The chemical mechanism of Field, Körös, and Noyes for the oscillatory Belousov reaction has been generalized by a model composed of five steps involving three independent chemical intermediates. The behavior of the resulting differential equations has been examined numerically, and it has been shown that the system traces a stable closed trajectory in three dimensional phase space. The same trajectory is attained from other phase points and even from the point corresponding to steady state solution of the differential equations. The model appears to exhibit limit cycle behavior. By stiffly coupling the concentrations of two of the intermediates, the limit cycle model can be simplified to a system described by two independent variables; this coupled system is amenable to analysis by theoretical techniques already developed for such systems.

1.
A. J.
Lotka
,
J. Am. Chem. Soc.
42
,
1595
(
1920
).
2.
N. Minorsky, Nonlinear Oscillations (Van Nostrand, Princeton, NJ, 1962).
3.
P. Glandsdorff and I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations (Wiley, New York, 1971).
4.
G.
Nicolis
,
Advan. Chem. Phys.
19
,
209
(
1971
).
5.
P. J.
Ortoleva
and
J.
Ross
,
J. Chem. Phys.
55
,
4378
(
1971
);
P. J.
Ortoleva
and
J.
Ross
,
56
,
287
,
293
(
1972
).,
J. Chem. Phys.
R. G. Gilbert, H. Hahn, P. J. Ortoleva, and J. Ross, “Chemical Oscillations and Multiple Steady States Due to Variable Boundary Permeability” (to be published).
A. Nitzan and J. Ross, “ Oscillations, Multiple Steady States and Instabilities in Illuminated Systems” (to be published).
6.
A. M.
Turing
,
Phil. Trans. R. Soc. Lond. B
273
,
37
(
1952
).
7.
J. J.
Tyson
,
J. Chem. Phys.
58
,
3919
(
1973
).
8.
H.
Degn
,
J. Chem. Ed.
49
,
302
(
1972
).
9.
G.
Nicolis
and
J.
Portnow
,
Chem. Rev.
73
,
365
(
1973
).
10.
Some substitutions are possible for the cerium ion and malonic acid, but the requirement for bromate ion is specific. The sulfuric acid is about 1M. See Ref. 13 for further details.
11.
B. P. Belousov, Sb. Ref. Radiats. Med., 1958, Medgiz, Moscow, 145 (1959).
12.
A. M. Zhabotinskii, Oscillatory Processes in Biological and Chemical Systems (Science Publ., Moscow, 1967), p. 149;
A. N.
Zaikin
and
A. M.
Zhabotinsky
,
Nature
225
,
535
(
1970
).
13.
R. J.
Field
,
E.
Körös
, and
R. M.
Noyes
,
J. Am. Chem. Soc.
94
,
8649
(
1972
).
14.
R. J.
Field
and
R. M.
Noyes
,
Nature
237
,
390
(
1972
). A quantitative model is in preparation.
15.
R. M. Mazo, University of Oregon (unpublished results).
16.
C. F.
Curtiss
and
J. O.
Hirschfelder
,
Proc. Natl. Acad. Sci. (USA)
38
,
235
(
1952
).
17.
J. O.
Hirschfelder
,
J. Chem. Phys.
26
,
271
(
1957
).
18.
D.
Edelson
,
J. Comput. Phys.
11
,
455
(
1973
).
19.
R. J.
Gelinas
,
J. Comput. Phys.
9
,
222
(
1972
).
20.
C. W. Gear, Numerical Initial Value Problems in Ordinary Differential Equations (Prentice‐Hall, Englewood Cliffs, NJ, 1971), p. 209ff.
21.
A. C. Hindmarsh, Lawrence Livermore Laboratory, Livermore, California 94550.
22.
J. J. Jwo, University of Oregon (unpublished results).
23.
A. T.
Winfree
,
Science
175
,
634
(
1972
).
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