×

zbMATH — the first resource for mathematics

Symmetry and phaselocking in chains of weakly coupled oscillators. (English) Zbl 0596.92011
Weakly coupled chains of oscillators with nearest-neighbor interactions are analyzed for phaselocked solutions. It is shown that the symmetry properties of the coupling affect the qualitative form of the phaselocked solutions and the scaling behavior of the system as the number of oscillators grows without bound. It is also shown that qualitative behavior of these solutions depends on whether the coupling is ”diffusive” or ”synaptic”, terms defined in the paper.
The methods include the demonstration that the equations for phaselocked solutions can be approximated by a singularly perturbed two-point (continuum) boundary value problem that is easier to analyze; the issue of convergence of the phaselocked solutions to solutions of the continuum equation is closely related to questions involving numerical entropy in computation schemes for a conservation law. An application to the neurophysiology of motor behaviour is discussed briefly.

MSC:
92Cxx Physiological, cellular and medical topics
92B05 General biology and biomathematics
35Q99 Partial differential equations of mathematical physics and other areas of application
35B25 Singular perturbations in context of PDEs
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Fenichel, Indiana Univ. Math. J. 21 pp 193– (1971)
[2] Ermentrout, J. Math. Anal. 15 pp 215– (1984)
[3] and , Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer, New York, 1983. · Zbl 0515.34001 · doi:10.1007/978-1-4612-1140-2
[4] Howard, Studies in Appl. Math. 56 pp 95– (1977) · Zbl 0349.35070 · doi:10.1002/sapm197756295
[5] Hagen, Adv. in Appl. Math. 2 pp 400– (1981)
[6] Neu, SIAM J. Appl. Math. 36 pp 509– (1979)
[7] and , Mathematics Applied to Deterministic Problems in the Natural Sciences, MacMillan, New York, 1974.
[8] Linear Algebra and its Applications, Second Edition, Academic Press, New York, 1980.
[9] Harten, Comm. Pure and Appl. Math. 29 (1976)
[10] Le Roux, Math. of Comp. 31 pp 848– (1977)
[11] Engquist, Math. of Comp. 36 pp 321– (1981)
[12] Osher, SIAM J. Numer. Anal. 18 pp 129– (1981)
[13] Traub, Neuroscience 7 pp 1233– (1982)
[14] Wilson, Kybernetik 13 pp 55– (1973)
[15] Kopell, Ann. N.Y. Acad. Sci. 357 pp 397– (1980)
[16] Grillner, Physiol. Rev. 55 pp 247– (1975)
[17] and , On the generation and performance of swimming in fish, in Neural Control of Locomotion, , , and , eds., Plenum, New York, 1976.
[18] Grillner, Exp. Brain Res. 20 pp 459– (1974)
[19] Models in neurobiology, in Nonlinear Phenomena in Physics and Biology, , , and , eds., Plenum, New York, 1981.
[20] Grillner, Science 228 pp 143– (1985)
[21] Cohen, J. Math. Biol. 13 pp 345– (1982)
[22] Toward a theory of modelling central pattern generators, to appear in Neural Control of Rhythmic Movements, , and , eds., J. Wiley, New York.
[23] Coupled oscillators and locomotion by fish, to appear in Oscillators in Chemistry and Biology, 1985, ed., Lectures in Biomath., 66, Springer-Verlag, New York, 1986.
[24] Kopell, Studies in Appl. Math. 52 pp 291– (1973) · Zbl 0305.35081 · doi:10.1002/sapm1973524291
[25] Kopell, Adv. in Appl. Math. 2 pp 389– (1981)
[26] Hagan, Adv. in Appl. Math. 2 pp 400– (1981)
[27] and , Methods of Bifurcation Theory, Springer-Verlag, New York, 1982. · Zbl 0487.47039 · doi:10.1007/978-1-4613-8159-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.