Abstract
Mathematical examples are presented of oscillators with two variables which do not oscillate in isolation, but which do oscillate stably when coupled with a twin via difiusion. Two examples are presented, the LefeverPrigogine Brusselator and a system used to model glycolytic oscillations. The mathematical method is not the usual bifurcation theory, but rather a type of singular perturbation theory combined with bifurcation theory. For both examples, it is shown that all stationary solutions are unstable for appropriate parameter settings. In the case of the Brusselator, it is further shown that there exist limit cycles; i.e. stable oscillations, in this parameter range. A numerical example is presented.
Similar content being viewed by others
References
Alexander, J. C., Yorke, J. A.: Global bifurcation of periodic orbits. Am. J. Math. 100 263–292 (1978)
shkenazi, M., Othmer, H. G.: Spatial patterns in coupled biochemical oscillators, J. Math. Biol. 5, 305–350 (1978)
Hassard, B. D., Kazarinofi, N. D., Wan, V. H.: Theory and applications of Hopf bifurcation. Cambridge Univ. Press 1981
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)
Lefever, R., Prigogine, I.: Symmetry-breaking instabilities in dissipative systems II. J. Chem. Phys. 48, 1695–1700 (1968)
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)
Author information
Authors and Affiliations
Additional information
Partially supported by NSF
Rights and permissions
About this article
Cite this article
Alexander, J.C. Spontaneous oscillations in two 2-component cells coupled by diffusion. J. Math. Biology 23, 205–219 (1986). https://doi.org/10.1007/BF00276957
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00276957