Abstract
Many chemical and biological phenomena are modeled as systems of coupled limit cycle oscillators. These models are inherently complex in that they involve often large numbers of coupled ordinary and partial differential equations. To understand any of the behavior of these systems, simplifying assumptions are made. One such assumption is that the individual oscillators are nearly identical and weakly coupled. In this case only the phase of the individual oscillators matters and so the coupled system becomes a smaller system on a k-torus. This technique has been applied to discrete systems [1–3] as well as to reaction-diffusion equation [4,5]. Many interesting aspects of chemical and biological systems can be understood by studying the simple phase-models [6–8]. For example, see the paper by Kopell in this volume.
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References
N. Levinson, “Small periodic perturbations of an autonomous system with a stable orbit”, Ann. Math. 52 (1950), 727–738.
G. B. Ermentrout and N. Kopell, “Frequency plateaus in a chain of weakly coupled oscillators, I.”, SIAM J. Math. Anal. 15 (1984), 215–237.
J. Neu, “Coupled chemical oscillators”, SIAM J. Appl. Math. 37 (1979), 307–315.
J. Neu, “Nonlinear oscillators in discrete and continuous systems”, Ph.D. Thesis, Cal. Tech (1978), Chapter 5.
P. Hagan, “Target patterns in reaction-diffusion systems”, Adv. Appl. Math. 2 (1981), 400–416.
S. Daan and C. Berde, “Two coupled oscillators: Simulations of the circadian pacemaker in mammalian activity rhythms”, J. Theor. Biol. 70 (1978), 297–314.
A. H. Cohen, P. J. Holmes and R. J. Rand, “The nature of coupling between segmental oscillators of the lamprey spinal generator”, J. Math. Biol. 13 (1982), 345–369.
N. Kopell and G. B. Ermentrout, “Symmetry and phaselocking in a chain of weakly coupled oscillators, preprint.
K. Bar-Eli, “On the stability of coupled chemical oscillators”, Physica 14D (1985), 242–252.
I. Schreiber and M. Marek, “Strange attractors in coupled reaction-diffusion cells”, Physica 15D (1982), 258–272.
G. B. Ermentrout, S. P. Hastings and W. C. Troy, “Large amplitude stationary waves in an excitable lateral-inhibitory medium”, SIAM J. Appl. Math. 44 (1984), 1133–1149.
P. S. Hagan, “Spiral waves in reaction-diffusion equations”, SIAM J. Appl. Math. 42 (1983), 762–786.
G. B. Ermentrout, “Stable small-amplitude solutions in reaction-diffusion systems”, Quart. Appl. Math., April (1981), 61–86.
G. B. Ermentrout, D. Aronson and N. Kopell, (in preparation).
V. Torre, “A theory of synchronization of two heart pace-maker cells”, J. Theor. Biol. 61 (1976), 55–71.
N. Kopell and L. N. Howard, “Plane wave solutions to reaction-diffusion equations”, Stud. Appl. Math. 52 (1973), 291–328.
N. Kopell, “Target pattern solutions to reaction-diffusion equations in the presence of imparities”, Adv. Appl. Math. 2 (1981), 389–399.
H. Meinhart, personal communication.
A. J. Winfree, The Geometry of Biological Time, Springer-Verlag, New York (1980), 328–329.
G. B. Ermentrout and W. C. Troy, “Phaselocking in a reaction-diffusion svstem with a linear frequency gradient”, preprint (1985).
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© 1986 Springer-Verlag Berlin Heidelberg
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Ermentrout, B. (1986). Losing Amplitude and Saving Phase. In: Othmer, H.G. (eds) Nonlinear Oscillations in Biology and Chemistry. Lecture Notes in Biomathematics, vol 66. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-93318-9_6
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DOI: https://doi.org/10.1007/978-3-642-93318-9_6
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