Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X

   
 
 

 

On linearly coupled relaxation oscillations


Authors: Jacques Bélair and Philip Holmes
Journal: Quart. Appl. Math. 42 (1984), 193-219
MSC: Primary 58F10; Secondary 34C15, 34E15, 70K05
DOI: https://doi.org/10.1090/qam/745099
MathSciNet review: 745099
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Abstract: We study the dynamical behavior of a pair of linearly coupled relaxation oscillators. In such systems vastly different time scales play a crucial rôle, and solutions may be viewed as consisting of portions of slow drift linked by rapid jumps. This feature enables us to reduce the analysis from four dimensional phase space to that of a two dimensional system with discontinuous but well determined behavior at certain points on the phase plane. We determine the existence and stability of periodic motions for identical oscillators and oscillators with an uncoupled frequency ratio of $ 1:\omega $. We give additional details on nonperiodic motions for the special case of $ \omega = 2$.


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  • [J] Bélair [1983a], Phase locking in linearly coupled relaxation oscillators, Ph. D. Thesis, Cornell Univ.
  • [J] Bélair [1983b], Une application de l'analyse nonstandard dans l'etude d'oscillateurs de relaxation, preprint
  • [E] Benoit, J.--L. Callot, F. Diener and M. Diener [1980], Chasse au canard, IRMA
  • [R] Bowen [1975] Equilibrium states and the ergodic theory of Axiom A diffeomorphism, Lecture Notes in Math., vol. 470, Springer, Heidelberg MR 0442989
  • [M] Cartwright and J. E. Littlewood [1945], On nonlinear differential equations of the second order: I. The equation $ \ddot y - k(1 - {y^2})\dot y + y = b\lambda k\cos (\lambda t + \alpha )$, $ k$ large, J. London Math. Soc. 20, 180-189 MR 0016789
  • [E] A. Coddington and N. Levinson [1955], Theory of ordinary differential equations, McGraw Hill, New York MR 0069338
  • [M] Davis [1977], Applied nonstandard analysis, Wiley, New York MR 0505473
  • [J] Flaherty and F. Hoppensteadt [1978], Frequency entrainment of a forced van der Pol oscillator, Studies in Appl. Math. 58, 5-15 MR 0499449
  • [J] --P. Gollub, T. O. Brunner and B. G. Danly [1978], Periodicity and chaos in coupled nonlinear oscillators, Science 200, 48-50
  • [J] Grasman and M. J. W. Jansen [1979], Mutually synchronized relaxation oscillators and prototypes and oscillating systems in biology, J. Math. Biol. 7 171-197 MR 648978
  • [J] Grasman, H. Nijmeijer and E. J. M. Velig [1982], Singular perturbations and a mapping on an interval for the forced van der Pol relaxation oscillator (preprint TW 221/82, Mathematisch Centrum, 413 Kruislaan, Amsterdam)
  • [J] Grasman, E. J. M. Velig and G. M. Willems [1976], Relaxation oscillators governed by a van der Pol equation with periodic forcing terms, SIAM J. Appl. Math. 31, 667--676 MR 0432975
  • [J] Guckenheimer [1980], Bifurcations of dynamical systems Dynamical Systems, C.I.M.E. Lectures Bressanone, Italy, June 1978, Progress in Mathematics #8, Birkhauser, Boston MR 589591
  • [J] Haag [1943], Etude asymptotique des oscillators de relaxation Ann. Sci. Ecole Norm. Sup. (3); 60, 35-111 MR 0014538
  • [M] Jirsch and S. Smale [1974], Differential equations, dynamical systems, and linear algebra, Academic Press, New York MR 0486784
  • [J] Kevorkian and J. D. Cole [1981], Perturbation methods in applied mathematics, Appl. Math. Sci. 34, Springer, New York MR 608029
  • [N] Levinson [1949], A second order differential equation with singular solutions, Ann. Math. 50, 127-153 MR 0030079
  • [M] Levi [1981], Qualitative analysis of the periodically forced relaxation oscillations, Memoirs of the AMS 32, #244, Providence MR 617687
  • [R] Lutz and M. Goze [1981], Nonstandard analysis, Lecture Notes in Math., Vol. 881, Springer, Heidelberg MR 643624
  • [G] Reeb [1974], Seance-debat sur l' Analyse Non-standard, Gazette des Mathematicians 8, 8-14
  • [A] Robinson [1974], Nonstandard analysis, 2nd edition, American Elsevier, New York
  • [J] J. Stoker [1950], Nonlinear vibrations in mechanical and electrical systems, Interscience, New York MR 0034932
  • 1. -[1980], Periodic forced vibrations of systems of relaxation oscillators, Comm. Pure Math. 33, 215-240
  • [K] Stroyan and W. A. J. Luxembourg [1975], Introduction to the theory of infinitesimals, Academic Press, New York
  • [F] Takens [1976], Constrained equations: a study of implicit differential equatins and their discontinuous solutions, Structural Stability, the Theory of Catastrophes, and Applications in the Sciences (P. Hilton, ed.) Lecture Notes in Math. Vol. 525, Springer, Heidelberg, 147-243 MR 0478236
  • [B] van der Pol [1926], On relaxation-oscillations, Phil. Mag., 7th Ser. 2, 978-992
  • [B] van der Pol and J. van der Mark [1928], The heartbeat considered as a relaxation oscillator, and an electrical model of the heart, Phil. Mag., 7th Ser. 6, 763-775
  • [E] C. Zeeman [1973], Differential equations for the heartbeat and nerve impulse, Dynamical Systems (M. M. Peixoto, ed.) Academic Press, New York, 683-741 MR 0342207

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DOI: https://doi.org/10.1090/qam/745099
Article copyright: © Copyright 1984 American Mathematical Society

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