J . Phys. Chem. 1989, 93, 2833-2836
2833
Critical Slowing Down, Phase Relations, and Dissipation in Driven Oscillatory Systems George E. Tsarouhast and John ROSS* Department of Chemistry. Stanford University, Stanford, California 94305 (Received: September I , 1988)
Three dynamical properties of forced nonlinear systems are discussed with approximate analytic solutions obtained from the dynamic equations for oscillatory systems, near a supercritical Hopf bifurcation, driven by periodic perturbations of small amplitude. With these solutions we first obtain the phase difference between the response of the system and the periodic perturbation and its dependence on the parameters, and hence the mechanism, of the system. Second, we derive expressions for critical slowing down near edges of entrainment bands, with consideration of possible variation of both the radius and phase of the perturbed limit cycle with the amplitude of perturbation. Third, we show by analysis the previously numerically calculated variation of the dissipation within entrainment bands, which depends linearly on the square of the amplitude of the response of the perturbed system.
I. Introduction In two prior articles1,2we investigated some aspects of the dynamics of a driven oscillatory reaction based on the analysis of the normal form of the differential equations of such systems. In ref 1 we discussed the response of an oscillatory system close to a supercritical Hopf bifurcation in the region where a response of the system is quasi-periodic and we presented approximate, but general, analytic solutions. In ref 2 we made a similar study in regard to the periodic response of the system within entrainment In the present work we continue our study of the dynamics of a driven oscillatory system and seek approximate analytic solutions of the dynamic equations of motion in normal form to discuss three further topics. ( 1 ) The phase difference between an external periodic perturbation of one of the variables of the system and the response of that same or another variable. We obtain an analytic expression for the relation of this phase difference to the nonlinearities of the system close to a supercritical Hopf bifurcation. If the dependence of this phase angle on the amplitude of the perturbation is negligible, which is the case for small amplitudes, then that angle, if measured, provides one number which is system-dependent and useful in the complex task of the determination of reaction mechanism. ( 2 ) We derive the known phenomenon of critical slowing d o ~ n ~with v ~ the , ~ contribution of an analytical expression valid for all entrainment bands. (3) We discuss the dissipation in driven oscillatory systems both within and outside of entrainment bands and present approximate analytical formulas for these quantities. The theory predicts variability7-I0of the dissipation within entrainment bands and improves on earlier numerical calculations7 marred near the edges of entrainment bands by the phenomenon of critical slowing down. 11. Analysis We consider a system with the chemical (or physical) variables y , , y2, ...,y n which oscillates near a supercritical Hopf bifurcation. Due to the stable oscillations of the system, we may use the center manifold theorem" to reduce this n-variable problem to two variables. We write the kinetic equations for the two-variable system i 1 i 2
=
=
(YIIxI
azixi
+
+ f l ( ~ 1 , ~+Z )2a1Q COS ut +
2 ( ~ 1 ~ 2 )2a2Q COS
(ut + 4 ) (1)
in which the functionsf, express the nonlinearities in the system and the last term in each equation corresponds to the imposed external periodic perturbation with amplitude Q and frequency w . The bifurcation parameter for this system is labeled p and is related to the eigenvalues of the linearized system A1.2
=
CL
* i%b)
zl = z , [ p + iw,
+ bl(0,0,1)Q2+ bl(0,1,0)zlz2]+ z2q-lz3pYl(l ,O,O) (3)
for the entrainment p / q and for the fundamental entrainment band (1/1) 21 =
Z
I
+ ~iw, + z1~2b1(O,l,O)l + z,2yI(l,0,0)
(4)
The equations for variable z2 are, in the corresponding cases, the complex conjugates of eq 3 and 4. The coefficients b , , yl, etc., are given in Table 1 of ref 2 in terms of the nonlinearities of the given system. The additional variables z3 and z4 have been introduced to deal effectively with the sinusoidal form of the external perturbation; we further introduce polar coordinates z1 =
r2 = relP,
z3 =
z4 = Qel(wf-T/2)
(5)
For the fundamental entrainment, (w,/w) = ( l / l ) , the solutions for the additional variables are z3 = 2, = (Q/2)'/2et(wf-"/2). In addition to the polar coordinate 9, which increases steadily, we also introduce the angle 0 0 = qp - p ( w t - a / 2 )
(6)
which can be associated with a stroboscopic map. Hence, we write the kinetic equations as
+ rQ cos f + b2Q2 + blr21 f = 2(6 - yQ sin E + b'2Q2+ Y l r 2 ) i. = r [ p
(7)
for the entrainment band (1/2) = (w,/w), as i. = r [ p
+ blr2]+ ( y / 2 ) Q
cos (
r f = r[6
+ b',r2] - ( y / 2 ) Q
sin f
for the fundamental entrainment band ( l / l ) , and as
(~12x2
+ a22~2+ f
The frequency of the autonomous system is denoted by w,. The development in ref 1 and 2 proceeds by a transformation of variables to put the kinetic equation ( 1 ) into normal form. This is not be repeated here, but we do write again the equations in normal form
(2)
'On leave from the Department of Theoretical Mechanics, University of Thessaloniki, Greece.
0022-3654/89/2093-2833$01.50/0
(1) Tsarouhas, G.; Ross, J. J . Chem. Phys. 1987,87, 6538. (2) Tsarouhas, G.; Ross, J. J . Chem. Phys., in press. (3) Rehmus, P.; Ross, J. J . Chem. Phys. 1983, 78, 3747. (4) Pugh, S.; DeKock, B.; Ross, J. J . Chem. Phys. 1986, 85, 879. Pugh, S.;Schell, M.; Ross, J. J . Chem. Phys. 1986, 85, 868. (5) Schneider, F. W. Annu. Rev. Phys. Chem. 1985, 36, 347. (6) Schell, M.; Ross, J. Annu. Rev. Biophys. Biophys. Chem. 1987, 16,401. (7) Richter, P.; Rehnus, P.; Ross, J. Prog. Theor. Phys. 1981, 66(2), 385. (8) Escher, C.; Ross, J. J . Chem. Phys. 1985, 82, 2453. (9) Richter, P.; Ross, J. J . Chem. Phys. 1978, 69, 5521. (IO) Schell, M.; Schram, H . M.; Ross, J. J . Chem. Phys. 1988,69, 2730. ( 1 1) Carr, J. Applications of Center Manifold Theory; Springer: New York, 1981.
0 1989 American Chemical Society
2834
Tsarouhas and Ross
The Journal of Physical Chemistry, Vol. 93, No. 7, 1989 i = r[p
+ b2Q2 + blr2] + yQZ cos E
r i = r[6
+ b’2Q2+ b’,r2] - yQ2 sin f
where the radius of the un erturbed limit cycle near the Hopf bifurcation is Ro = lp/bIl1 2. The correction term p o is (see eq
P
(9)
11)
for the (2/ 1) entrainment band, with the additional definitions
E
= 0 - Er(l,o,o,, 6 = u, - ( p / q ) w , bl = Re b1(0,1,0) b2 = Re bl(O,O,l), b’, = Im bl(O,l,O) b12 = Im bl(O,O,l), Y = 1~1(1,0,0)1 tan tr(i,o,o)= [Re 71(1,0,0)3/[Im + ~ i ( l ~ o , oEr(i,o,o) )l~ Er (10)
for the (1/2) entrainment band PO
J
Again the quantities b and y are found in Table I of ref 2. We restrict the study to these three entrainment bands; there are no additional difficulties in treating other entrainment bands. The fixed points of the kinetic equations (7-9) are obtained by setting the time derivatives to zero, and they correspond to periodic orbits of the variables x (eq I). For the entrainment band (1 /2) we find ro2 = (F + y Q
6 = 1.1 tan
u-
COS
Eo + b2Q2)/lbil
Q2(bl2- b2 tan u )
(1 la)
+ ( r Q sin (Eo + u))/cos
u
PO
, , ,a 6,,
= b’,/b,
(12)
+ b1r021 = -(y/2)Q
6 = p tan u
COS
Eo
+ (1/2)- ro rQ sin (to+ u ) cos u
+ b2Q2+ blro2)= - y e 2 cos Eo
6= p tan u - Q2(br2- 6 2 tan u )
for the (1/2) band (25a)
6,,,
= p tan u - (1 /2)yQ/cos u
(25b)
6,,,
= p tan
+ yQ2/(Ro cos u )
(26b)
= 0 (15)
for the band ( 1 /2)
u-
Q2(br2- b2 tan u )
for the (2/1) band. For the purpose of further discussion we transform the solutions obtained to the original variables, x ~ ,which ~ , to first-order terms including the approximations (20-23) are XI
+ 2 Q l ~ l COS l (wt + 91) cos [ ( p ( w t - r / 2 ) + o o / q + 9 0 1 + +
= 2ro COS [@(ut- ~ / 2 ) Bo/q]
x2 = -2r&q,/cu,2ll/2
2Ql4
COS
( u t + 9 2 ) (27)
for the (2/1) and (1/2) bands (and the same equations with u I = u2 = 0 for the (1 / 1) band). The expressions for the phases 9 0 , pl,and cpz in terms of already defined quantities are given in ref 2. The quantity Bo is defined as Bo = Eo l p . The solutions to the same order outside of entrainment bands are
+
-ylbllQro cos (Eo
+ U)/COSu = 0
(16)
for the band ( l / l ) , and X2 - 2X(p
u
(26a)
+ b2Q2 + 2b,ro2) +
+ 2b1r02) + ( ~ Q / r o )+~
= p tan u
= p tan u - Q2(bt2- b2 tan a) - yQ2/(Ro cos u )
A linear stability analysis of the nontrivial, that is, nonzero, fixed points given in eq 11-13 yields
X2 - 2X(p
+ (1 /2)rQ/cos
6,,
6,,
+ (yQ2/(ro cos u ) ) sin (40 + a)
u
(24b)
(13b)
(14a)
2y1b11Qro2cos (lo+ u)/cos
(24a)
for the (1 / 1) band, and
( 1 4b)
X2 - 2X(p
u
(13a)
and for the entrainment band (2/1) ro(p
(23)
+ Q2(b’2 - b2 tan a) + yQ/cos u = p tan u + Q2(bt2- b2 tan a) - yQ/cos u
= p tan
These equations hold except for the trivial case ro = 0. Similarly, we find for the fundamental entrainment band (1 / 1) r o k
(rQz/2~L)COS Eo
for the (2/1) entrainment band. With the use of these approximations in eq 1I , 13, and 14, we may show that w and ro form an ellipse in w,ro space, and we can easily find the limits of entrainment bands by noting that at the limits of stability given in eq 18 and 19 the term cos (Eo + u ) is 0 and hence the term sin (Eo + u ) in eq 1 lb, 13b, and 14b is f l . That gives the extrema, and hence the extents of the entrainment bands are
where u
(22)
for the (1/1) entrainment band, and
(1 1b) tan
( y Q / 4 ~ )COS Eo
x l = 2r0 cos (w,t) x2 = -2rola21/a12)’/2 cos
+ b2Q2 + 2b1r02) + ( ~ Q ’ / r o ) ~+
+ 2Qlull cos ( w t + p,)
+
(ant
(00)
+ 2Qlu21 cos (wt + p2) (28)
where ro is obtained as a solution of i. = r [ p
for the case ( 2 / l ) . From these characteristic polynominals we find (for small Q,(Q/p3I2) 0
(18)
and a/2
-
Eo
u I
I3 ~ / 2-
U,
for cos
u
