structure of strange attractor and homoclinic bifurcation of two-dimensional cubic map

Volume 104A, number 4 PHYSICS LETTERS 27 August 1984

STRUCTURE OF STRANGE ATTRACTOR

AND HOMOCLINIC BIFURCATION OF TWO-DIMENSIONAL CUBIC MAP

Y. YAMAGUCHI a,1 and N. MISHIMA b a Service de Chimie Physique II, Universit~ Libre de Bruxelles, Boulevard du Triomphe, Brussels, Belgium b Department of Physics, Tokyo Gakugei University, Nukuikita-machL Koganei, Tokyo, Japan

Received 5 June 1984

The structure of a strange attractor of the two-dimensional cubic map with jacobian J is investigated in the cases J ~ 0 and J -.~ 1. The strange attractor has a self similar three-belt structure. The threshold of homoelinic bifurcation is calculated f o r J ~ 1.

Recently Holmes [ 1] has constructed a simple two- dimensional cubic map M to study the behavior of the Poincar6 map of the Duffing equation with nega- tive linear stiffness:

M: Xn+l =Yn, Yn+l =aYn _ y 3 _ J x n , (1)

where J denotes the jacobian and a is a bifurcation pa- rameter. This map shares much of the behavior of the Poincar6 return map of Duffing's equation. In the lim- i t J = 0, this map can be reduced to a simple one-dimensional cubic map studied by May [2] and by Testa and Held [3] * 1. The universal metric properties have been investigated by Hu and Mao [4]. In this paper we shall study the structure of the strange attractor shown in fig. 1 (the Lyapunov exponents are )~1 = 0.595 and )~2 = -2 .204 , and the Lyapunov dimension is a = 1.269 at a = 2.77 and J = 0.2) and the homoclinic bifurcation.

First, let us consider the case J :# 0, but small. It is reasonable to say that the strange attractor is localized on the unstable manifold starting from the unstable saddle point (0, 0). To find it, we use the approach of perturbation method introduced by Bridges and Rowlands [5] and Daido [6]. Let us suppose that the invariant curve is expressed b y y = F(x) . It satisfies a

1 Address after September 1984: Center for Studies in Sta- tistical Mechanics, University of Texas at Austin, Austin, TX 78712, USA.

t l They studied the cubic map: Yn+l = (1 - a)y n + aY3n .

0.375-9601/84/$ 03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

functional equation:

F(F(x)) = - J x + aF(x) - F3(x) . (2)

As J is a small quantity, a solution is sought in the form

F(x) = ~ JkFk(X ) . (3) k=0

Inserting eq. (3) into eq. (2), and equating terms of the same order on both sides, we obtain

Fo(Fo(x) ) = aFo(x ) - F~(x) , (4)

FI(X)F'o(Fo(x)) + F 1 (Fo(x))

= - x + aF 1 (x) - 3F 2 (x)F 1 (x), (5)

where the prime denotes d/dF 0. As a solution for the zeroth order, we easily obtain

Fo(x) = ax - x 3 (type I) , (6)

= 0, + x / a - 1 (type II) , (7)

where the type I and II (F 0 = 0) solutions correspond respectively to the unstable and stable manifold starting from the origin.

For type I, eq. (5) can be rewritten as

F I (F0) = - x ( F o ) . (8)

The first-order function Fl(X ) is given by

179


J =0 .2 a = 2 .77

Y

~b~

.... • -...../ \., \ ..` , ¢ ~ g r ' " '~ ~ . ~ 1 ~ • .

t ' / / \ ~ '..

X

~C,

. . . .

• . . . . . -

• . - • . •

s

, ' k . a . . * : . I • • . . . . . . . • .

(a)

• • . . . . • • . •

. . : . . , . . . . • . - . . . - . . . . . . .

. . : . ° ' - : . . . . . . .

• . . . . • . . . . . -

. .

Fig. 1. Strange attractor in the phase space (x, y) at J = 0.2 and a = 2.77. (b) and (c) show the magnifications of the rectangles in (a) and (b), respectively.

cos ( ] 0 )

F I ( X ) = 2 - c o s ( ½ 0 - ~ r r ) (x2~<4a3/27) (9)

- c o s 0 *

and

F l (X ) = _(p1+/3 + pl_/3) (x 2 > 4a3 /27) , (10)

where 0 (x) = arctan [ 4 a 3 -x2)l/Z/x], and P+ = 1 [ - x + (x 2 - 4 a3)l /Z] . Roughly ~peaking the attractor is confined in the region x 2 ~< ~ a 3, and thus we may consider only the case of eq. (9).

180

Using type II, the first-order correction F 1 for the stable manifold is given by

Fl(X)=x/a. (11)

Here we find the characteristic results that the unstable manifold W u is a "three-valued" function in the region x 2 ~ ~ a 3 and that the stable manifold W s remains a "single-valued" function.

To see more fine structure of Wu, let us compute the higher-order corrections. These are given by

F 2 ( F 0) = - F 1 (x)F' 1 (Fo ) , (12)


Fn(F o) = - F n _ I ( X ) F i (Fo) + .... (13)

Here we find that the function F 2 is a "nine-valued" function with respect to F 0 because x is a "three-valued" function o f F 0 and that the function F 1 is also a "three-valued" one. This means that one branch of three-belts expressed by eq. (9) splits into three-belts. Repeating this discussion, we have the result that the single-belt attractor splits into "3~-belt ' ' structure w h e n J =/= 0.

Next for the opposite case that 6 = 1 - J is small, we shall approach the structure of the strange attractor by using the Fokker-Planck equation [7]. This method is useful in the limit that the previous method is not applicable. The map M can be immediately cast into the dissipative standard map M' :

M': xn+ 1 = (1 - 6 ) x n +f(Yn), Yn+l=Yn +Xn+l , (14)

where f(y) = ay _ y 3 , a = a - 2 + 8, and the variables x and y are the canonical ones in the limit 8 = 0. Com- paring the standard mapping [8] and M', we call the variable y the phase.

For the steady state, the phase averaged Fokke r - Planck equation [7] for the initial distribution p(0) becomes

1 - B P (°) + g dDP(°)/dx = 0 , (15)

where the dynamical friction B and the diffusion constant D are determined by

Ym

B=2__~ ml f [ - 6 x + r C ~ ) l dy = - 6 x , (16) - Ym

Ym

m 8 =62 2 + v r6 m r6 m (17)

Here we make the random phase approximation for the variable y where the attractor is assumed to be spread over from - X m ( - Y m ) to X m (Ym) being of order o(1). We assume that 62x 2 "~ a~s Y6 m.

The distribution as an initial guess is easily obtained

t gO) (x) = (48/DrO 1/2 e x p ( - 8x2 /D) , (18)

where the constant has been found from the normal-

ization. For the dissipative system, the final distribution P(=) is given by iteration according to the follow- ing relation:

P(i+l)(xi+ 1 ,Yi+l )=P(i)(xi ,Yi)/det M' , (19a)

xi = [xi+t - f 0 ' i + 1 - X , + l ) l / ( 1 - 6 ) ,

Yi = Yi+ 1 -- xi+ 1 ' (19b)

with initial guess p(0). Here the old variables x i and Yi are given by the inverse m a p M ' -1 where de tM' = 1 - 6.

Using eqs. (18) and (19), the first-order distribution p(1) is derived as

p(1) ( x , y ) = (1 - 8) -1 (48/DTr) 1/2

X e x p { - If/D(1 - 8 ) 2 1 [ x - f ( y - x ) ] 2 } . (20)

The positions of the center of P (1) are determined by

x - f ( y - x ) = 0 . (21)

For giveny (lYI< 2 [½(1 + c0] 3/2), there are three dif- ferent solutions. This fact means that the single peak splits into three peaks with the same intensity. Re- peating this discussion, the actual distribution P(**) has a "3 =-peak' ' structure.

We have showed that the strange attractor of the two-dimensional cubic map has the "3 ~*-belt or peak" structure at 0 < J < 1. The hierarchy of the strange attractor originates from the many-valued properties of the inverse function of the one-dimensional map. Using this result, it is shown that the map T

. 2m+1 _ J x n T: Xn+l =Yn, Yn+l =aYn - Y n

(m = 1,2 .... )

has the "3°%belt" structure, and the map S

S: Xn+l =Yn, Yn+l = 1 - a y 2m - J x n

(m = 1,2 .... )

has the "2°°-belt ' ' structure. The "3 °~-belt'' structure is intrinsically equal to the set of the quinary represen- tation without numerals 1 and 3. On the other hand, the "2~-belt ' ' structure is well known as the middle- thirds Cantor set.

Finally, the homoclinic bifurcation is discussed using Melnikov's theory [91. The dissipative standard map can be written as the equation of motion [8]

181


q=l

dy/dn = x, (23)

where the iteration number n plays the role of time.

Assuming that 6 and the harmonic mode with q = 1 are of order e ~ 1, with all higher harmonics being negligible, eqs. (22) and (23) become

dx/dn ~ f ( y ) + [ - fix + 2f(y) cos 27rn] , (24)

dy/dn =x, (25) where the terms in brackets are treated as perturba-

tions. The lowest-order equations in eqs. (24) and (25)

have homoclinic orbits at a > 2 - 6. The value a s = 2 - /5 corresponds with the threshold of the saddle-node

bifurcation (see the dash-dotted curve in fig. 2). The

fixed point (0, 0) is stable at a < a s, and other fixed

points (0, + x/d) appear at a > a s. The time dependence of homoclinic orbits based

on the initial conditions x 0 = 0 and Y0 = -+ x/~'ff at n = n o are expressed by

YO(n - n o ) = + , v ~ sech [x/'~ (n - n0)l , (26)

xo(n - n o ) = -v- ,v/'2c~ sech [x/~(n - no)]

X tanh [X/~ (n - no)] . (27)

Using Melnikov's theory, we can calculate the distance/x (no) between the stable and unstable mani-

folds starting from the origin:

o~

A(no) = f dn xo(n- n o )

× [ - 5x0(n - no) + 2f(Y0(n - no) ) cos 27rn] . (28)

The first integral is elementary, and the second can be

integrated by the method of residues yielding

A(n0 ) = --~4 a3/2 [6 + h l ( a ) sin 27rn0] , (29)

h 1 (or) = (27r3/,v/'~)(27r3/e - 1) cosech (Tr2/~v/-~). (30)

For the conservative case 6 = 0, the perturbed system has the transversal intersection since A(n0) has a simple zero at a > 2. While the quadratic tangencies occur for the dissipative system 6 4= 0 because A(n0) has a quadratic zero at a = a c, defined by

6 = h l (ac). (31)

The threshold a c is shown in fig. 2 as a function of 6.

3 -

a 2 -

al a ~

ao

as ~ \

\ \

\ \

\

I 1 I t I I l l l r I I I I I I I I I J I I I I [ I

Fig. 2. State diagram. The dash-dotted curve shows the saddle- node bifurcation. The threshold of homoclinic tangency a c is shown by the dashed curve (theoretical result) and by the solid curve (numerical calculation). The period-doubling bifurcation starts at a = a c and the period-4 cycle appears at a = a c. The accumulation point is expressed by a*.

The theoretical result is shown by the dashed curve and the numerical one by the solid curve. In the limit 6 -+ 0, our results agree with numerical calculations.

In fig. 2 other thresholds are also shown. The period- doubling sequence starts at a = a 1 = 2(1 +J) and the period-four orbit appears at a = a 2 = (5J 2 + 8,I + 5) 1/2 .

The accumulation point a* is calculated by the method of renormalization group [4]. The strange attrac-

tor appears at a > a*. Fora s < a < a c, the distance A(n0) is always neg-

ative. This means that the unstable manifold remains

in the confined region defined by the stable manifold (see fig. 3a). The orbit rapidly approaches one of the fixed points. When a exceeds a c, the unstable mani-

fold overlaps the region defined by the stable manifold and the orbit has a sensitive dependence on initial conditions due to the appearance of a horseshoe [10]. The basins of attraction for two fixed points have a fractal boundary. A detailed discussion will be reported elsewhere. The tangled structures of W u and W s are shown in fig. 3 which are the results of iterating 200 initial points on the unstable manifold of the map M' and of the inverse map M ' -1 near the origin. As we shall see, the unstable and stable manifolds oscillate

182


(a) Y

a-2.4 1

< 7 7.. / . { Wu }!

/

!l ,/¢'

~,t //!

w----"t,_ S?[ i x -1

%~!:)X

(b)

a-2.5

. ~i~ I( ~"

f ' /

"\ I

"~ \ : . . .. .,,- Ws ., , : j

/

Y

,,,i

i f

: . ? •

0: 2 : . :q:

X

Fig. 3. Tangled structure of the unstable W u and stable W s manifolds before (a) and after (b) the homoclinic tangency. The threshold value of homoclinic tangency isa c = 2.425... and ti is set to be 0.01.

infinitely often near the origin. But the orbit starting from arbitrary initial conditions does not oscillate and

it deviates from the stable manifold, exponentially. The orbit approaches one of the attractors (0, -+ vr& ") after a transient chaotic motion.

Let us discuss the case J ~ 0. From fig. 2, the homoclinic bifurcation occurs after the period-doubling bifurcation. Then the strange attractors localized in the region y < 0 or y > 0 (for map M') suddenly merge each other at a = a e. The homoclinic tangency gives rise to a crisis of the chaotic at tractor [ 11 ] .

One of the authors (Y.Y.) thanks Professor H. Nagashima and Dr. K. Sakai for helpful discussions. He would like to thank Professor I. Prigogine for his kind hospitali ty at the Universit6 Libre de Bruxelles.

References

[1] P. Holmes, Philos. Trans. R. Soc. A292 (1979) 419. [2] P. May, Ann. N. Y. Acad. Sci. 316 (1979) 517. [3] J. Testa and G.A. Held, Preprint. [4] B. Hu and J.M. Mao, Phys. Rev. A27 (1983) 1700. [5] R. Bridges and G. Rowlands, Phys. Lett. 63A (1977) 189. [6] H. Daido, Prog. Theoro Phys. 63 (1980) 1190. [7] A.J. Lichtenberg and M.A. Lieberman, Regular and

stochastic motion (Springer, Berlin, 1983)~ [81 B.V. Chrikov, Phys. Rep. 52 (1979) 263. [9] V.K. Melnikov, Trans. Moscow Math. Soc. 12 (1963) 1.

[10] S. Smale, Bull. Math. Soc. 73 (1967) 747. [ 11 ] C. Grebogi, E. Ott and J.A. Yorke, Phys. Rev. Lett. 48

(1982) 1507.

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structure of strange attractor and homoclinic bifurcation of two-dimensional cubic map

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