Quasiperiodic to bursting oscillations transition in the model of the Belousov¡À Zhabotinsky reaction Oleg V. Noskov,a Aleksandr D. Karavaev,*a Valery P. Kazakova and Semen I. Spivakb aInstitute of Organic Chemistry, Ufa Scientific Centre of the Russian Academy of Sciences, 450054 Ufa, Russian Federation. Fax: +7 347 235 6066; e-mail: lang@chemorg.bashkiria.su bDepartment of Mathematics, Bashkirian State University, 450074 Ufa, Russian Federation.Complex dynamics of the BZ-oscillator: the transition from quasiperiodic to bursting regimes was simulated based on an eleven- - stage reaction scheme; the transition was realized in two forms: (i) smooth deformation of the T T2-tore due to its interaction with h an unstable stationary point; (ii) a sharp change over between the regimes.. Experimental studies of the auto-oscillation Belousov¡ÀZhabo- tinsky (BZ) reaction have revealed a variety of dynamics: from simple quasisinusoidal oscillations to quasiperiodic, bursting, complex periodic and various chaotic ones.1¡À4 Some types of these regimes were reproduced with more or less success in various models1,5,6 based on oversimplified schemes.However, studies of the finer effects (in particular transitions between regimes) were faced with some difficulties. Thus6 quasiperiodic and mixed-mode oscillations were simulated and studied in detail, but the continuous transition between themwas not seen. Recently7,8 we showed that, based on the detailed eleven-stage scheme of Ruoff and Noyes, not only can all known types of BZ oscillations be modelled but the experimentally observable alternation order could also be reproduced: SS¡¦QS¡¦QP¡¦B¡¦CP¡¦B¡¦QP¡¦QS¡¦SS; where SS= stationary state; QS= quasisinusoidal oscilla- tions of low amplitudes; QP= quasiperiodic regimes; B = bursting oscillations; CP= a complex succession of chaotic and periodic regimes. This allowed us to study the mechanisms of the BZ dynamic transformations in more detail.This paper concerns the transition from quasiperiodicity to bursting. The reaction scheme under consideration9 is as follows: BrO73 +Br7+2H+! HBrO2+HOBr (1) HBrO2+Br7 + H+ !2HOBr (2) BrO73 +HBrO2 + H+! 2BrO . 2+H2O (3) BrO . 2 +M(n¡À1)+ + H+! HBrO2 +Mn+ (4) 2HBrO2! BrO73 +HOBr + H+ (5) HOBr+Br7 + H+! Br2 + H2O (6) RH+Br2!RBr +Br7 + H+ (7) HOBr+R .!ROH+Br . (8) RH+Br . !Br7 + H+ + R . (9) RH+Mn+!M(n¡À1)+ + H+ + R . (10) 2R . + H2O !RH+ROH (11) where RH stands for malonic acid, and Mn+ for metal ions. As in refs. 7¡À9, modification of the closed system with permanent the initial concentrations of BrO73 and RH were calculated which allow one to obtain stationary regimes. Calculations were carried out by the m,k-method10 with a relative precision of er=1075¡À1076 and an integration step of 0.1¡À1 s.The constant k8 was used as a variable parameter which determines the rate of Br7 formation. The other parameters used are as follows:8 k1=2.1 dm9 mol73 s71; k71=1104 dm3 mol71 s71; k2=3106 dm6 mol72 s71; k3=42 dm6 mol72 s71; k73=4.2107 dm3 mol71 s71; k4=8104 dm6 mol72 s71; k74=8.9103 dm3 mol71 s71; k5=3103 dm3 mol71 s71; k6=8109 dm6 mol72 s71; k76=110 s71; k7=4.61073 dm3 mol71 s71; k8=106 ¡À107 dm3 mol71 s71; k9=106 dm3 mol71 s71; k10=0.2 dm3 mol71 s71; k11=3.2109 dm3 mol71 s71; [Br7]0=1075 mol dm73; [Mn+]0=51074 mol dm73; constants: [BrO73 ]=0.08 mol dm73; [RH]=0.2 mol dm73; [H+]=1 mol dm73.There are two areas of each kind of oscillation's existence in the model, corresponding to the oxidized (with low k8 values) and reduced (higher k8 values) form of the catalyst.Accordingly, two areas of quasiperiodic oscillations (with toroidal attractor D T2-tore11 D in a phase space) and two forms of transition to bursting oscillationsD`mild' and `rigid' are observed. Figures 1 and 2 describe the first case. Quasiperiodic regimes, arising in the model as a result of secondary Hopf bifurcation, are considered to be two-frequency and look like high frequency oscillations modulated by a low frequency component on the kinetic curves.Moving fromthe bifurcation point (when k8 increases) the amplitude of a second frequency grows which leads to an increase in the modulation depth (Figure 1). Thus, phase portraits demonstrate a decrease of the T2-tore internal diameter.Simultaneously, the tore increases and approaches an unstable saddle focus stationary point (Figure 2). This evolution is caused by their interaction (`attraction' of phase trajectories by a stable manifold of the saddle focus and `rejection' by a two-dimensional unstable one). The approach of the phase point to the stationary point is accompanied by a slowing down of its movement, and it begins to spend more and more time in the vicinity of the latter.Subsequently, sloping areas are observed on kinetic curves whose length rises as k8 increases up to the appearance of pronounced bursting oscillations (Figure 3). This process coincides with the formation of an homoclinic structure12 in the system, causing complex organization of phase space and chaotic character of bursting attractors (Figure 3c). The continuous character of the quasiperiodic regimes evolution described above is in good agreement with experimental observations.4 Thus, the left hand side of the diagram shows a smooth transition without abrupt changes of the attractor's structure.Similar T2-tore evolution was observed previously in simpler model.6 However, as the authors of ref. 6 noted, they could not monitor the transition to bursting oscillations due to calculation difficulties. Mendeleev Communications Electronic Version, Issue 1, 1997 (pp. 1�C46)In the right hand side of the diagram this transition is very sharp. The quasiperiodic and bursting regimes in Figure 4 were obtained with k8 values differing by less than 1078%.Probably, these oscillating states coexist in a range of k8 values, and when the parameter changes the sharp transition from one regime to another occurs. The existence area for quasiperiodic regimes was calculated to be much less (by 1±2 orders of magnitude) than that for bursting oscillations. Apparently, this is why the direct transition from bursting to stationary state, by passing quasiperiodicity, is often observed in experiments.3,4 [Br2]103 [Br7]105 (a) (c) . 0 1.790 1.7835 1.771 0.087 0.068 . 0 1.790 1.775 1.771 0.105 0.06 [Br2]103 [Br7]105 1.790 1.790 .0 1.754 1.772 0.105 0.059 [Br2]103 [Br7]105 .0 1.754 0.092 0.065 [Br2]103 [Br7]105 (b) (d) Figure 2 Tore transformation during `mild' transition to bursting oscillations.k81076=2.117 (a); 2.118 (b); 2.119 (c); 2.120 (d) dm3 mol71 s71. 0 is an unstable stationary point (saddle focus). 1.0 1.05 1.1 0.06 0.08 0.1 [Br7]105 t1075 / s (c) 1.0 1.05 1.1 0.06 0.075 0.09 [Br7]105 t1075 / s (a) 1.0 1.1 1.2 0.06 0.075 0.09 [Br7]105 t1075 / s (b) (d) 1.0 1.1 1.2 0.05 0.08 0.11 [Br7]105 t1075 / s Figure 1 Kinetic curves of quasiperiodic oscillations.k81076=2.117 (a); 2.118 (b); 2.119 (c); 2.120 (d) dm3 mol71 s71. Mendeleev Communications Electronic Version, Issue 1, 1997 (pp. 1–46)0.14 0.22 0.06 6.0 4.0 [Br7]105 t1074/s (a) (b) [Br7] [Br2] [Mn+] 0.14 0.22 [Br7]105 0.06 7.0 6.0 t1074/s (d) (c) Figure 4 `Rigid' transition between bursting [(a), (b), k8=7.370306040106 dm3 mol71 s71] and quasiperiodic [(c), (d), k8=7.370306041106 dm3 mol71 s71] regimes.(a), (c) kinetic curves; (b), (d) phase portraits. 1.78 1.72 0.0930 0.048 [Br2]103 [Mn+]103 (c) 0.8 0.04 0.095 0.15 t1075 / s 1.0 (a) [Br7]105 1.784 1.715 0.15 0.05 [Br2]103 [Br7]105 1.7495 0 (b) Figure 3 Kinetics (a), phase portrait (b), and Poincare section (c) of bursting oscillations. (k8=2.14106 dm3 mol71 s71 , 0 is unstable stationary point).Mendeleev Communications Electronic Version, Issue 1, 1997 (pp. 1–46)This research was supported by a grant from the Russian Foundation for Basic Research (no. 93-03-18165). References 1 Kolebaniya i begushchie volny v khimicheskikh sistemakh (Oscillations and traveli waves in chemical systems), eds. R. 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Shil'nikov, Dokl. Akad. Nauk SSSR, 1965, 160, 1261 (in Russian). Received: Moscow, 10th January 1996 Cambridge, 22nd April 1996; Com. 6/00312E Mendeleev Communications Electronic Version, Issue 1, 1997 (pp. 1–46)