Oregonator

• Dr. Richard J. Field, Department of Chemistry, The University of Montana, Missoula, MT 59812

Figure 1: Limit cycle attractor in the oregonator model.

The Oregonator (Orygunator) is the simplest realistic model of the chemical dynamics of the oscillatory (Zhabotinsky, 1991; Gray and Scott, 1991; Epstein and Pojman, 1998) Belousov-Zhabotinsky (BZ) reaction. It was devised by Field and Noyes (1974) working at the University of Oregon and is composed of five coupled elementary chemical stoichiometries. This network is obtained by reduction of the complex chemical mechanism of the BZ reaction suggested by Field, Korös and Noyes (1974) and referred to as the FKN mechanism. Reduction is accomplished by application of standard methods of chemical kinetics, especially the rate-determining-step approximation (Espenson, 1995).

Form

The underlying Oregonator dynamics is an activator/inhibitor system (Epstein and Pojman, 1998) containing both an autocatalytic step and a delayed negative feedback loop. It is composed of the five coupled stoichiometries shown below, together with the mass-action rate expression for each step.

\(A + Y \rightarrow X + P\)	Rate \(= k_1AY\)	\(k_1 = k_{R3}\)	[H+]2	(a)
\(X + Y \rightarrow 2P\)	Rate \(= k_2XY\)	\(k_2 = k_{R2}\)	[H+]	(b)
\(A + X \rightarrow 2X + 2Z\)	Rate \(= k_3AX\)	\(k_3 = k_{R5}\)	[H+]	(c)
\(X + X \rightarrow A + P\)	Rate \(=k_4X^2\)	\(k_4 = k_{R4}\)	[H+]	(d)
\(B + Z \rightarrow 1/2f Y\)	Rate \(= k_cBZ\)	\(k_c\)		(e)

Oregonator species concentrations and parameters are italicized in Eqs. a–e. Species identifications with respect to the FKN mechanism are X = HBrO₂, Y = Br^-, Z = Ce(IV), A = BrO₃^-, B = CH₂(COOH)₂, and P = HOBr or BrCH(COOH)₂. The reactant and product species A, B and P are normally present in much higher concentrations than the dynamic intermediate species X, Y and Z and are assumed to be constant on the time scale of a few oscillations. Steps a–d correspond to reactions \(R2 - R5\) in the FKN mechanism given in Table 1. Step e (C in Table 1) is a net stoichiometry where \(f\) is an adjustable stiochiometric factor, and \(k_c\) is an adjustable rate parameter. The quantities \(k_1 - k_4\) are the actual, experimentally derived (Tyson, 1985; Field and Försterling, 1986), FKN forward rate constants \(k_{R2} - k_{R5}\) (Table 1) adjusted for the acidity of the medium. The rate parameter \(k_c\) is scaled by \(B\) in the dynamic equation in order to adjust for the total concentration of organic material, CH₂(COOH)₂ plus BrCH(COOH)₂, present. The activator species is X, and the inhibitor species is Z. The inhibition process (negative feedback) occurs via the sequence, Step c\(\rightarrow\)Step e\(\rightarrow\)Step b, which inhibits autocatalytic production of X in Step c. Various more complex forms of the Oregonator have been developed in order to better represent the FKN chemistry or to understand particular observed behaviors of the BZ reaction.

Dynamic Equations

The Oregonator Mass-Action dynamics in a well-stirred, homogeneous system is given by Eqs. (1)–(3). \[\tag{1} dX/dt = k_1AY - k_2XY + k_3AX - 2k_4X^2\]

\[\tag{2} dY/dt = -k_1AY - k_2XY + 1/2k_c f BZ\]

\[\tag{3} dZ/dt = 2k_3AX - k_cBZ\]

Eqs. (1)–(3) are typically scaled (Tyson, 1985; Scott, 1994) as Eqs. (4)–(6). \[\tag{4} \epsilon(dx/d\tau) = qy - xy +x(1 - x)\]

\[\tag{5} \epsilon'(dy/d\tau) = -qy -xy +fz\]

\[\tag{6} dz/d\tau = x - z\]

The scaling relationships and parameters in Eqs. (4)–(6) are \(x = 2k_4X/(k_3A)\ ,\) \(y = k_2Y/(k_3A)\ ,\) \(z = k_ck_4BZ/(k_3A)^2\ ,\) \(\tau = k_cBt\ ,\) \(\epsilon = k_cB/(k_3A) = 9.90 \times 10^{-3}\ ,\) \(\epsilon' = 2k_ck_4B/(k_2k_3A) = 1.98 \times 10^{-5}\ ,\) \(q = 2k_1k_4/(k_2k_3) = 7.62 \times 10^{-5}\) for the rate constant values defined in Step a–Step d and with \(A = 0.06 M, B = 0.02 M\ ,\) \([H^+\)]\( = 0.8 M\ ,\) and \(k_c = 1 M^{-1}s^{-1}\ .\) Equations (4)–(6) are sometimes referred to as the Field-Noyes equations.

Oscillation

Figure 2. Numerical integration of Eqs. (4)–(6) for the above parameter values. \(A\) = 0.06 M, \(B\) = 0.02 M. \(k_1 = 1.28 M^{-1}s^{-1}\ ,\) \(k_2 = 2.4\times 10^6 M^{-1}s^{-1}\ ,\) \(k_3 = 33.6 M^{-1}s^{-1}\ ,\) \(k_4 = 2400 M^{-1}s^{-1}\ .\) \(k_c = 1 M^{-1}s^{-1}\) and \(f = 1\ .\) The values of \(k_1-k_4\) are calculated from the values of \(k_{R2}, k_{R3}, k_{R4}\ ,\) and \(k_{R5}\) in Table 1 with [H⁺] = 0.8 M. Initial conditions are \(x_0=y_0=z_0=1\ .\)

Simple Modifications

The relatively small value of \(\epsilon\)' compared to \(\epsilon\) and to one allows the steady-state approximation (Tyson and Fife, 1980; Espenson, 1995) to be made for \(y\) in Eq. (5), yielding

\[\tag{7} y = y_{ss} = fz/(q + x)\]

and the dynamic equations

\[\tag{8} \epsilon(dx/d\tau) = x(1 - x) - fz(x - q)/(q + x)\]

\[\tag{9} dz/d\tau = x - z.\]

It is possible to make the steady-state approximation for x rather than y, especially using the original Field and Noyes (1974) scaling, but the result is not of such convenient form as Eqs. (8) and (9). The Field-Noyes scaling does emphasize the role of x (Br^-) as the "control variable" switching the system back and forth between activator and inhibitor dominated states. Numerical agreement between Eqs. (4)–(6) and Eqs. (8)–(9) is excellent. Equations (8)–(9) are the simplest form of the Oregonator dynamics and are often used in analytical work or in computationally demanding applications such as the reaction-diffusion equations used to describe spatial phenomena, including traveling waves and stationary patterns (particularly in two or three spatial dimensions) observed in the unstirred BZ reaction.

Introduction of the cerium-ion mass balance, \(C_0 =\) [Ce(III)] + [Ce(IV)], into Eqs. (4) and (6) controls unlimited growth of \(z\) and leads to Eqs. (10)–(12) with \(c_0 = C_0k_ck_4B/(k_3A)^2\ .\)

\[\tag{10} \epsilon(dx/d\tau) = qy -xy + (1 - z/c_0)x - x^2\]

\[\tag{11} \epsilon'(dy/d\tau) = -qy - xy + fz\]

\[\tag{12} dz/d\tau = (1 - z/c_0)x - z\]

Only about 20% of Ce(III) is oxidized to Ce(IV) in Eqs. (10)–(12) for the parameters in Fig.1, as is observed experimentally.

A reversible Oregonator was suggested by Field (1975) in order to investigate how distance from chemical equilibrium affects its behavior. The oscillations are a far-from-equilibrium phenomena (Nicolis and Prigogine, 1977). A modification due to Showalter et al.(1978) is often used to better represent the FKN chemistry and observed complexity of the BZ reaction.

Bifurcations

Reasonable BZ/FKN acidities lie in the range [H⁺] ~ (0.1–2) M, for which chemically reasonable ranges of the expendable variables \(k_c\) and \(f\) are ~ (0.1–10) M^-1s^-1 and 0–3, respectively. The concentrations A and B may vary through the chemically realistic range of ~ (0.01–1) M. The effective rate in Step c (activation) is given by \(k_3AX\) and in Step e (inhibition) by \(k_cBZ\ .\)

There are two real steady states of Eqs. (4)–(6), \([x_{ss},y_{ss},z_{ss}]\ ,\) throughout the above parameter ranges, one being positive and the other negative.

Linear stability analysis about the chemically acceptable positive state for the parameters in Fig.1 shows in Fig. 3 a roughly triangular area of instability in \([k_c,f]\ .\) As \(k_c \rightarrow 0\) the instability range approaches 0.5 < f < 1 + 2^1/2. The trajectory in Fig. 1 originates at the unstable steady state and rapidly approaches the limit cycle.

The borders of the region of instability are Hopf bifurcations where the real part of a pair of complex-conjugate eigenvalues passes through zero. The third eigenvalue is always negative. The unstable steady state is surrounded by a strongly attracting, large-amplitude limit cycle. The Oregonator numerics and the real BZ chemistry are both so sensitive in the vicinity of the bifurcation lines that it is difficult to determine details of the complex behavior occurring there (Brons and Bar-Eli, 1991; Mazzoti et al., 1995). Most mathematical work has been done using the the reduced Oregonator, Eqs. (8) and (9). The Hopf bifurcations are found to be subcritical or supercritical, largely depending upon A/B, a measure of the relative rates of the activator and inhibitor processes. Often the low-f Hopf is subcritical while the high-f Hopf is supercritical, although for some parameter values, both bifurcations are super or subcritical. Canard explosions occur near to the supercritical Hopfs. In all cases, growth of the period and amplitude of the limit cycle is very sharp near to the Hopf bifurcation lines, in keeping with the observation that the BZ oscillations appear full blown at low-f and disappear abruptly from large-amplitude at high-f. In keeping with the presence of subcritical and supercritical Hopfs associated with a canard explosion, the Oregonator is excitable (Alonso et al.,2006) and shows bursting near to the bifurcation borders (Janz et al., 1980).

CSTR/Bistability/Chaos

The basic chemistry of the BZ oscillations involves jumps between high and low [HBrO₂] (\(x\)) states, which is reflected in the relaxation oscillator nature of the Oregonator (Fig. 2). This fundamental bistability may be stabilized in a flow reactor (CSTR) with reactants and Br^- in the feed stream. Hysteresis between the two states is observed both experimentally and in the Oregonator (Ruoff and Noyes, 1986; Gaspar et al.,1985). Quasiperiodicity and chaos also are observed in CSTR (Turner et al.,1981) and can be modeled by the Oregonator (Richetti et al.,1987) with a reversible Step 3 and an expanded form of Step 5 (Györgyi et al., 1991).

Reaction-Diffusion

Traveling waves of chemical activation (high \(x\)) are observed in the BZ reaction, especially when ferroin is used as the metal-ion catalyst, and the chemical reagent is spread in a thin layer in contact with air (Zaikin and Zhabotinsky, 1970). The Oregonator reaction-diffusion equations support traveling wave solutions (Armstrong et al.,2004) when D_x ~ D_z. Recall that \(x\) is the activator species and \(z\) is the inhibitor species. Turing patterns appear in the Oregonator when D_z > D_x (Becker and Field, 1985) but have not yet been observed in the BZ reaction itself.

References

• S. Alonso, F. Sagues, A. S. Mikhailov, Negative tension instability of scroll waves and Winfree turbulence in the Oregonator model, J. Phys. Chem. A 110(2006)12063-71.
• G. R. Armstrong, A. Taylor, S. K. Scott, V. Gaspar, Modelling wave propagation across a series of gaps, Phys. Chem. Chem. Phys. 6(2004)4677-4681.
• P. Becker, R. J. Field, Stationary concentration patterns in the Oregonator model of the Belousov-Zhabotinsky reaction, J. Phys. Chem. 89(1985)118-28.
• M. Brons, K. Bar-Eli, Canard explosion and excitation in a model of the Belousov-Zhabotinsky reaction, J. Phys. Chem. 95(1991)8706-13.
• I. R. Epstein, J. A. Pojman, An introduction to nonlinear chemical dynamics (Topics in Physical Chemistry), Oxford University Press, New York, 1998.
• J. H. Espenson, Chemical kinetics and reaction mechanisms, 2nd. Ed., McGraw-Hill, Inc. New York, 1995.
• R. J. Field, Limit cycle oscillations in the reversible Oregonator, J. Chem. Phys. 63(1975)2289-96.
• R. J. Field, H.-D. Försterling, On the oxybromine chemistry rate constants with cerium ions in the Field-Korös-Noyes mechanism of the Belousov-Zhabotinsky reaction: The equilibrium HBrO₂ + BrO₃^- <-->2BrO₂ + H₂O, J. Phys. Chem. 90(1986)5400-07.
• R. J. Field, R. M. Noyes, Oscillations in Chemical Systems IV. Limit cycle behavior in a model of a real chemical reaction, J. Chem. Phys. 60(1974)1877-84.
• R. J. Field, E. Korös, R. M. Noyes, Oscillations in Chemical Systems II. Thorough analysis of temporal oscillations in the Ce-BrO₃^- - malonic acid system, J. Am. Chem. Soc. 94(1972)8649-64.
• V. Gaspar, G. Bazsa, M. T. Beck, Bistability and bromide controlled oscillation during bromate oxidation of ferroin in a continuous-flow stirred tank reactor, J. Phys. Chem. 89(1985)5495-9.
• P. Gray, S. K. Scott, Chemical oscillations and instabilities: Nonlinear chemical kinetics, Oxford University Press, New York, 1990.
• L. Györgyi, S. L. Rempe, R. J. Field, A novel model for the simulation of chaos in low-flow-rate experiments with the Belousov-Zhabotinsky reaction: A chemical mechanism for two-frequency oscillations, J. Phys. Chem. 95(1991)3159-65.
• R. D. Janz, D. J. Vanecek, R. J. Field, Composite Double Oscillations in a modified version of the Oregonator model of the Belousov-Zhabotinsky reaction, J. Chem. Phys. 73(1980)3132-8.
• M. Mazzotti, M. Morbidelli, G. Sarraville, Bifurcation analysis of the Oregonator model in the 3-D space bromate/malonic acid/stoichiometric coefficient, J. Phys. Chem. 99(1995)4501-11.
• G. Nicolis, I. Prigogine, Self-organization in nonequilibrium systems, John Wiley & Sons, Inc., New York, 1977.
• P. Richetti, J.-C. Roux, F. Argoul, A. Arneodo, From quasiperiodicity to chaos in the Belousov-Zhabotinsky reaction. II Modeling and theory, J. Chem. Phys. 86(1987)3339-57.
• P. Ruoff, R. M. Noyes, An amplified oregonator model simulating alternate excitabilities, transitions in types of oscillation, and temporary bistability in a closed system, J. Chem. Phys. 84(1986)1413-23.
• S. K. Scott, Oscillations, waves and chaos in chemical kinetics, Oxford University Press, New York, 1995.
• K. Showalter, R. M. Noyes, K. Bar-Eli, A modified Oregonator model exhibiting complicated limit cycle behavior in a flow system, J. Chem. Phys. 69(1978)2514-24.
• J. Turner, J.-C. Roux, W. D. McCormick, H. L. Swinney, Alternating periodic and chaotic regimes in a chemical reaction - experiments and theory, Physics Letters A, 85A(1981)9-12.
• J. J. Tyson; P. C. Fife, Target patterns in a realistic model of the Belousov-Zhabotinskii reaction, J. Chem. Phys., 73(1980)2224-37.
• J. J. Tyson, A quantitative account of oscillations, bistability, and traveling waves in the Belousov-Zhabotinsky reaction, In Oscillations and Traveling Waves in Chemical systems, R. J. Field, M. Burger, Eds., John Wiley & Sons, New York, 1985.
• A. N. Zaikin, A. M. Zhabotinsky, Concentration wave propagation in a two-dimensional, liquid-phase self oscillating system, Nature, 225(1970)535-7.
• A. M. Zhabotinsky, A history of chemical oscillations and waves, Chaos, 1(1991)379-86.

Internal references

• Yuri A. Kuznetsov (2006) Andronov-Hopf bifurcation. Scholarpedia, 1(10):1858.
• Anatol M. Zhabotinsky (2007) Belousov-Zhabotinsky reaction. Scholarpedia, 2(9):1435.
• John Guckenheimer (2007) Bifurcation. Scholarpedia, 2(6):1517.
• Eugene M. Izhikevich (2006) Bursting. Scholarpedia, 1(3):1300.
• Martin Wechselberger (2007) Canards. Scholarpedia, 2(4):1356.
• Olaf Sporns (2007) Complexity. Scholarpedia, 2(10):1623.
• James Meiss (2007) Dynamical systems. Scholarpedia, 2(2):1629.
• Eugene M. Izhikevich and Richard FitzHugh (2006) FitzHugh-Nagumo model. Scholarpedia, 1(9):1349.
• Peter Jonas and Gyorgy Buzsaki (2007) Neural inhibition. Scholarpedia, 2(9):3286.
• Jeff Moehlis, Kresimir Josic, Eric T. Shea-Brown (2006) Periodic orbit. Scholarpedia, 1(7):1358.
• Anatoly M. Samoilenko (2007) Quasiperiodic oscillations. Scholarpedia, 2(5):1783.
• Gregoire Nicolis and Anne De Wit (2007) Reaction-diffusion systems. Scholarpedia, 2(9):1475.
• Philip Holmes and Eric T. Shea-Brown (2006) Stability. Scholarpedia, 1(10):1838.

See Also

Belousov-Zhabotinsky Reaction, Brusselator, Excitable Media, FitzHugh-Nagumo Model, Reaction-Diffusion Systems, Traveling Waves

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Accepted on: 2007-05-04 20:57:42 GMT