Dynamical structure functions at critical bifurcations in a Bonhoeffer-van der Pol equation

Chaos. Sohrom

L Fructds Vol. 7, No. 11, pp 1799%1805, 1YYo Copyright 0 lYY6 Elscvier Science Ltd Printed m Great Britain. All rights resened 0960.0779/96 $lS.Gil + I).011

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Dynamical Structure Functions at Critical Bifurcations in a Bonhoeffer-van der Pol Equation S. RAJASEKAR Department of Physics, Manonmaniam Sundaranar University, Tirunelveli 627 002, Tamilnadu. India (Accepted 23 April 1996)

Abstract-Chaotic attractors at various bifurcations in a Bonhoeffer-van der Pol (BvP) equation are studied in terms of a,(q)--the variance of fluctuations of the coarse-grained local expansion rates of nearby orbits. For all the chaotic attractors of the BvP equation the o,(q) versus q plot exhibits a peak at q = qa. We show that additional peaks, however, occur only for the attractors just before and after the bifurcations such as crisis (sudden widening of a chaotic attractor), band merging and type-1 intermittency. Copyright 0 1996 Elsevier Science Ltd.

In a dynamical system as the control parameter is varied the chaotic attractor makes drastic changes at several points. This is mainly due to the segregation or aggregation of various co-existing attractors and repellers in the phase space. Thus, it would be important to elucidate the transitions between the various forms of the chaotic motion, that is, bifurcation of chaos. Recently, features associated with the fluctuations of coarse-grained local expansion rates for various attractors have been studied [l-7]. In this paper we study the chaotic attractors of the Bonhoeffer-van der Pol (BvP) equation [8, 91 i =x

-x3/3

- y + fcos@,

(14

j = c(x + a - by), c$ = n,

(lb)

(mod 2n)

(lc) at various bifurcation points using dynamical structure functions. The BvP oscillator is an interesting dynamical system of considerable biological and physical significance [8] and it exhibits chaotic motion, phase-locking, and so on [9-121. Let {x,}, (n = 1,2,. . .) be a chaotic orbit generated by a two-dimensional Poincar& map of the BvP oscillator and define the coarse-grained expansion rates [6]

ht(Xl> = (l/4 i A(Xm), IF?=1

(2)

where /%(X,) is the local expansion rate of nearby orbits at X, along the unstable manifold and X = (x, y). The values of A,(X,) for different X,s are randomly distributed between a maximum value Amax and a minimum value Amin. As II + ~0, A,(X,) converges to a positive Lyapunov number A”. The probability function P(A; n) for A,(X,) to take a value around A takes the scaling form P(A; n) = exp[-nq(A)]P(A”;

n).

(3) To describe the fluctuations of the local expansion rates we consider the partition function

z,(q) = (exp[-n(q - l)~,(Xdl)~ 1799

(--co < q < 00)

(44

1800

S. RAJASEKAR

where ( . . . ) denotes the long time average (G(X,))

= JiFm(1/N);G(X,) i=l

(4b)

and the temporal scaling exponents &l(q) = -(l/n)ln

Z,(q),

(5a)

At(q) = d@n(q)/dq~

(Sb)

fin(q) = -%(d/dq:

!5c)

v(A) = 4(q) + (1 - 4).

04

The dynamical structure functions A,,(q) (the q-weighted average of the coarse-grained local expansion rates of nearby orbits), a,(q) (the variance of fluctuations of A,(X,) around An(q)) and q(A) (which describes fluctuations of A,(X,) around A”-the long time average value of local expansion rates) are calculated for parametric values far away and near the bifurcations such as sudden widening of a chaotic attractor, band merging and type-1 intermittency for the BvP equation (1). For all the chaotic attractors of (1) the o,(q) versus q plot shows a peak at q = qn. Additional peaks are found only for the attractors just before and after the bifurcations. To obtain the dynamical structure functions defined by (5) the local expansion rate 1.(X,) along the unstable manifold at X, is calculated in the Poincark map. Taking an initial point X,, = (x(to), y(to)) and a unit vector (u(to), u(Q) we numerically integrate the variation equations ti = u - x224- v,

(ha)

ti = c(u - bu),

(6b)

along with (1) until tl = 21~/& and then calculate A(X,) = ln[l~(t,)1~ + Iv(t,)/*]‘~*. We again integrate (1) and (6) starting from the point (x(tl), y(tl)) and unit vector [u(tJ/ exp (A(XI))~ f4hYexp @(Xdjl until r2 = 2(271/Q). Then we obtain X(t,) and A(X2). Repeating this process we calculate k(X,) and the coarse-grained expansion rates (2) and the dynamic structure functions (5). We fix the parameters a, b, c and Q at 0.7, 0.8, 0.1 and 1, respectively. For f > 0 the system (1) shows variety of behaviours such as period doubling phenomenon, crisis, intermittency and so on. Figure 1 shows the bifurcation structure of the BvP equation for f E (1.2855, 1.289). A band merging crisis is found to occur at f = f,, = 1.28653. As f is decreased to f, a three band chaotic attractor merges together and forms a single band chaotic attractor. Figure 2 shows the dynamical structure functions o,,(q) and v(A) at f = 1.287, far from the bifurcation point fm, where we have used II = 21 and N = 8000. The a,(q) has a peak at q = q. = 1.65 and q(A) has a linear part to the left side of minimum with slope s, = 1 - qa = -0.65. The dynamical structure functions just before the band merging are shown in Fig. 3. I$(A) has two linear parts whose slopes are given by s = s,= -1.4 and sB= 0.6, and o,(q) has two peaks at q = qn = 2.4 and 4 = qp = 0.1. Two peaks are observed for f values just after the band merging also. For example, for f = 1.285 peaks at qm= 1.85 and yp = -0.075 are observed. Thus, just near the band merging there occur two remarkable peaks. Further, the function q)(A) is a concave function of A and takes a minimum value zero at A = A”. For q = 1, A,(l) = A” and 4,(w) = 4mn anr’ 4,,(-~0) = Amax. Therefore, a,(q) with q > 1 and q < 1 can explicitly describe negative and positive large fluctuations of &(x1), respectively. As q is varied, the coherent large fluctuations of A result in discontinuous phase transitions of A,,(q) at certain

i,1

Chaotic attractors in a BvP equation

1801

x

I i.

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:

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; ,

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!I::: . '

.

-0.85

* : :

: : ;

t..:. * , .. .. . I :

. .

j

. . .

I 1 *

: ! : . i . ! !.!I

‘!

-1.1

i

j

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:

: . .

.

. ! 9 :

:

.

,

I!; : ! f ,

: *. i i :*i.::.,

.* . . I

i

;

I

l

I

i . * i

0

!

iiti;::

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.

: : 1

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:

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. I . I

-. : .. .

:

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*

i

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i

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1:

;

.:.

)

:

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'

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i

1

!

i

a

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I ’

* I ! iI ; . I :.:1 ! 1, * I. 9, ! : : . , ’ . * ‘ .* . I:*‘:t ,:.a; .i * . : I

!

!

* .-.‘,l

;

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11

1;

T

I 1 ^“rc

. ^-,--

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i : . ;

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i: ! .

I i i iI :!

- : - . .. ,. : : . f

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Fig. 1. Bifurcation structure showing band merging crisis at f, = 1.28653.

0.4 h(A)

(b)

Fig. 2. The dynamic structure functions o,(q) and y(A) for f = 1.287 far from band merging, where n = 21, N=8000.

S. RAJASEKAR

1802 0.15

(a)

o-“(q)

4 -

0.4.

(b)

yin(A)

:

-0.2

0.0

0.2

0.4

Fig. 3. a,(q) and @(A) for f = 1.2866 just before the band merging, where n = 5 and N = 8000.

values of q, for example, at q = qa for far from and far away from the band merging and at q = qa and q = q8 for just before and just after the band merging bifurcation; in the limit IZ-+ to. a,(q) diverges at the transition points. In Figs 2(a) and 3(a) o,(q) does not diverge and has finite height peaks. This is because n is finite. Figure 4 shows the bifurcation phenomenon in the interval f E (0.73,0.76). A crisis, sudden expansion in the size of the attractor, is observed at f = f,,, = 0.747486. Figure 5(a) shows numerically computed a,(q) for f = 0.73, far from the crisis, where n = 5 and N = 8000. The q-weighted variance a,(q) has only one peak at q = qa = 1.55. I@(A) is found to possess a linear part to the left side of minimum with slope S, = -0.55. Just before the crisis, corresponding to f = 0.747482, the variance a,(q) has two peaks at 4 = qru= 2 and .q = q8 = 0. In this case, q(A) is found to possess two linear parts on both sides of its minimum at A = A”. In BvP equation type-1 intermittent chaos is observed for f E (1.092,1.094). For f values just below 1.092 a window region with period-4 attractor is found. In the intermittency region the time series consists of regular laminar motions interrupted by irregular bursts. The laminar motions between two successive bursts have different durations which are randomly distributed over the time series. As the bifurcation point f = 1.094 is approached

Chaotic attractors in a BvP equation

1801 . .

Y .:a.............. ..:-.

-...:.. ,

f.

.

.

. .

.

'

. _-.

1

1

I

1

0.73

0.74

f,

0.75

-

0.76

Fig. 4. Bifurcation diagram of the BvP equation for f E (0.73, 0.76) showing the sudden expansion in the size of the chaotic attractor for f,+ - 0.747486. 4

(a)

a;,(q)

-6 I.5

-6

-3

-

0

qa

3

a;n((4)

-3

6

(b)

0

3

6

Fig. 5. Dynamic structure function u,,(q) around the crisis point f = fW. (a) Far from the crisis, where f = 0.73; (b) just before the crisis, where f = 0.747482.

S. RAJASEKAR

1804

the length of the laminar region decreases. Thus, in the intermittency region the chaotic attractor has two types of local structures which produce the laminar motions and turbulent bursts, respectively. Figure 6(a) shows a,(q) versus q in the intermittency region. The f value used was 1.0922 and 12= 10, N = 8000. The o,(q) has two peaks at q = qa = 1.75 and q = qD = 0.5. Figure 6(b) shows the result for f = 1.096 far after intermittency at which fully developed chaotic motion is observed. a,(q) has only one peak at q = qu = 0.6. In summary, we have studied the dynamical structure functions near critical bifurcations in the BvP equation. The chaotic attractors just near the bifurcations such as band merging, sudden widening of the attractor and intermittency exhibit large fluctuations of the coarse-grained local expansion rates A. These fluctuations lead to the linear parts of q(A). Further, the o,(q) versus q plot exhibits a peak at q = qa for all the chaotic attractors. However, additional peaks are found near the critical bifurcations only. Different phases of a,(q) represent different local structures of chaotic attractors. Thus, o,(q) is useful for characterizing the chaotic attractors at their bifurcation points. Even though it is not clear how to distinguish different bifurcations using dynamical structure functions they are useful to identify the parametric values at which bifurcations of chaos occur. One may ask

1.5

0-n (9)

0-u

l.O-

4

5

oq B 9,

-5

r cm(q)

(b)

1.0 -

4

Fig. 6.

a,(q) versus q for (a) in the intermittency

region, where f = 1.0922 and (b) far after intermittency, f = 1.096.

where

Chaotic attractors in a BvP equation

1x0s

whether the present analysis is of importance since the computer generated trajectory is in general not the true trajectory of the system for a given initial condition. We wish to note that for low-dimensional systems a computer generated pseudo-trajectory can be deformed into a true trajectory by suitable shadowing methods [13-171 and thus computer simulation is still considered as providing meaningful information. Acknowledgement-The

present work forms part of a UGC minor research project

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