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Miyakawa, Kenji

Mizoguchi, Michiko

Okabe, Tadao

Sakamoto, Fumitaka

Kyoto University Research Information Repository

TitleSYNCHRONIZATION AND CHAOS IN THE DISCRETECHEMICAL OSCILLATION SYSTEM(Session I : Cross-Disciplinary Physics, The 1st Tohwa University InternationalMeeting on Statistical Physics Theories, Experiments andComputer Simulations)Author(s)Miyakawa, Kenji; Okabe, Tadao; Mizoguchi, Michiko;Sakamoto, FumitakaCitation物性研究 (1996), 66(3): 376-377Issue Date1996-06-20URL http://hdl.handle.net/2433/95843RightType Departmental Bulletin PaperTextversionpublisherKyoto UniversitySYNCHRONIZATION AND CHAOS IN THE DISCRErE CHEMICAL OSCILLATION SYSTEMKenji Miyakawa, Tadao Okabe, Michiko Mizoguchi and Fumitaka SakamotoDepartment of Applied Physics, Faculty of ScienceFukuoka University, Fukuoka 814-01, JapanFig.2. Phase diagram of the coupling state. The ordinate axis denotesthe natural frequency ratio of faster oscillator to slower one, OJ2/OJ l'and the abscissa axis denotes the. distance between tWO oscillators,corresponding to the coupling strength. Symbols denote therepresentative frequency ratios in the coupled state.1/14/35/43/22/15/23/1500 600o•200III +150100 200 300 400t (s)100d(!Am)a50Fig.I. Light intensity averaged over the fixed area on asingle bead as a function of time, where peaks correspondto the oxidized state.•a431Coupled chemical oscillator systems have been studied to provide insights in understandingimportant features of biological phenomena such as heart contraction and collective oscillations inintracellular calcium concentration. In particular, coupling in the Belousov-Zabotinskii (BZ) reactionsystem has been extensively studied, using two or three continuous stirred tank reactors (CSTR's)[1-5]. Unfortunately, in the case of a CSTR, a extension to the system with many oscillators is notso easy. In this work, we investigate the properties of the coupling in the discrete BZ reactionsystem consisting of cation exchange beads loaded with ferroin. Our attention is particularly focusedon the coupling between two beads in no contactwith each other.The cation exchange beads of about 500 !Am indiameter were loaded with ferroin of a specifiedconcentration. Then those beads were suspendedin the BZ solution placed in a Petri dish, wherethe temperature of BZ solution was maintained at23.0+0.05t. The redox reaction confined tothe surface of the beads was monitored through aCCD camera attached to a microscope andrecorded on a video tape recorder. The changein color was detected as the change in light intensity through an image processor, where the intensitywas obtained by averaging the gray level of the fixed small area on the bead surface, as shown inFig.l. Thus we regarded the cation exchange bead as a chemical oscillator.The coupling is accomplished by a mass diffusion, so that a time delay is inherent in this coupling.Here the coupling strength can be controlled by varying the distance d 'between two beads: the largerd, the smaller the couplingstrength. Here the distance drefers to the spacing betweenbeads, so that d=O means thatbeads are in contact with eachother." In contrast to usualcoupled oscillators such asCSTRs, the coupled frequencywas found to be higher than thatof the faster oscillator. Here theoscillator acting as a pacemakerwas spontaneously determineddepending on the phase of eachoscillator at the beginning ofcoupling. These results suggestthat the synchronization is notattributed to a simple entrainmentof the slower oscillator by thefaster one. Two oscillators mayfall into a new type of coupledstate which is probably self-- 376-1401201000!----2~50~--5~00~--7~50~-"""":1~OOO-::~50 0 100. 200 . 300 400 500FREQUENCY(mHz)organized with some interactions through excitable media.The features of coupled states were investigated as a function of d and 0>2/<01, where <01 and O>zare the natural frequencies of slower and faster oscillators, respectively. It was found that there arethree coupling regimes depending on d: a tight coupling regime, a weak coupling regime and auncoupled regime. In the tight coupling regime ( d <120 flm), two oscillators cause synchronizationwith the fixed phase difference, that is, an entrainment of 1: 1. In the region of d >200 flm, eachoscillator independently oscillates with its own natural frequency, corresponding to the uncoupledregime. At an intermediate distance, the entrainment occasionally occurs and consequently thefrequency ratio takes intermediate values between those in tight coupling and uncoupled regimes,corresponding to the weak coupling regime. The coupling in this regime occasionally occurred, andits state was metastable. Various coupling states were found in the weak coupling regime, dependingon both natural frequencies of two oscillators and d, as shown in Fig.2. The region I corresponds tothe tight coupli~g regime. The boundary curve separating the region I from another region is seen torise rapidly with decreasing d. There may exist the critical distance dc, below which the entrainmentof 1: 1 necessarily occurs, no matter how large the natural frequency ratio. In regions II and III, thefrequency ratio takes n+1/1 with n being an integer. These regions including the region I aredominant in the whole diagram, and those coupling states are quite stable.In region IV which is located in the boundary between stably coupled regions I and II, variousentrainments were observed depending sensitively on d. The coupled frequency ratio takes n+1/nwith n being an integer of not less than 2. Furthermore, the regular coexistence of two frequencyratios n+1/n and n+2/n+1, for example, 2/1 and 3/2, was also observed. In the stable couplingstate, the pacemaker is usually fixed on either of two oscillators. In the coexistence state of twomodes, in contrast, the alternative replacement of the pacemaker occurred. This indicates that such acoupling state is metastable. Such analternative replacement may come from acompensation effect for the weakly frustrated -.oscillators as observed in the coupled system :icdusing CSTR's. On increasing 0>2/<01, the ~boundary region between regions I and IIbecomes narrow, as seen in Fig.2 In thisregion we observed the irregular behaviors inboth interval and height of peak, as shown in .-~Fig.3 (a). The corresponding power ~spectrum is shown in Fig.3(b). Sharp p..peaks and associated harmonics entirelydisappear, while those clearly appear in thepower spectrum of the periodic oscillation,shown in Fig.I. This irregular behavior is Fig.3.Time series (a) and power spectrum (b) in the boundaryprobably chaotic, which may correspond to region between regions I and II in Fig.2.one predicted by Tomita and Kai [6].REFEREN'CES[1] M. Marek and I. Stuchl, Biophys. Chern. 3, 241 '1975).[2] K. Nakajima and Y. Sawada, J. Chern. Phys. 72,2231 (1980).[3] M. F. Crowley and R. J. Field, J. Phys. Chern. 90, 1907 (1986).[4] M. F. Crowley and I. R. Epstein, J. Phys. Chern. 93, 2496 (1989).[5] M. Yoshimoto, K. Yoshikawa and Y. Moli, Physical Review E 47,864 (1993).[6] T. Kai and K. Tomita, Prog. Theor. Phys. 61,54 (1979).- 377-

SYNCHRONIZATION AND CHAOS IN THE DISCRETE CHEMICAL OSCILLATION SYSTEM(Session I : Cross-Disciplinary Physics, The 1st Tohwa University International Meeting on Statistical Physics Theories, Experiments and Computer Simulations)

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SYNCHRONIZATION AND CHAOS IN THE DISCRETE CHEMICAL OSCILLATION SYSTEM(Session I : Cross-Disciplinary Physics, The 1st Tohwa University International Meeting on Statistical Physics Theories, Experiments and Computer Simulations)

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