Controlling chaos in map models

Mechanics Research Communications, Vol. 26, No. 1, pp. 13-20, 1999 Copyright (~ 1999 Elsevier Science Ltd Printed in the USA. All rights reserved 0093-6413/99/S-see front matter

Pergamon

PII S0093-6413(98)00094-9

CONTROLLING CHAOS IN MAP MODELS

N. D. Caranicolas Department of Physics, Section of Astrophysics, Astronomy and Mechanics, University of Thessaloniki 540 06 Thessaloniki, Greece E-mail: [email protected]

(Received 12 June 1998, accepted for print 17 October 1998) Introduction In the present article we examine the regular and chaotic motion in map models with one or two parameters. We start with the monoparametric map model. First we fmd the properties of this map. In particular we investigate the evolution of the chaotic regions as the perturbation parameter K increases. With the help of the Hamitonian, corresponding to the map, we fred the value of K where the last KAM curve between the period 1 and period 2 islands is destroyed and global stochasticity occurs. In other words we find the value of K where all the chaotic regions join together in order to produce a unified chaotic sea.. Furthermore, we find numerically the Lyapunov characteristc exponent (LCE) in the chaotic region for different values of K and obtain an analytical expression for the (LCE) in the case when the entire phase plane is chaotic. In the next step we present a mechanism for the control of chaos in map models. We add a new parameter ct to the above map. Changing this parameter properly we are able to create regular regions in a completely chaotic phase plane. We are also able to increase, decrease or transfer these regular regions in the J-0 phase plane. The mechanism is simple and work in two levels: (i) creates stable periodic points in a completely chaotic phase plane and (ii) increases or decreases the stability indices of the elliptic period-1 fixed points. The behaviour of the system can be explained using the properties of the map and the stability conditions.

The monoparametric map

Let us take the map

J.+l = J . + K s i n 2 0 . (1) 0.+~ = 0. +J.+~,

13

14

N.D. CARANICOLAS

where both J and 0 are considered modulo 2n. The period -1 fixed points of ( 1 ) are at

(i) J = 2er m,

0=0,:r

(2; {ii) J - 2 7 c m ,

0:a/2,37c/2

.

while the period -2 fixed points o f ( l ) are at

0 -0,2-

(i) J = ( 2 m + l ) e r ,

(3~ (it) .l -

2m+

l)Tz-,

O= x / 2 ,

3x:2

,

where m Js an integer, l'he distance ~,J between the tixed points (2) is 2n w h i b the distance 6J, between fixed points (2) and (3) is n. Fig. 1 shows the J-0 phase plane, when K - 0 . 4 . As one can see there is a chaotic layer along the two separatrices. Also note that the structure of the phase space near the period-1 and period-2 fixed points is the same. Let us now come to the stability, of the period 1 fixed points ( 2 ). It is well known ( see [1] ) that the periodic points are stable i f 6 = ]TrA]<2, where 'IrA is the trace A~l +A22 of the matrix

I

",, .... . . . . .

, .j

",...---J

'~,.

.;.-

8

In

trig. 1. l h e J-0 phase plane when K-0.4.

CHAOS IN MAP MODELS

15

F~J.+I OJ.+l IAI,

A12]=/ C J ~

0(p.

L 0J.

,

(4)

c7t,o.

evaluated at the fixed points. Writing if=0., we fmd A,,=I

,

AI_, = 2 K c o s 2 0

(5) A.~j =1

,

A,~ = l + 2 K c o s 2 0 .

Using (2) and (5) we fmd that the stability index 61,62, for the period-1 fixed points (i) and (ii) is

c~, = I 2 + 2 K I

,

52

=

I2-2KI

(6)

respectively. Thus for K>0 the periodic points (i) are always unstable while periodic points (ii) are stable when K<2. The value of K where the transition to global stochasticity occurs can be found semi-analytically using the unperturbed Hamiltonian (see [1] for details ) corresponding to the map (1) which is H = l J2 + K c o s 2 0 2 2

(7)

The maximum excursion of J, near the elliptic fixed points, is equal to the half of the separatrix width, that is AJmax = ( 2 K ) 1/-~

(8)

Observing that the last KAM surface is at a distance of about rd3 from the elliptic fixed points (2) and (3), ( see Fig. 1) we set AJr~=rd3 yielding

7/-2

K = -18

.

(9)

The value K=0.55 given by (9) is very good agreement with the value K=0.57 found by numerical experiments. Fig. 2 shows the J-0 phase plane when K=0.57. Here one can see that the two chaotic

16

N.D. CARANICOLAS

layers have.joined together to produced a large chaotic sea. t)n the other hand when K>2.9 the entire phase plane is chaotic, l:ig. 3 shows the J-0 phase plane \qaen K=2.9.

..~ . .~. "

•:

,

,

,~

0

~.5

:,..~...:.:<~

(

c;/A /

.-. ~;,." .-? {J~,Z;'

,.'2 ~

.....

~:

.

~!_-~a_

e-.

,

\X3

'

/

~ /

.:

'~/'~4""

¢:'::~ '~'.'

,

.~"71 , " ; r ' ~ "

~.~.:'~s-,. -.\

"

~r

c"

'

'

it

',

."2."

(

"-'%1

8

2n

Fig. 2. l h e J-0 phase plane when K=0.57

Zn[-

01

~n

0

tqg. 3. rheJ-O phase plane vvrhen K=2.9

C H A O S IN M A P M O D E L S

17

0.g ,i

o.8

'?'~

:;

'!.~ i, ~' '

L

"

,,/

0.7

~,¢,,,,,~,r-----

7

0.6

~!

0

/I

~ , ,

h

~

'T'

k

0.5 I ' , N~/ I : . ,~' ' ; '

~ , ~ , ~ ,~ ri i /~

o.a°"4 I""::.,'i

r~. Pc" w -

X~

:

L-------'--3 2 1

0 10

100

1000

10000

n

100000

Fig. 4. LCE for several values of K. Plots 1,2,3,4 and 5 correspond to K=0.2, 0.4, 0.6, 1.0 and 1.6 respectively.

Fig. 4 shows the Lyapunov characteristic exponents o ( see [1,2] ) for orbits starting near the hyperbolic fixed points (2). The number of iterations in all cases is 50.000. Note that as K increases o increases as well. It is interesting that the LCE for large values of K, where the entire J-0 phase plane is chaotic, can be found analytically. Working as in [1] p.279 we obtain cr = l n K

(10)

Table 1 gives analytical and numerical values of o for different values of K. We can see that the agreement between numerical and analytical results is good.

Table 1

K

o (analytical)

o (numerical)

3.00

1.0986

1.1441

3.50

1.2528

1.2806

4.00

1.3863

1.4045

4.50

1.5041

1.5202

5.00

1.6094

1.6164

8

N.D. CARANICOLAS

Chaos control

An interesting question, which one may ask, is: Is there a way to decrease the area covered by chaotic orbits in J-0 plane while leaving K unchanged? A simple way to do this is by introducing a new suitable parameter to the map (1). On this basis let us consider the two parametric map

J,,+~ = J,, + K s i n 2 0 , , (11) O,+t = O, +J,+i - c~ sin[J,+ I ]

where ct>0 is the new parameter. The advantage of map ( 11 ) is that it has the same period -1 and period-2 fixed points given by equations (2) and (3). l'he presence of the new parameter ct can bring back regularity in a completely chaotic J-0 phase plane as that shown in Fig. 3.The picture is s h o ~ in Fig. 5 where K-3. a-0.5. Note the large regular regions near the new born elliptic period -1 fixed points.

~ i~' ~i"~

"

~:~,'

~

'~

'

,,J~

~

,

Fig. 5. The J-0 phase plane when K=3, a=0.5

C H A O S IN M A P M O D E L S

19

An explanation, for the above behavior of the system, can be given by studying the stability conditions of the period -1 fixed points (2). The new indices of stability are

8~,~2+2K(1-a)l

, 8~2:12-2K(1-a)l

(12)

A plot of 8al, 8 ~ , as a function of~t, for K=3, is shown in Fig. 6. As one can see the period-1 fixed poins (ii) are stable when 1/3
61

6i 4 2

I

0.5

1

1.5

(]

2

Fig. 6. Stability indices, as a function of ct, when K=3

Discussion

In this article we presented results on regular and chaotic motion in map models. Map models have been proved useful in the study of

Hamiltonian systems describing galactic [3, 4] or asteroid motion

[5,6,7,8]. We started with the map (1), which is a map similar to the standard map containing a basic parameter, that is K. For this map we have found, semi-analytically, the value of K where the transition

20

N.D. CARANICOLAS

to global stochasticib' occurs. By the tema global stochasticity, which is extensively used in [1], we mean that the chaotic layers are joined together to produce a unified chaotic sea.

An

analytic

expression for the LCE c~, when the entire J-O phase plane chaotic, is also given. The above results are in good agreement with the results given numerically. Adding a new parameter ct to the system we obtain the map (11). Now, by changing ~t, one can create regular regions in the chaotic phase plane and can also decrease or increase these regular regions. In other words, introducing the auxiliary control parameter a, is like introducing a catalyst or a ~'anti-chaotic'" medicine in the system. This seems to be a way of controlling chaos. We shall close this article with one more comment. One may ask : "What is the use of a new parameter as far as one can increase or decrease chaos using only the basic parameter K? This is true, but we must also take into account that chaos occurs, not only in Physics but also, in many other human activities such as industrial, medical, social etc. In many of those cases one needs to take into consideration more than one parameters. Furthermore, there are cases where it is not easy or possible to change one specific parameter and one have to introduce an auxiliary parameter in order to control chaos.

References [1 ] A.J. Lichtenberg, M. A. Lieberman, Regular and Stochastic Motion, Springer, Berlin (1983) [2] N. Saito, A. Ichimura , Stochastic Behaviour in Classical and Quantum Hamiltonian Systems, Eds. G. Casati and J. Ford, Springer, Berlin, Heidelberg, New York p. 137. [3] N. D. Caranicolas, Cel. Mech. 47, 87 (1990) [14] N. D. Caranicolas, A&A 291,754 (1994) [5] J. D. Hadjidemetriou, Mapping Models Jbr Hamiltonian Systems with application to Resonant Asteroid Motion, b7 PredictabiliO,, StabiliO, and Chaos it7 N-Body Dynamical 5"),stems, A.E. Roy (ed.) p. 157, Kluwer Publ. ( 1991 ) [6] J. D. Hadjidemetriou Cel. Mech. 56, 563 (1993) [7] J. D. Hadjidemetriou, A. Lemaitre, The dynamical behaviour of our planetary System, p. 277. ( 1997) Kluwer Academic Publishers [8] M. Sidlichovsky A&A 259,341 (1992)