Symbolic kinetic equation for a chaotic attractor

Volume 156, number 7,8

PHYSICS LETTERS A

1 July 1991

Symbolic kinetic equation for a chaotic attractor Alexander B. Rechester Department of Earth Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

and Roscoe B. White Plasma Physics Laboratory, Princeton University, Princeton, NJ 08543, USA Received 19 February 1991; revised manuscript received 23 April 1991; accepted for publication 25 April 1991 Communicated by AR. Bishop

We used symbolic dynamics to partition into “states” the phase space (coarse graining) of a chaotic attractor. The time evolution equation for a distribution function has been discretized and a closed system of coupled linear equations with constant coefficients has been obtained. We calculated the invariant distribution function and correlation function using these equations and foundan excellent agreement with direct time average computations.

The purpose of this paper is to derive a symbolic kinetic equation which describes a relaxation process toward a steady state defined by an invariant distribution function on a chaotic attractor. The symbolic kinetic equation gives an algorithm for the construction of a course grain invariant distribution function and a means of calculating statistical properties of chaotic systems in course grain approximation. The rigorous mathematical justification of our algorithm follows from Sinai—Bowen—RuelIe theory The we method appears to be quiteusing general. In [1,2]. this paper illustrate its application the quadratic map. The quadratic map is a simple example exhibiting chaotic attractors [31, x,~ = ax, (1 —x,) = F(x,, a)

.

(1)

Here t =0, 1, 2, can be thought ofas a discrete time. We call the sequence x 0, x1, x2 x~_1 {x,} T an orbit. We assume that 0< a ~ 4 and study orbits which are bounded in the interval 0< x, < 1. There are two types of orbits depending on the value of the parameter a: asymptotically periodic and chaotic (nonperiodic) orbits. Define a Lyapunov number, ...

0375-9601/9l/$ 03.50 © 1991

—

,~

2 = urn T+oo

~ in dx,~1 T ,=o dx,

-~-

(2)

Then in the case of a periodic attractor we have 2<0, and orbits exponentially converge to stable fixed points given by x,=F~(x1)

(3)

.

Here i= 1, 2, p, p is the period ofthe attractor and 2~ (x)= ~ denotes p iterations, for example, F(F(x)). In the case ofa chaotic attractorF~ 1>0, and nearby orbits exponentially diverge from each other. We will see later that this divergence is the main reason why a probabilistic description appears in the time evolution equation for a distributiort function of the chaotic attractor in spite of the deterministic nature of eq. (1) and it does not appear in the case of a periodic attractor. ...,

We would like to calculate the average properties of chaotic attractors. For example, the two-point correlation function is defined by C(x) = ~ ~ >2 (4) — <.~,

Here the time averaging is defined by

Elsevier Science Publishers B.V. (North-Holland)

419

Volume 156, number 7,8

=

1

lim T-~oo

PHYSICS LETTERS A

will have exactly the same first N symbols {S,}N. In the following we will use A({S}N) to refer not only to the interval (Xa, Xb) but also to its length XbXa>O.

T—

~ x,.

t0

In order to use analytic methods for calculating averages, we introduce a density distribution of orbits f(x, t). The evolution of f(x, t) from time “t” to time “t+ 1” is given by the Perron—Frobenius equation [3] f(x,t+l)=

t

1 July 1991

ö(x—F(y))f(y t)dy.

(5)

This is the analog of the Liouville equation for continuous flow. The invariant distribution is defined by [3] f(x)= limf(x,t). (6)

The meaning will be clear from the context. In this way we can divide the interval (0, 1) into the small subintervals A ( {sfl}N). Obviously there are many ways to introduce symbolic dynamics. The most important criterion for the choice of symbols s, is that cells A({s}N) have to be simply connected, so that there are not two or more different intervals correspond. ing to the same sequence {S}N. It is convenient to introduce instead ofthe sequence {sfl}Njust one integer 1, which characterizes the sequence uniquely. Clearly this can be done for any bounded sequence of bounded elements ~tk~For the quadratic map it is convenient to use N-I

~

1=

Questions concerning the existence of the invariant distribution and its uniqueness are quite difficult and are thoroughly discussed in the mathematical literature on measure theory [1,2]. Using the invariant distribution we can substitute averaging over time by a phase space average. For example

j

(x,÷~x,>= F~(x)xf(x)dx.

(7)

o

In order to discretize eq. (5) we have to discretize the phase space (coarse graining). The usual way of doing this is to divide the interval (0, 1) into a large number of equal subintervals (see for example ref. [2]). But the resulting set of linear equations have a much simpler matrix (see eqs. (24)—(26)) if we use coarse graining based on a symbolic description of orbits, which we introduce now. Define symbols s,,

—0

•fxt

—

=

1

,

2

if x,> ~

(10)

2N_I_kmk,

k=o

m,, =

(~

mod 2

s

(11)

)

\n=o

From I we can reconstruct the sequence {sfl}N. The . index 1 corresponds to the ordering of the x 0 vanable, so that if x01 eA1 and x02EA,2 and l~>12, then x01>x02 [3,4]. The maximum number of different sequences {Sfl}N is equal to 2N but not all of them are permitted by the dynamics. For example, if we cornpute a long chaotic orbit in a symbolic representation for the case of a= 3.61 we will find that except for some initial transients only combinations of 01 and 1101 are possible in the formation of {s,}, for example it can have the form 01011101 .The rules which determine the transitions of point orbits between different intervals A ( {sfl}N) in one time step results in a mapping of the intervals A, into one an...

other. A, is related In theto particular only twocase intervals ofthe at quadratic the previous map time each

‘

(8)

.

Orbits with x, = are connected with the so-called kneading orbit [3,4] and will be treated in detail in a longer paper graining) [5]. Let us now discretize phase space (coarse according to the our following rule [1,2]: define the interval

and two intervals at the next time. This relation can be written as follows, A —A ~

~

51

S~j~/

(9)

/

5N— I, SN /3~l~ ...,

Ah(l—so, S

(12) 5N—I, 15N).

1

(xa,xb)=A(so,sI,...,SNI)

~

5N_I)~~A/4(5I,

~,

(13)

such that all orbits with initial conditions x 0e (Xa, Xb) 420

Notice that according to eqs. (10) and (11) (see also

Volume 156, number 7,8

PHYSICS LETTERS A

1 July 1991

eqs. (28) and (29)) we have 14131=l, thus in-

A,.

tervals 4,, and 4/4 are adjacent to each other in x space. The non-uniqueness of the transition rules (12) and (13) is due to the fact that interval A,, as well as ~/2 is stretched to the length of 4/3 +4/4. Some-

r,1.,, =~‘12-*13= 4 +4,~

times one of the sequences /3 or /4 is prohibited by the rules of symbolic dynamics and then such transitions are unique. But in the case of stable periodic attractors they are always unique. Now we are in the position to discretize eq. (5). We introduce a coarse grained distribution,

~‘/I/4

(21)

/3

And similarly 4/4 =

=f’/2~/4

A

13

(22)

+4/4

We interpret these coefficients as transition probabilities, or branching ratios, because they are correctly normalized according to eqs. (19) and (20). It is at this point that a probabilistic description is introduced. Notice that if we consider just a single

Xb

p,,=

Jf(x, t)

dx=f,,4,

(14)

Xa

and integrate eq. (5) over the interval to get

Xb3

—xa3 =

4/3

P13~4~ _p,11r11.,3+p,211’,2.,3.

Here

$ J

(15) (16)

f(y, I) dy/P121.

(17)

then approximation (21) and (22) is not correct because what we call transition probabilities would be functions of time according to eqs. (16) and (17). Thus eqs. (15)—(20) take a simple form, P,3,,±,=y,3(P,1,+P,2,,) (24) P,41~1=y,4(P,1~,+P,2,,),

Integration in eqs. (16) and (17) is done over the intervals Aj,~and 4,ç,~which are the pre-images of interval 4/3 in the intervals 4,~and ~,2, respectively. A similar equation can be written by integrating eq. (5) over the interval x~4— Xa4 = A P,4,,+i =P,,,1F,~.,4+P,2,,F12.14

(18)

.

The equations which define F,~14and 24 are similar to eqs. (16) and (17). It is easy to check that ,

(19)

112_.13+T’12~14_l

(20)

r/l_.,3+r,I.,41

.

Consider now the limit ~ 4~—~0 then if we assume thatf(x, 0) is a continuous function of x then according to eq. (5) it remains a continuous function of x for all times “t” (but not in the limit t—~oc) andwecanapproximateeqs. (16) and (17) as [5] ~,

One could show that if4,(N-. no) #0 for any I then there exists a stable periodic orbit(s). Obviously in this caseeqs. (21) and (22) and all which follows do not apply.

(25)

(26) In the case when the “state P’ is not permitted then y/3+y,4=l

.

P,,w 0. For example, if

~l

(23)

,

,

F,~,3= f(y,t)dy/P,11, =

orbit, that is f(x,i)=ö(x—F”~(x0))

/4

is not permitted then

~V~40

and y,3 = 1. The coupled system of equations (24)—(26) for the time evolution ofnumber density P~,1of orbits in every permitted state / we call a kinetic equation. In order to write a complete set of equations for all permitted transitions we will use the following relations, (27) (28)

122l/1, /3 = 2/1 , /42/1+1

(29)

.

Here we assume without loss of generality that 0/<2N—1 These relations can be easily derived from eqs. (12), (13) and (10), (11). Notice that the normalization condition ~ p,, = u (30) /

is preserved by eqs. (24)—(26) for all times t if it is valid for I = 0. Consider the limit P1 = lim P1,,

.

(31)

421

Volume 156, number 7,8

PHYSICS LETTERS A

I July 1991

According to eqs. (6) and (14), P, is simply related to the coarse grained invariant distribution f,

otic attractor with ct=3.71 and N=6. The numbers on this diagram correspond to the I states while the arrows indicate the transitions between different states in one time step. For the example of fig. 1 there is only one closed network diagram, which corresponds to one attractor at the interval (0, 1) [3].In this case the solution of eqs. (24)—(26) is uniquely determined by normalization condition (30). But for a more general case the network diagram could consist of several disconnected graphs, which conrespond to several coexisting attractors. In such a case a solution ofthe system (24)—(26) is not unique because different attractors could have different normalization weights depending on the initial distributionf(x, 1=0). Eqs. (24)— (26) define an algorithm for constructing the course grain distribution function f,. This method is readily generalized to one-dimensional

f/ — P/A / l~

(32)

maps require moreHénon than two symbols and also to thewhich two-dimensional map and the Chin-

P, is a steady state solution ofeqs. (24)— (26). A non-

kov—Taylor, standard map [5]. For these more general cases, eqs. (24)—(26) will take the form

53

~ 21 0

42 43

45

n=6

41

22 38 52

50 25

26 Fig. 1. Network diagram of all permitted transitions forthe case a=3.71,N=6.

zero solution exists if the determinant of the cornplete set of equations (24)—(26) is equal to zero. It is easy to prove that this is true using eqs. (24)—(26). Is this solution unique? The answer to this question depends on the structure of the system of equations (24)—(26). This structure could be represented graphically as a network diagram. In fig. 1 we give an example of such a network for the case of a cha-

P1,1÷1= ~ y,,.P,,,

(33)

1

(34)

~

Yie

=

,

and eq. (34) guarantees the existence of a steady state solution to eq. (33), i.e. the existence of invariant

Table 1 Results fora=3.7l, N=6.

2 {s,} 011110 011111 011101 010101 010111 110111 110101 111101 111111 111110 111010 111011 101011 101010 101110 101111

422

I 20 21 22 25 26 37 38 41 42 43 44 45 50 51 52 53

A,x10 2.7 4.0 5.1 6.2 9.2 9.2 6.2 5.1 4.0 2.7 2.9 3.9 3.5 2.2 1.7 2.1

y,

P,(th.)x102

P,(exp.)x102

0.40 0.60 1.00 1.00 1.00 1.00 1.00 1.00 0.60 0.40 0.43 0.57 0.62 0.38 0.45 0.55

3.3 4.9 6.6 4.0 6.3 8.4 6.3 8.2 7.3 4.9 6.3 8.5 6.3 3.9 6.6 8.2

2.8 5.9 5.0 3.5 7.3 6.3 7.3 8.7 9.1 5.9 7.3 6.3 7.3 3.5 5.0 8.7

Volume 156, number 7,8

PHYSICS LETTERS A

number densities P,. We would like to emphasize again that matrix y,,., based on symbolic coarse graining, is much simpler, i.e. contains many more zeros, than one obtained using equal size domains. A practical way of computing steady state solutions P, for a large system ofcoupled equations (24)— (26) is to take initially P,,0=const for all permitted / states and then iterate the kinetic equations. This procedure rapidly converges to a steady state solution. In contrast to this the method using equal size domains converges very slowly (see ref. [2]). Now we can easily compute any averaged quantity. For example the correlation function defined by eq. (4) can be expressed in coarse grained approximation using eq. (7) as /

C’ r) —

x p

v

—

~

o~ /0

—

t~

8

6

F(x) 4

2 0 0

0.2

0.4

0.6

0.8

1.0

\2

Iv ,~

1 July 1991

p

1

(35)

x,

Fig. 2. Invariant distribution: comparison of time average computations (solid line) with the theory (dotted line) for the case a=3.7l, N= 14.

where the first sum is over all allowed paths, /~,/1, 1~,and x, is defined as XtXa+~Ai.

(36)

The only free parameter of this theory is N, which defines the degree of coarse graining. The coefficients A,, y, etc. can be easily computed from the equation of motion (1). We now briefly present the results of application of this method for the specific case of a=3.71. In this case eq. (1) exhibits a chaotic attractor with A = 0.363. In table 1 we give as an example the data for N= 6. We list permitted symbolic sequences (wedo not list here transient permitted states for /<20 and 1> 53 such that ~ these states would be appendices on the network diagram of fig. 1), the corresponding / numbers,4,, and y,. P,(th.) has been calculated from the kinetic equations, while P,(exp.) results from taking time averages. In figs. 2 and 3 we compare the invariant distribution and correlation functions calculated by the method of the kinetic equations with N= 14, and direct numerical computations of these quantities using time averages. It is clear that as N increases our theory becomes quite accurate. We should emphasize here that throughout the paper we assumed that the invariant distribution exists and we were mostly concerned with approximating it using our method.

0.02 001

0 c(t) -0.01

-0.02 -0.03 0

5

10

15

20

Fig. 3. Correlation function: comparison of time average cornputations (solid line) with the theory (dotted line) for the case ofa=3.7l, N= 14.

A.B. Rechester is very thankful to Professor Edward N. Lorenz for many useful discussions and encouragement. He also would like to thank Robert L. Devaney, David Kazhdan, and M. Yakobson for useful discussions, Stuart Johnson for providing us with reference of his review paper, and Dr. Leslie Bromberg for help with computing. R.B. White ac423

Volume 156, number 7,8

PHYSICS LETTERS A

1 July 1991

knowledges useful discussions with C.F.F. Karney. This work is supported by the U.S. Department of the Navy under contract No. N000 1 4-88-K-0282 and the U.S. Department of Energy under contract No. DE-ACO2-76CH03073.

D. Ruelle, Thermodynamic formalism (Addison-Wesley, Reading, 1978); M. Yakobson, Commun. Math. Phys. 81(1981) 39. S. Johnson and J. Guckenheimer, Ann. Math. 132 (1990) 72. [21T.-Y. Li, J. Approx. Theory 17 (1976) 177. [31P. Coullet and J.P. Eckmann, Iterated maps on the interval as dynamical systems (Birkhäuser, Basel, 1980); R.L. Devaney, An introduction to chaotic dynamical systems

References

(Benjamin/Cummings, Menlo Park, 1986). [4] P. Cvitanovic, Phys. Rev. Lett. 61(1988)2729. [5] A.B. Rechester and RB. White, Princeton University preprints.

[11 Ya.G. Sinai, Russ. Math. Surv. 27 (1972) 21; R. Bowen, Springer lecture notes in mathematics, Vol. 470. Equilibrium states and the ergodic theory of Anosov diffeomorphisms (Springer, Berlin, 1975);

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