Intermittency route to chaos in a periodically forced model reaction system

Intermittency route to chaos in a p-dodically

pevrokidti

I. INTRODLCTTION

Studies on dissipative and conservative systems have shown that temporal hehaviour in c+rrai~~ paratneter regionr can that may evcntwlly end eahiMt ccnnplcx phane trajcctorits up in chaos (e.g. Lorenz, 1963; Rossler, 1976; Peng et al., 1990). The presence of periodic forcing adds further complexities to the problem, although in a few cases it has been observed that the for& Periodic operation of chemical reactors improves cwversio~ enhances seleclitity, impam stability and reduces parametric sensitivity (Bailey, 1977). Under the influence of an external periodic forcq non-linear systems can exhibit transition to chaos and it would be inkrating to characterize the transition from among the vslrious possible TOUIIB such as @&doubling cuscade (Tomha. 1982: Raj&ar and Lakshmanan, ‘J988; Cardonier et al., 19SQ, frequency locking (Minorsky. 1962) and intermittmcy (JetTries and Pcre& ISSZ; Prict and Mullin, 1991). we analyse a model catalytic, isoIn this communication thermal reaction under conditions of periodic forcing. The reaction scheme given by nA+sS---+pq

presence

1 -y-

system possesxs a unique unstable solution giving rise to s&tied periodic oscillations (limit cycle). (2) For the iame magnitu&,of a, 8, y, k, as in (1) with k, = 0.05, k, = 0.01 and 8.0544 < $ < 8.2116, the systern possasesthree steady stat=, of which the Imer two arc uestable and the remaining one linearly stable. The system would peridrm sustined osdllations as long as the trajector+ remain outside the attracting region of the stable steady state.

We have chosen one set ol parameter values from =h of regions 1.2, respz+tiveIy, and fixing w &pal to I, examined the effects of forcing amplitude on the dynamic behaviour of eqs (SHS). The parameter # was fixed at 4,15 and 8.06 to represent regjonc 1 and 2, resJx&vely, with a, fl, y, k,. Ikl and k,, taking the corresponding vAw.s as listed eadier. Heretier we shall call the iwo parameter sets with respect to ~=4.15and8.06~stland2,~pectivcly.For9=4.15 and 8.06, the time peribda oC sustained oscilltions of the unfonxd system have beFn determined FS 17.48 and 34.399, respectively. The numerical integration OF the ordinary differcntial equations(forccd and unforced) was wormed with the help of the Runge-Kutta-Memnalgorithm, with acon: stant step size ol Q.tXB. The forced system was studied by varying forcing amplitude A, over a range beginning with small values. We report the results lor parameter set 1. The complete bifurcation behaviour of the forced system has ken shown in Fig. I(a finding the minimas of x whik

q5RCX.Y)

denoted

CES

48:15-,I

Number

%633.

in the

and 4.042 -C # c 5,267, tie

and the dimcnsionlesv

Communication

version

is given by

(1) for m = o-z, B = 5, y = 1, k, = 1. k, = 0.125, k, = 0.M

where

tN.C.L.

forcing

where A, and w reptesent the forein~ amplitude and forcing Frequency. respectively. Bruns eral. (1973) have perfanned detailed linear stability anaJyti ofeqs (2) and (3) and found the existence of two parameter regions wherein the system exhibits a limit cycle and multiplicity behavlour. The two regions are

(1)

concentrations of spties S aud A are by x. y, re$pe6tireky. The parameters k,. k, ad L rqmsmt the male1 constmntfand z the dimerrsto~lees fime. Defmitiolis of the other model parameters can be found in the Notation section. The specific rate form described by q. (4) is also valid if the reactioh is inhibited by high wucwtrations of A. We consider a caw wherein the governing dynamical q. (2) for qxcies S is modifial to a-urn for an external periodic forcing. The non-autonomous set of equations can he cm-~rte.d to the autonomous farm by defining new vat+ ables Z,, Z2 with initial conditions 2, (0) = 1 and Z,(O) = 0

C$al., f986). The fully autonomous

of perimlic dx

where S represents a non-ionizing substrate and A an acid substrate, was previously studied by Bruns et nL, (1973). The reaction is assumed to be. carVied out in continuous stirred Lank reactor (CSTR). The concentrations of speck A and S are maintained conslant via exchange through impermeable membranes. Under such conditions, this system MII represent a simple model roar a living ccl1 (Denbigh et al., 1948; Othmer and Striven, 1971) or that or an immobili& enzyme reactur. Additionally, smne of the commonly employed Langmuir-Hinshelwood kinetic forms used to deSxilx non-biological catalysis are also mathematically equivalent to the Michaelis-Menten ,type rate form cmsidered here. The governing dynamical equations in dimensionless form are given by

!g_

forced model reaction system7

numerically

integrating

eqs {5)-(g). As can be wn,

there are

six regions corrcaasponding to (i) quasi-per&Cc, (ii) sustained three-peak periodic (3Ph (iii) chaotic, (iv) 1I-peak periodic (IIP), (v) quasi-periodic and (vi) sing@eak pcsioddic (t P) oscillatiom~in the A, rane O-1. The A, ranges mrresponcing to each of these temporal behaviours are listed in Table 1. In the range OXl.098 lor A,, a quasi-p&& &h&our is encountered. In this region the system khavtour is weakly

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2.0#0

LWOO

n

1.2mO

D.EWO

x

Fig

2. Phase

plane plot (x vs y) for A, = 0.503 corresponding to 3P 0sEillatians.

Lyapunov exponents

of the two-dimensional system represthe algorithm suggested by Wolf r~ ul. (1985). Figure qc) shows the estimated maximal Lyapunov exponent, A,,,, as a l-unction of forcing amplitude, wherein the positive. values of a,,,., correspond to the chaos. To have a better distinction betwem chaos and quasiperi&ic oscillations. a fine bifurcation diagram for A, lying between 0.5 and 0.78 is shown in Fig. l(b). The maximal Lyapunov exponents for 0.6001 < A, G 0.78 were found to attain small (0 > L. > - 0.001) values suggesting the presencc of quasi-periodic osciilations. ented by eqs (5) and (6) employing

Fig. 1. (a) Bifurcation diwam of the forced model [cqc (5)-(S)] obtained by +tting minimas ol x as function of A,, (b) fine bifumtion diagram of the forced model Ceqs I5HN)l.

chaotic. as OonGrmed by computing the Lyapunov exp+ nants which aie in the range 10-*-10-5. It was found that. for 0.098 < A, c 0.X)3. the f+cecl system shows sustained plot (x vs J) 3P oscillations: Figure 2 showa the phase +ne of s&h oscillations. As A, is increased to 0.504. fully developed chaotic ttajcctoti emerge and are portrayed as the phm plane plot in Fig. 3(a). The Poincare section at Z, = O+5 in the x-y plane following Heaon’s prescription (Henon. 1982) is shown in Fig 3(b). Further characterization of the chaotic region was performed by computing the

Table meter

1. Bifurmdon pattems.in the AI rangr: O--l {pareset I; 4=4.15, ruzO.2, j-5, y-l. k,=l. k, = 0.125, k, - Oar)

Pattern

Dynamic behaviour

no. 0.0 < A, < 0.09%

0.098 < Al 0.503 A, < zz 0.503 0.593 0.593 < A, r 0.601 0.601 < Ai CO.78 0.78 < A, G 1.0

Quari-periodic 3P Chaos IlP Quasi-periodic

IP

2.1. Comptit&nR of frracsaI dimension The phase space volume enclosed by the dissipative chaotic system representad by cqs (5) (8) contracts during its evolution in time. As time tends to infinity the system tr-+ctories converge to an attracting region commonly known 8s the strange ortractor. This attracting set is chatacteriti in terms of the fractal dimension (D) and describes the amount of necessary information to characterize the attractor within a spacificd accuracy. For a two-dimensional system, Kaplan and Yorke (1983) sugsested a relation connecting the fractal dimension and the Lyapunov exponent spectrum given by .=I+fi_

a (9)

For A, = 0.504, the magnitudes of &,_. and L,,. were found to be 0.018 ( k 0.001) and - 0.336 ( f O.oOl), respectively. Employing eq. (91, we get the fractal dimension {D) al the strange attractor as 1.05357.

For 0.098 < A, < 0.503, the forced system exhibits threepeak periodic osclllatims, whereas for 0.503 < A, & 0.78, the system shows chaotic. 1 I -peak periodic and quasi-pcriodic behaviour. Our interest lies in knowing the route to chaog From Fig l(a), it is very clear that the system does nbt evolve via the period4oubling bifurcations route to chaos. The dynamic behaviour in the range 0.503-0.5045 for A,, show that the regular 3P cycles arc interrupted abruptly by “bursts’“. As Ai is incremented, thm bursts occur more frequently. Such behaviour is characteristic of an intermittency route to chaos (for detailed discuwion see Pomcau and Mannetillc. 1980; Price and Mullin. 1991; Rasband, 1990). When the system follows ii tyw 1 intemittmcy mute to chaog there is a “memory” of the vanished 3P cycles and as a result trajectoriw near the fixed point OCa Poincarc map

Sbvrter

C*mmunicationx

2119

Iteration

no

Fig. 4. Laminar phaws ol the forced system where every third iterate OI-Ithe Pvincare section (2, = 0.5) is plotted as a function of iteration number for A, = 0.504.

Fig. 3. (a) Phase p&me plot corresponding to the chaotic (A, = O.W4), (b) Poincare sCction of chaotic trajectories attractor at 2, = 0.5 and A, = MU4, (c) estimated maximal Lyapunov exponent ( *m.s ) a~ a function of .4 I.

remain in the neighbourhood for a long time. This behavinur is depicted in Fig. 4, in which x at every third iteration on the Poincare se&on was plotted as a function of the itemtion number. It can be yeen that there are three stretches, each corresponding to original 3P solutions, wherein every third point comes back tv nearly the fame stretch. The stretches are termed “laminar phases” and the chaotic “bursts” are the regions which connect that phases.

Type 1 inttrmittency, a charaiAe.ristic of tanent (or fold) bifurcation, is defined as an eigenvalue of a fixed point of the Poincare return map passing through the unit circle at + 1. If the Poincarc map is projected onto the eigenvector associawd with the dgenvalue, then such a plot contains the essential information regarding the intermittent transition. Thus. predictions of type 1 infctmittency can be made by local analysis of a nne-dimensional restriction of a Poincare map of arbitrary dimension (Price and Mullin, 1991). The intermittency of the behaviour is clearly illustrated in Fig. 5(a), which shows the full return map of every third iterate for A, = dS0325. To obtain such a plot, we first get the I and y values on the Poincare section defined by Zi = 0.5. One iteration is assumed to bz ~plete when the trajectory passes through the PoinFare plane fot the third time. Figure 5(a) portrays x.+> [sometimes represented as J’(X)] vs xlf where II rdeerr; to the iteration number. In the figure the degetwrdte fixed point is shown a8 xi. The enlarged we-w of the area surrounding X, iu shvwn in Fig. S(b). As can be wcen, the return map touches the line representing the identity map at x;. Any trajectory traversing through the basin tif attraction of this fixed point would eventually wnverge to it. For higher values of the bifurcation parameter A, the return map does not intersect the identity map, indicating that the fixed point solutions in the neighlwurhood of x6 have vanished. Under such circumstances, the trajjectory slowly moves upwards through the channel between the quadratic return map and the identity map. Figure S(c) shows such a channel {the region between the rcwrn map c1and identity map b) through which the trajectories escape. The bifuwtiun behaviour for parameter set 2 was studied by varying Al in the range O- 1, and Table 2 lists the various bifurcation patterns exhibited by the forced system In the A, domain of 0.398-0.404, an intermittency bhaviour analogous to parameter set I is observed. For A, = 0.404. the values of the Lyapunov exponents were found to be = 0.0015 & 0.003 and J.,,, = - 45.5 k 0.1. The small > pZtive i,,, indicates that the system behaviour is weakly chaotic. The fractal dimension vl this strange attractor using eq. (9) Gomev to l.oooO33. In order to avoid unnecessaryduplication in characterizing the route from SP state to chaos in a manner similar lo that of the rvute from 3P state Lo chaos, we give only one plot (Fig. 6) wherein five laminar phasra corresponding to A, = 0.4t.M are clearly discernible. The existence of Iaminar phases confirms that the transition Trvm the 5P state to chaos also occurs via a type 1 intermittmcy route.

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Table 2. Bifurcation patterns in the A, range O-l (parameter set 2, cp = 8.06, a = O.&B = 5,q = 1, k, = 1, k, = 0.05. k, = 0.01)

Pattern no. I 23 4 5 6

Dynamic bchaviour

A, mnge O& < A, 4 0.301 0.301 0.398 0.429 0.509 0.510

< < < <

Al A, A, Al Al

6 G G 6 G

IP

0.398 0.429 0.509 0.510 1.0

2, 1P 11P 1P

i

5

,Fig. 6. Laminar phases of the forced system where cvcry fifth iterate on the- Poincare section (Z, = 0.5) is platted as a function of iteration number for A, - 0.404.

ing upbn the magnitude of the forcing amplitude the systrm shows transitions from a three-peak periodic state to chaos and from a five-peak periodic state to chaos. The transitions were found to follow an intermittency route to chaos and the intennittcncy was charactcriaed as type 1.The chaotic behaviour was characterized in terms of the Lyapunov exponents and the fractal dimension. SANJEEV S. TAMBE B. D. KVLKARNI’

63

NOTATION

x.

stoichiometric efficient of A concentration of species A concentration of s@es S initial conmtration of spcies A initial concentration of species S inhibition rate constan!s dimensionlass inhibition rate constants defined as K&LO. &/CAO Michaeelia-Mcnten rate constant dimensionless Michaclis-Mentetl rate constant

Fig. 5. (a) Full return map plotted

at every third iterate on tlu P&care section defined by Zr = 0.5 for Ar = 0.50325. Solid line indicates identity mapping and the fixed point is shown as X, . (b) Enlarged view of (a) in the noighbourh& of x,.(c) Enlarged view of thefulf return map in the vicinity of*, for A, = O.sws.

defined 88 KJC.40

3. CONCXUSION In this paper, we have studied a model reaction system carried out in CSTR, in the presenti of periodiclllly forced input. The unforced model, in two dii%rent parameter re&ns was known to crhibit limit cycle oscillationsad a multiplicity of steady states We have shown that depend-

stoichiometric time characteristic

?Author

coefficient

of S

time for species S

to whom Gotrespondence

should be addressed.

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characteristic time for specie A maximum reaction vclb~ity dimensionI& concmtration ol spsciea S defined aa Cs/C,a dimensionless concentration of species A dcfmai as C.&z,,

REFERENCES

Bailey, J. E., 1977, Chemical R,xx~ar Theory: Reukw (Edited by L. Lapidus and N. R. Amundson). PrmticsHall, En&wood-Cli& NJ. Bruns, D. D., Bailey, J. E. and Luss. D., 1973. Ste%dy state multiplicity and stability of enzymatic reaction systems. BiotechllclI. Bioengng 1s. 1131-l 145. t%rdomier, G. A., Schmidt, L. D. add Arig R.. 1990, Forced oscillationa of chemical tea~t*r~ with multiple steady stati, Gti Elrrrng Qi. 45.1659-1675, Denbigh. K. G, Hicks, M.-and Page, F. M., 1948, Kinet& of open reaction systems. T~ans Furuduy Sac. 4d,479-494. Henon. M.. 1982, On the numerical computation of Poincare maps. Pkystca D S, 412-414. Jeff-. C. and Perez, I, 1982. Obsmation of Pomeau-Manneville intermittent route to chaos in a nonlinear oscillator. Phys. Rev. 26, 2117-2122. Kaplan, J. and Yorke, 1.. 1983, Funrtiod Difmntid Equa(Edited by H. 0. tions ad Approxinza&w~.~ a_fFixed hitis Peitgen and H. 0. Walther). Springer, Berlin.

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Kevrekidis, I. G., Schimidt, L. D. and Aris. R.. 1986, Some common features of peri~ically forced reacting systems. c!zem. Engng sc1.41. 1263-1276. Lore% E. N, 1983, Deterministic non-periodic flow. J. &mos. sci. 20, 130-141.

Othmer, H. G. and Striven, L. E., 1971, Instability and dynam& in ~llulrrt network& J. Tremor. Viol. 32,507-537. Peng, B., &tt, S. K. and Showalter, K., 1990, Period doubling and chaos in three variable autocatilator. J. Pkys. Ckem. W, S243 -5416. Pomeau, Y. and Manneville, P.. 1980, Intermittent transition to turbulence in diss@ative dynamical systems. Cnnvntut. Math. Pkys. 74, 189-197. Price, T. J. and Mullin, T., 1991, An experimental obseervation of a new type of intermittency. Physica D 48, 29-52. Rajekar, S. and takshmaman, M., 1988, Perioddoubting bifurcations, chaos, phase-locking and devil’s staiin Ph~sicu D 32, a BonhaRtr-van der Poll oscillator. 146-l 52. Rasband, S. Neil, 19X1, Ckudc Dymmics af Nonlinear Sysrms. p. 60. Wiley Interscience, New York. Ruler, 0. E.. 1976, Chaotic behaviour in simple reaction systems. Z. Natw-forsch 31 A, 259-264. Sinha, S. and Ramaswamy, R., 1987, On the dynamics of controlled metabolic network and ozllular behaviour. BOOsystt?ms 2& 34-354. Tomita, K., 1982, Chaotic response of nonlinear oscillators. Phys. Rep. 86, 113-167. Wolf, A.. Swift, J. B., Swinney, H. L. and Yastano. J. A., 1985, Determining Lyapunov exponents from a time series, PRysica D 16, 285-317.

caw-25w93 sum •F oml D 1993Pcr#ammRe@aLtd

And&g circuit for simulation

of prwwre swing adwrption: khtetic model+

Rapid pressure swing adsorption (RPSA, Keller and Jones, 1980) is characterized by signi6canL pnasure drop through the column; thcreforc, those mathematical trcatmentrs of pressure swing adsorption (PSA] which neglect the axial p-ure drop cannot be applied directly to describe RPSA behavior. In addition, prtiuci storage and fluid nxistan~e cause surge of boundary conditions at the two ends of the adsorption cdum. calling for additional mathematical treatment. Along with those who have researched in this field, such as ‘Furno& and Kadlec (1971), Kowler and Kadlec (1972). Sundaram and Wankat (19gS). Doong and Yang (1988)and q uzanowski et al. (198P), we have developed an eIectric analog circuit for simulation of PSA (Guan and Ye, 1990). The analog circuit defines the physical process

+Praject Foundatiofi

supported ol Chti

by

the

National

Natural

Sclenrre

parameters in terms ol electric quiualtits, leading b firstorder ordinary diffxential equations to depict the axial distribution of pressure and the change&l< boundary wnditions in RE’SA. tiowever. in our previous article (Guan and Ye, 199Oh only the local equilibrium theory (LET) was UIMXI to solve the derived equations to simulate the RPSA pro.oess for produdg oxygen-nriched air. In our p-t work, the effect of axial dispemion (AxiDA and the extemaI 6lm ma11s transfer (FilTJ and macropore diffusion (MacD) Mistan-, will be taken into account. The analog circuit for simulating B ~~~~lumn RPSA, which involves finite mass transfer rate when gas flows through the packed column, may bc designsd as shown in Fig 1. Switch K, turned to “A”, touching neither “‘A” nor *Et”, or turned to YB*. writs. res@veIy, adsorption, delay or desorption. In section j ol the column, adsorbate k must overcome the mass transfer resistance RJ,h WOE ahorption. In the same way ar in Guan and Ye (l%W), for