J. Phys. Chem. 1993,97, 6945-6947
6945
Transition from Quasiperiodicity to Chaos in Coupled Oscillators Sory I. Doumbouya and Friedemann W. Schneider’ Institut fur Physikalische Chemie, Marcusstrasse 9-1 1 , 8700 Wurzburg, Germany Received: April 6, 1993; In Final Form: May 19, 1993
W e present the experimental observation of a quasiperiodic route to chaos followed by phase locking in the second of two serially mass coupled reactors containing the Belousov-Zhabotinsky reaction. We report on the evidence of a hysteresis phenomenon and of both a primary and a secondary Hopf bifurcation leading to quasiperiodicity .
QP PI
Introduction In a recent communication’ we reported period doubling routes to chaos in one of two serially coupled chemical oscillators. We found that the coupling constant Qc, which measures the extent of the mass exchange between the reactors, depends upon factors such as the stirring rate in the lower reactor and the volume ratio between the reactors. It is interesting to observe that a change of the volume ratio from v1/v2 = l / 3 in the previous study’ to 1 in the present study leads to another transition to chaos, namely, the quasiperiodic route. Earlier Richetti et al.273 reported a quasiperiodic route to chaos in the free-running Belousov-Zhabotinsky (BZ) reaction. They interpreted this as a result of the interaction of two elementary instabilities, a Hopf bifurcation and a hysteresis bifurcation. The latter accounts for the phenomenon of bistability. In theoretical work, Kaneko coupled two logistic maps? which led to the observation of a quasiperiodic route to chaos accompanied by frequency locking. In this work we present experimental evidence of a Hopf and hysteresis bifurcation, the quasiperiodic transition to chaos and phase locking in the second of two serially mass coupled BZ oscillators. The time series are analyzed using the tools of dynamical system theory: Fourier spectra (see Figure 6 in ref l ) , attractor reconstruction (using the singular value decomposition (SVD) method), Poincart section, return map, Lyapunov exponents, and generalized dimensions. Experimental Section The apparatus and materials are the same as those used in previous experiments’ except for the volume ratio. The coupled CSTR apparatus consists of two vertically stacked reactors with a common wall bearing an aperture. Reagent solutions flow into the lower reactor (VI = 4.5 mL). The reaction mixture from VI then enters the upper reactor (Vz = 4.5 mL) through the aperture. Both convection and diffusion occur between the two coupled reactors through the aperture. The rate of convection was determined in the absence of flow according to a procedure described previously.’ The outflow occurs from V2. The upper reactor was stirred a t 600 rpm, and the lower reactor was stirred at 900 rpm. The entire apparatus was lowered into a water bath and thermostated a t 28 OC. Thelower reactor in this arrangement is completely submerged in the water bath, making it inaccessible to a redox electrode, and thereby precluding direct observation of the dynamical behavior in the lower CSTR. A self-designed precision piston pump driven by a stepping motor was used to control the flow of reactants into the lower CSTR. Three gas tight Hamilton syringes (50 mL each) served as the reagent reservoirs. The input concentrations are 0.75 M malonic acid and 0.0025 M cerous sulfate, 0.3 M potassium
* Author to whom correspondence should be addressed. 0022-365419312097-6945$04.00/0
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Figure 1. Schematic representation of the experimental bifurcation diagram (flow rate versus states of the system), where kf refers to the total flow rate through both CSTRs. U I / U Z = 1.0. SS1 = steady state 1, SSZ= steady state 2, PL = phase locking, C = chaos, QP = quasiperiodic oscillation,PI = period 1 oscillation.
bromate, and0.6 M sulfuricacid. Each step of the motor advanced the pistons by 100 nm into the syringes. Virtually constant flow of solutions into the CSTR is ensured by high stepping frequencies from 5 to 85 Hz, corresponding to residence times of 7-120 min. Results and Discussion Starting at values of kf > 36.0 X min-1 and decreasing flow rate, a period doubling bifurcation from PI to PZthen to P4 was observed (Figure 1). At a flow rate of kf = 30.0 X 10-3min-l the system reaches a stationary state (SSI)instead of an expected chaotic state. A further decrease of the flow rate around kf = 25 X min-I leads to the emergence of relaxation oscillations with increasing amplitudes (PI)via a primary Hopf b i f u r c a t i ~ n . ~ ~ A slight decrease of the flow rate from kf = 25.0 X 10-3 min-’ to kf = 23.3 X l t 3 min-I leads to the transition from a periodic (PI) to a quasiperiodic regime (QP) via a secondary Hopf bifurcation.10-12 The time series (not shown) exhibit a modulation of the PI oscillation. The Q P region persists to about kf = 8.3 X 10” min-I if the flow rate is changed continuously in the same direction (upper branch in Figure 1). However, a discontinuous decrease of the flow rate from kf = 23.3 X 10-3 min-1 to kf = 8.3 X min-I brings the system to a steady state (SS2). On reversing (increasing) the flow rate the system passes from SS2 to a phase locking state (PL) at kf = 13.3 X 10-3 min-1. Then it undergoes transitions between bursts and phase locking (C PL) states up to a kfvalue around 18.3 X min-I. A further increase of the flow rate brings the system to the chaotic state (C) at kf = 22.2 X l t 3min-I, then to the quasiperiodic (QP) and periodic ( P I )states on the upper branch a t kfvalues around 23.3 X 10-3-25.0 X 10-3 min-I. This series of transitions exhibits the phenomenon of hysteresis. Chaos on a Torus. A detailed analysis of the time series (Figure 6 in ref 1) at kf = 22.2 X min-1 gives the following information: (1) The attractor (Figure 2a) is a “strange” attractor, and the Poincarisection (Figure 2b) and the one dimensional map (Figure 2c) indicate the presence of folds on the surface of the torus. This process of stretching and folding on the torus surface is an indication of the fractal13 nature of the process.
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0 1993 American Chemical Society
Letters
The Journal of Physical Chemistry, Vol. 97, No. 27, 1993
6946
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spectrum (Figure 3b). The frequencies are commensurate (@I/ w2 = l / ~ ) which , is indicative of a phase-locking motion.
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F i p e 2. (a, top) "Strange" attractor (state C at kf= 22.2 X l t 3m i d ) the first three SVD dimensions are plotted ( x y , ~ )(b, ; middle) Poincard section ( x versus y ) ; (c, bottom) return map ( x , versus x,,+l).
(2) The D, spectrum calculated using the near-neighbor analysis
Thus extending the volume ratio leads to the observation of a transition from quasiperiodicity to chaos. The observed hysteresis behavior supports the existence of bistability. The presence of a secondary Hopf bifurcation leading to quasiperiodicity is supported by the introduction of an incommensurate frequency in the Fourier spectrum. The D, values are nearly 2.0 (torus P) and the set of the Lyapunov exponents is (O,O,-). Furthermore, the PoincarC section and the one-dimensional map are clearly closed curves with smooth elliptical shapes. This is a strong indication that the dynamics occurs on a torus. The observation of phase locked state provides additional indication that a breakup of the torus has occurred. The Fourier spectrum shows commensurate frequencies in this case. In general, the transitions from quasiperiodicity to chaos and then to phase locking can be described according to the following scenario: (1) The torus arises via a secondary Hopf bifurcation. (2) The form of the torus is distorted (Figure 2a) and the dimension of the a t t r a ~ t o r ~ ~increases. J*J~ This is well correlated with an increase of the maximum Lyapunov exponent which becomes positive. (3) As process (2) continues, the region of phase locking increases. All processes (1-3) have been observed in the present experiment. Acknowledgment. This work was supported by the Deutscher Akademischer Austauschdienst (S.I.D.), the Stiftung Volkswagenwerk and the Fonds der Chemischen Ind. We thank A. F. Miinster for helpful discussion.
(NNA)method1&17also shows values of the Hausdorff dimension DO= 2.45, theinformationdimension D1= 2.33 and the correlation dimension4 = 2.28 that arecharacteristicofa chaoticattractor. There is a positive value in the set of the Lyapunov exponents (not shown), which is another indication of a chaotic motion. Phase-Locking State. Figure 3a shows the observed phase lockingstateat kf= 13.3 X 10-3min-l. There aretwo frequencies at WI = 1.5 X I t 2 Hz and w2 = 4.45 X H z in its Fourier
References and Notes (1) Doumbouya, S.I.; MBnster, A. F.; Doona, C. J.; Schneider, F. W. J . Phys. Chem. 1993, 97, 1025. (2) Richetti, P.; Roux, J. C.; Argoul, F.;Ameodo, A. J . Chem. Phys. 1987.. 86.. 3339. (3) Argoul, F.; Ameodo, A.; Richetti, P.; Roux, J. J . Chem. Phys. 1987, 86, 3325. (4) Kaneko, K. Prog. Theor. Phys. 1983, 69, 1427,
Letters (5) Guckenheimer, J.; Holmes, P. Nonlinear Oscillations, Dynamical Systems and Bifurcations of Yectors Fields; Springer: Berlin, 1984 and references therein. (6) Marsden, J. E.; Mccracken, M. HopfBifurcation andits Application; Applied Math. Sciences 19; Springer: New York, 1976. (7) Arnold, V. I. Supplementary Chapter to the Theory of Differential Equations; Nauka: Moscow, 1978 and references therein. (8) Ioos, G.; Joseph, D. D. Elementary Stability and Bifurcation Theory; Springer: New York, 1980. (9) Coullet, P. H. Phys. Rep. 1984, 95, 103. (10) Langford, W. F. Nonlinear Dynamics and Turbulence; Ioos, G., Joseph, D. D., Eds.; Pitman: New York.
The Journal of Physical Chemistry, Vol. 97, No. 27, 1993 6947 (1 1) Ioos, G. Bifurcation of Maps and Application; North-Holland Amsterdam, 1979. (12) Ioos,G.;Langford, W. F. Nonlinear Dynumics; Helleman, R. H. G., Ed.; Academy of Sciences: New York, 1980; p 149. (1 3) Mandelbrot, B. B. The Fractal Geometry of Nature; Freeman: San Francisco, 1982; pp 193-198. (14) Renyi, A. Probability Theory; North-Holland: Amsterdam, 1970. (15) Cenys, A.; Pyraga, K. Phys. Lett. 1988,12BA, 227. (16) Grasabcrger, P. Phys. Lett. 1985, 107A, 101. (17) Eckmann, J. P.; Ruelle, D. Rcv. Mod. Phys. 1985, 57, 617. (18) Mori, H. Prog. Theor. Phys. 1980, 63, 1044. (19) Kaplan, J.; Yorke, J. Springer Lecture Notes Math. 1979,730,204.
