Precursor transition in dynamical systems undergoing period doubling

PHYSICAL REVIEW A

VOLUME 35, NUMBER 5

MARCH 1, 1987

Precursor transition in dynamical systems undergoing Krishna Kumar, Arvind Department

of Physics,

K. Agarwal, Jayanta K. Bhattacharjee,

period doubling and Kalyan Banerjee

Indian Institute of Technology, Kanpur 208 016, Uttar Pradesh, India (Received 2 September 1986)

We show that when a system of coupled ordinary nonlinear differential equations undergoes a transition before period doubling, the threshold of the symmetry-breaking symmetry-breaking transition can be found from harmonic balance.

In a variety of dissipative systems the period-doubling bifurcations are preceded by a precursor transition where a symmetric limit cycle bifurcates to an asymmetric one. Although it is a common phenomenon, it has been proven by Swift and Wiesenfeld' that it is not necessarily the case and an exceptional class for which it does not occur, exists. Novak and Frehlich found the pre'cursor transition in the Duffing oscillator, which does not belong to the exceptional class of Swift and Wiesenfeld, ' and estimated the threshold for the transition by applying harmonic balance. In this Brief Report we point out on the basis of numerical evidence that a class of coupled nonlinear differential equations which model forced convection in ordinary and double diffusive systems do not belong to the exceptional class of Swift and Wiesenfeld, ' and that the threshold for the symmetry-breaking transition in these systems can be estimated quite accurately by harmonic balance. The system of differential equations are the following. (a) The Lorenz system,

x =o.( — x

+y),

(la)

= —xz+ r [1+A cos(cot) ]x —y, z=xy —bz .

the natural frequency of the oscillatory instasystem, a limit cycle of frequency ~0 is stabi0.08. This symmetric limit cycle is prevented doubling because of the symmetry that causes to be invariant under the transformation —X, Y~ —Y, U~ —U, Z~Z, and t~t+m. /co0, V~V. As the value of 6 is raised, the limit cycle becomes asymmetric at 5, = 0. 21 (i.e., the Fourier transform of the cycle now includes the frequencies 2coo). Beyond 6„successive period doubling occurs. This is the result of numerical computation. We show below that 5, can be reasonably estimated by using the method of harmonic balance. ' We proceed by first reducing the system of Eqs. (2a) — (2e) to a one-variable equation. From Eq. (2a) we find where co0 is bility of the lized for from period the system

6)

X~

—

Y+ U= X +X,

(3)

while Eqs. (2b) and (2c) yield

Y+sU

= —XZ +XV + Ir&[1+6cos(cot)] —srp)X —X ——. ~ ~

(lb)

y

(4)

(lc)

For can=10, b=8/3, and r=28, the asymmetric limit cycle bifurcates as the magnitude of A is raised. (b) The double diffusive system,

X=cr( —X+ Y+ U),

(2a)

Y= —XZ+ r [1+5 cos(cot) ]X —Y, Z =XY —bZ, U= —XV —sr2X —sU,

(2b)

&

(2d)

V=XU —bs V . For 6=0, a. =10, s=0.4, and r2 ——6, Da Costa et al. observed period doubling as r was raised from the oscillatory convective threshold at 7. 18. The period doubling was 7.78. bifurcation at r I — preceded by a symmetry-breaking For 6&0, o. =10, s =0. 1, r2 —2, r& —1.32, and co=2co0, &

Equations (2c) and (2e) can be solved for the form

Z(t)=e

Z(t)

and V(t) in

"f e"X(t')Y(t')dt',

V(t) =e b"

0

eb" X(t')U(t')dt' .

(6)

The task now is to obtain U and Y from Eqs. (3) and (4) in terms of X and its derivatives, and substitute in Eq. (2a) to obtain an equation in X. At this point we note that U and Y satisfy integral equations. Closed-form solutions are not possible. However, analytic work is facilitated by the fact that we have chosen r& such that the system is This implies very close to the threshold of instability. that all amplitudes are going to be small and we can solve for Y and U in terms of a perturbation series in X, where we need to retain terms up to X alone. The integral equation for Y is [eliminating U from Eqs. (3) and (4)]

o'(1 — s) Y=crI ro[1+6 cos(cot)] — s(1+r2) IX —(cr+s)X —X+ —,' X

oX

0

e"" "— X(t') Y(t')dt'+

0

e"" 35

"X(t') Y(t')dt' + 2334

X—'" t

cr

2

e

1987

'IX (t')dt' .

The American Physical Society

35

BRIEF REPORTS The Liouville-Neuman

s)F=o cr(1 —

Ir p[1+5

series up to cos( ct)t)]

O(X

)

2335

is

—s(1+r2) IX —(o+s)X —X+ — j [o+s+ 1 $

'

—,

(b

+bs +1 — s)]X +2X X XI— I,

where

I=

o[rp

—s(1+r2)]+ b (b— +o+s) 2

0

e"" "X (t')dt'+ t

eh~(t'

o[rp

—s (1+r2)]+

—t)X2(ti)dt~+

p

We now obtain U from Eq. (3) in terms of X and its derivatives. for U and Y in terms of X, X, X, and X, we arrive at ~ ~

0~

~~

Differentiating

crrp5cos— (cot)]X+ I crs[r2+

= o.

gs

+s+ 1 (r2+1

2

0

eb (t —t)'X2(ti)d

ti

Eq. (3) once and inserting expressions

1

—,

—,

We note that in the absence of forcing and for negligible nonlinearities, Cc)p

+cr+s) — (1 —s)+o'(1 —s)

—rp —rp5cos(cot)]+cporp5 sin(cot)]X) I X— I . (10— XI X— +1 —s)+ ]X X+4XX +[cr+s+ (b +bs + 1 s))X — )

orp+— crsr2

=3[o+s+ —, (b +bs

(bs

eb(t —t)'X2(tr)d ti+

~ ~ ~

(1 — s)(X+(cr+s+ 1)X+ [cr+s+so ~

0

2

the system supports oscillatory motion with

—rp),

(1 la)

and

(o'+s)[o'+s

+ 1+o's(r2+ 1)]

(1 lb)

cr(cr+ 1) Re(A le Xp —

'+ A3e ' +

),

(12)

where the coefficients A ) and A3 can be found by applying harmonic balance to Eq. (10) with respect to the frequencies bifurcaand 3cop. The presence of odd harmonics implies that the limit cycle is symmetric. The symmetry-breaking tion occurs if an even harmonic is introduced in the time series. To find the value of the forcing amplitude at which this bifurcation occurs, we require to test the stability of the solution against perturbations of frequency 2cpp. To test the stability we write cop

X =Xp+g,

(13)

introduce it in Eq. (10), and linearize it to arrive at (1

—s)(2')'+ (o +s + 1)i) + [cop or p5 cos(2ct)pt )]2) +—[ (cr+s + 1)cop or p5[s cos(2cppt ) — — 2ct)psin(2copt = 3(B + )(2XQXQ71 +XQq ) +4(2XQXQ7J +X ()rj ) + 3BXQ7/ —[itIp+ rt(IQ + Ip ) ] —(Xp +Xp )(lllleal ized Pal t of I ) —Xp( lllleal ized Pal t of I )

) ] I 21 )

—,

where

B =o. +s+ —,'(b +bs + 1 — s), and Ip is the function I evaluated with X =Xp. The rest of the analysis proceeds along standard lines. We try for n (t) of the form a solution a (t)cos(2cppt) + b(t)sin(2copt), where a(t) and b(t) are slowly varying functions of time. Equating coefficients of cos(2ct)pt) and sin(2cppt) in Eq. (14) yields two differential equations in a(t) and b(t). Seeking solution of the form e ' for a(t) and b (t), the consistency condition forces the value of A, .

The threshold is obtained when

5=0. 19,

A,

=O. This yields (16)

quite close to the numerical value of 5=0.21. The technique outlined above is quite general and is capable of handling large-order differential equations. We note that in the system governed by Eqs. (2a) — (2e) it is observed that the threshold for precursor transition decreases if rz is decreased keeping o., s, and b fixed and setting co =2cop. This feature is reflected in the threshold obtained from harmonic balance.

2336

BRIEF REPORTS

J. W. Swift

and K. Wiesenfeld, Phys. Rev. Lett. 52, 705 (1984). For an experimental demonstration, see P. Bryant and C. Jeffries, Phys. Rev. Lett. 53, 250 (1984). ~S. Novak and R. G. Frehlich, Phys. Rev. A 26, 3660 (1982). See also D. D'Humieres, M. R. Beasley, B. A. Huberman, and A. Libchaber, Phys. Rev. A 26, 3438 (1982). L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Vol. 6 of

Course on Theoretical Physics (Pergamon, London, 1974)

~

4Y. Aizawa and T. Uezu, Prog. Theor. Phys. 68, 1543 (1982). 5L. N. Da Costa, E. Knobloch, and N. O. Weiss, J. Fluid Mech. 109, 25 (1982). A. K. Agarwal, J. K. Bhattacharjee, and K. Banerjee, Phys. Rev. Lett. 111A, 329 (1985). K. Wiesenfeld, E. Knobloch, R. F. Miraoky, and J. Clarke, Phys. Rev. A 29, 2102 (1984).