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Spontaneous oscillations in two 2-component cells coupled by diffusion. (English) Zbl 0588.92002
Mathematical examples are presented of oscillators with two variables which oscillate stably when coupled with a twin via diffusion. The mathematical method is a type of singular perturbation theory combined with bifurcation theory.
It is shown that all stationary solutions are unstable for appropriate parameter settings. Numerical examples are presented.
Reviewer: C.Drugarin

92C05 Biophysics
92B05 General biology and biomathematics
34D15 Singular perturbations of ordinary differential equations
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
92F05 Other natural sciences (mathematical treatment)
Full Text: DOI
[1] Alexander, J. C., Yorke, J. A.: Global bifurcation of periodic orbits. Am. J. Math. 100 263-292 (1978) · Zbl 0386.34040
[2] shkenazi, M., Othmer, H. G.: Spatial patterns in coupled biochemical oscillators, J. Math. Biol. 5, 305-350 (1978) · Zbl 0381.92006
[3] Hassard, B. D., Kazarinofi, N. D., Wan, V. H.: Theory and applications of Hopf bifurcation. Cambridge Univ. Press 1981 · Zbl 0474.34002
[4] Howard, L. N.: Nonlinear oscillations, In: Hoppensteadt, F. R. (ed.) Nonlinear oscillations in biology. American Mathematical Society Lectures in Applied Mathematics 17, 1-69 (1979)
[5] Lefever, R., Prigogine, I.: Symmetry-breaking instabilities in dissipative systems II. J. Chem. Phys. 48, 1695-1700 (1968)
[6] Smale, S.: A mathematical model of two cells via Turing’s equation, J. D. Cowan (ed.) Some mathematical questions in biology. V. American Mathematical Society Lectures on Mathematics in the Life Sciences 6, 15-26 (1974) · Zbl 0333.92002
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