.

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2. External perturbations2. External perturbationsof of n-n-dimensional mapsdimensional 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.

: ,G A A : ( ), .G a g a a A AExternal parametric perturbation:

: ( , ),aT ax f x1 1={ ,..., } , ={ ,..., }, .n nx x M f f a A x f

Dynamical system: where

In this case the perturbed map has the form:

: ( ), ,

( , ), ( ) ( , ), ( ) .

M

a a g a

T y h y y

y x h y f x

A

1 1( ), 1, 2,..., 1, ( ),

( ), , 1 .

i i

i j i

a g a i a g a

a a i j a i

A

3

Perturbations

We consider only periodic (cyclic) perturbations:

multiplicative force

1ˆ ˆ( ,..., ), .a a a a R

1

times

1

ˆ ˆ ˆ: ( ,..., ),

ˆ, 1 , , , ,..., , .i j

a a a a

a a i j i j a a

R

A A A A

A A

4

Perturbations with period

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

x1 x2 x3 x4 x5 x6

a1

a2

I

a

I

a

x1 x2 x3 x4 x5 x 6

a 1

a 2

I

a

I

a

t k

-cyclic transformation:

1

2

1 1

2 2,

: ( , ) ,

: ( , )

. . . . . . . . ,

: ( , ) .

a

a

a

T a

T a

T a

x f x f

x f x fT

x f x f

1 1 2 1

2 1 1 3 2

1 2 1

(1) ( ) (1) ( )1

( (... ( ( ))...)),

( ( (... ( ( ))...)),

. . . . . . . . . . . ,

( (... ( ( ))...)),

{ ,..., } and ,..., , ,..., .n nn i i i i ix x f f F F

F f f f f x

F f f f f f x

F f f f f x

x f F

6

Thus,7

1 1 1

2 2 1

1

: ( , ,..., ),

: ( , ,..., ),

. . . . . . . . ,

: ( , ,..., )

T a a

T a a

T a a

x F x

x F x

x F x

1 1 0 2 2 1

1 1 2

( ), ( ),...,

( ).

x f x x f x

x f x

with initial conditions:

Lemma [Loskutov & Rybalko, 1993]. If the map Tk has a periodic orbit of period t and the function fk(x) is a C0-function, then the map Tp, p=k+1 (mod ), also has a periodic orbit of the same period t. Moreover, if i) periodic orbit of the map Tk is stable then periodic orbit of the map Tp is stable as well; ii) if fk is a homeomorphism then the maps Tk and Tp are topologically equivalent.

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Analysis of the non-autonomous perturbed maps are reduced to the consideration of only one of autonomous maps Tp.

3. Stabilization of the required 3. Stabilization of the required periodic orbits and periodic orbits and

suppression of chaossuppression of chaos

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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.

c A A

ca A

3.1 Polymodal 13.1 Polymodal 1DD maps maps10

1 1( ), 1, 2,..., 1, ( ),

( ).

i i

i j

a g a i a g a

a a i j

: ,G A A : ( ), :G a g a a A A

: ( , ),aT x f x awhere .x I

Consider a one-dimensional transformation:

Introduce -periodic perturbation:

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( , ),

( ).

x f x a

a g a

T

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 x2,x3,...,x we can find such values x1 and a1,a2,...,a that the periodic orbit (x1,x2,...,x ) is stable for the perturbed map T at = (a1,a2,...,a ).

I 1 2,x x *

1 2( , )f x a xa A

cx ( , ) / | 0cx xf x a x

a

*a A

Example.

: ( , ) (1 ).aT x f a x ax x

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Theorem 2 [Loskutov 1992, 1994]. There exists a subset such that if then the perturbed quadratic familyof maps possesses stable periodic orbits.

ˆˆ da a Aˆ ˆd cA A

Logistic map

Introduce .Periodic parametric perturbations operating onlyin the chaotic subset can stabilize its dynamics:

( , 4]c aA

cA

For the quadratic family , where are the solution of is the intersection point of , i.e.

[ , ]b ex x

int int( , 4),x x x4 ( d) a1 ny x x y x

[ , ] [1/ 4,3 / 4].b ex x

,b ex x

2 1 1 1

3 2 2 2

1

(1 ),

(1 ),

. . . . . . . . . . . . . . . . . ,

(1 ).t t t

x a x x

x a x x

x a x x

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Suppose that this map possesses an orbit with period . Thus,1 2( , ,..., )tp x x x t

1

1 2( ) 1

1

ti

i i

xp

x

1 21, 0

2 1c

cc

xx

x

STABILITY:

= 2:

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

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Theorem 3 [Loskutov & Rybalko, 1998].

Suppose that and theperturbed map T at has a stableorbit of the period t, . Then , if

where i=1,2,...,t,

then this map also has stable t-periodic orbit at with

2( , )f x a C M A

1 2ˆ ( , ,..., )ta a a a1 2( , ,..., )tp x x x

1

1

1 ,t

t ii a a x x

i

a tS LS S

2

, ,max D ( , ) , max D ( , ) ,a a x

x a x aS f x a L f x a

,max D ( , ) ,x x

x aS f x a

1 2 2( , ,..., )c t tp x x x x x x 1 1 2 2ˆ ( , ,..., )t ta a a a a a a

11 .ti x xx LS

15Estimation of the admissible errors:

Quadratic family for :[0,4]a

, ,

1max ( , ) , max ( , ) 2,

4a xx a x a

S x a S x aa x

2

2,max ( , ) 8.

x aL x a

x

16

2

1

1 1, .

22 2

a x ttt i

i

t

Thus, the admissible errors:

3.2 Piecewise linear family3.2 Piecewise linear family

where

( ) ( ), 0 ,: ( , )

( )( 1), 1,a

q a x r x x aT x f x a

p a x a x

(0,1),

( ) (1 ) (2 ) ,

a

q a a a a

( ) 1 (2 ),

( ) 1 (1 ).

r a a

p a a

Ta

0 1 x

qx+r

p(x - 1)

a

17

2 (1/ 2), (1/ 2) 1/ 2a

Note:the quadratic map on the interval

is conjugated with this

family at .

The set is said to be a mixing attractor, if1) for any open set U in and any finite covering there exist and :

for all j.

2) : and

{ }i ,m m U 1r 1

0

rm i

ji

T T U

V ,V TV V

0

.i

i

T V

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Consider the perturbed family with period 2:

Introduce and suppose that .

2 1

1 2

1 1

2 2

: ( ) ,

: ( ) .

a a

a a

T x F x T T

T x F x T T

1 20 1a a 1 2, , 0a a a a

Theorem 4 [Loskutov & Rybalko, 1994]. For any the map has the mixing attractor .

(0,1)a aT

[0,1]

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Thus,

q2

p1

fa1

fa2

x

x

x

F1

q1p2

p p1 2q p1 2

q q1 2

p q1 2

Ta2

0 xJ1 J2

pq

p2

pq

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Now we can findparameters suchthat these orbitsbecomes stable:

1 2 1

2 1 2

(1- ) (2 - )(1- - ε) (ε),

(1- - ε) ( ε)(2 - - ε)(1- ε) (ε)

q p a a a a s

q p a a a s

1 1 1 2( , ) ( , )a a a a

2 1 2( , ) 1s a a 2-a

1-a-a

s ( )1

s ( )2

*

1

0

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3.3 23.3 2DD map with map withhyperbolic attractorhyperbolic attractor

Let be a square in : ( , ) : 1, 1Q x y x y ( , )x y

: ( , ) ( , ),T x y f x y

1 12

3 24

1( 1) 1, ( 1) 1 , ( , ) ,

( , )1

( 1) 1, ( 1) 1 , ( , ) ,

x y x y Q

f x y

x y x y Q

( ) :[ 1,1] [ 1,1]h x

1

2

( , ) : ( ) ,

( , ) : ( ) .

Q x y Q y h x

Q x y Q y h x

22

(-1,-1)

(1,1)

TQ1

TQ2

y

x

(1,1)

(-1,-1)

Q1

Q2

h(x)

y

x

We consider :

1 3 2 4 2( ) , ,1 1h x ax

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Thus,

where .

1 2

1 2

: ( , ) ( , )

( 1) 1, ( 1) 1 , ,

( 1) 1, ( 1) 1 , ,

T x y f x y

x y y ax

x y y ax

1a (1,1)

(-1,-1)

Q1

Q2

y

xy=ax

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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.

:f Q Q Q

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.

Q

Df sE uEsE

Df sEDf

Fully developed chaos

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For the hyperbolicity of the Belylh map:

2 1

1 20 ,1

2 1 1 a

2 11, 1

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.

1 21, 1a a

1 2,

2 2 1 12 1 2 1 1 2

1 1 2 21 1 2 2 1 2

( , ) , , , , ( , )

( , ) , , , , ( , )

x y f a f a x yT

x y f a f a x y

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The fixed points: (1,1) ( 1, 1)X Y

For : 1 1a 1 11 2 10 1 2, 1 2 1 a

For : 2 1a 2 22 1 20 1 2, 1 1 1 1 a

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2 11 1

2 12 2

2 1 *1 1 1

2 1 *2 2 2

0 0

0 0

0 0

0 0

DT

The differential:

Eigenvalues of are changed in: DT

* *1 2

2 1

1 10 , 0 .

1 1 1a a

* *1 21, 1

The fixed points X, Y become stable.

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The hyperbolic attractor is degenerated and

replaced with a simple attractor.

4. Dynamical systems with 4. Dynamical systems with continuous timecontinuous time

This fact gives us a method for the chaos suppression by an additive excitation of the system with a homoclinic structure.

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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.

4.1 The Melnikov method4.1 The Melnikov method

Consider a two-dimensional autonomous system having a unique saddle point:

0 1( ) ( )x f x f x

Chaotic motion appears only in the last case d.

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x0u

x0s

x0

а)

d)c)

b)

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.

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:

sx ux 0t0( )D t

0 0 1( , )D t t f f dt

If changes its sign then separatrices intersect each other and dynamics in this area is chaotic.

0( )D t

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4.2 Function of stabilisation4.2 Function of stabilisation

Consider the following equations:

( , ),

( , ) [ ( , ) ( , )],

x P x y

y Q x y f t F x y

where is a periodic perturbation andare some smooth functions.

( , )f t ( , ), ( , ), ( , )P x y Q x y F x y

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.0( )D t

0 0 0 0 0( , ) ( )[ ( , ) ( , )] [ ( , )],D t t y t t f t F x y dt I g

where 0 0( ) ( ).y t x t

For this equation the Melnikov distance is:

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To suppress chaos we should get a function of stabilisationa function of stabilisation: . Thus, consider the inverse problem.

*( , )f t

For systems for which it is possible to make an additional shift fromthe critical value of the Melnikov function , the externalstabilising perturbation has the explicit form (Loskutov and Janoev, 2002):

0( , )D t t

*

0 0

4 ( )( , ) cos( ),

( )

a tf t t

y t t

where is a Dirac delta-function.( )t

*( , ) Re{ ( )}i tf t e A t

*

0 0

( , ) Re ( [ ( , )] ) .( )

i ti te

f t I g t const e dy t t

Suppose that . In this case, omittingintermediate expressions, we finally get (Loskutov and Janoev, 2002):

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Example: Duffing oscillatorExample: Duffing oscillator

The Duffing equation with damping and parametric perturbation of the cubic term can be written as follows:

3[1 cos( )] [ cos( ) ],x x t x t x

where and are the amplitude and the frequency of the perturbation, respectively, and .

1

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):0( , )D t t

0( , ) 0D t t

min max4 2 2 4 2

6 26 sinh( / 2) 1 sinh( / 2),

( 4 ) ( 4 ) cos( / 2)

d

p

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In our analysis, we use the left-hand side of this inequality. Considerationof 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):

*

0 0

4 ( )( , ) cos( ),

( )

a tf t t

t t

0 2

2 sinh( )( )

cosh ( )

tt

t

where is a solution on the separatrix loop.

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-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 -0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

. X

X

100 150 200 250 300 350 400 450 500 -2

0

2

X

t

Numerical analysisNumerical 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.

0.145 8 0.01 0.14 1.1

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Introduce the stabilising perturbation . *( , )f t

( )x tThe corresponding phase portrait and the solution of the Duffing equation are shown here:

100 150 200 250 300 -2

-1

0

1

2

X

t

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4. Conclusion4. 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 beeffectively described by unimodal one-dimensional maps, the non-feedback control is possible. Therefore, in general, the obtained results allows us to risequestions 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.

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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.

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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.

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