Chaotic behavior in piecewise continuous difference equations

Keener, James P.

doi:10.1090/S0002-9947-1980-0580905-3

Chaotic behavior in piecewise continuous difference equations
HTML articles powered by AMS MathViewer

by James P. Keener PDF

Trans. Amer. Math. Soc. 261 (1980), 589-604 Request permission

Abstract:

A class of piecewise continuous mappings with positive slope, mapping the unit interval into itself is studied. Families of 1-1 mappings depending on some parameter have periodic orbits for most parameter values, but have an infinite invariant set which is a Cantor set for a Cantor set of parameter values. Mappings which are not 1-1 exhibit chaotic behavior in that the asymptotic behavior as measured by the rotation number covers an interval of values. The asymptotic behavior depends sensitively on initial data in that the rotation number is either a nowhere continuous function of initial data, or else it is a constant on all but a Cantor set of the unit interval.

References

O. M. Šarkovs′kiĭ, Co-existence of cycles of a continuous mapping of the line into itself, Ukrain. Mat. . 16 (1964), 61–71 (Russian, with English summary). MR 0159905
T. Y. Li and James A. Yorke, Period three implies chaos, Amer. Math. Monthly 82 (1975), no. 10, 985–992. MR 385028, DOI 10.2307/2318254
J. P. Keener, F. C. Hoppensteadt, and J. Rinzel, Integrate-and-fire models of nerve membrane response to oscillatory input, SIAM J. Appl. Math. 41 (1981), no. 3, 503–517. MR 639130, DOI 10.1137/0141042

A neuronal model for the discharge patterns produced by cyclic inputs

32

Leon Glass and Michael C. Mackey, A simple model for phase locking of biological oscillators, J. Math. Biol. 7 (1979), no. 4, 339–352. MR 648856, DOI 10.1007/BF00275153

Sur les courbes définies par les équations différentielles à la surface du tore

11

Earl A. Coddington and Norman Levinson, Theory of ordinary differential equations, McGraw-Hill Book Co., Inc., New York-Toronto-London, 1955. MR 0069338
Michael-Robert Herman, Mesure de Lebesgue et nombre de rotation, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976) Lecture Notes in Math., Vol. 597, Springer, Berlin, 1977, pp. 271–293 (French). MR 0458480
Jürgen Moser, Stable and random motions in dynamical systems, Annals of Mathematics Studies, No. 77, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1973. With special emphasis on celestial mechanics; Hermann Weyl Lectures, the Institute for Advanced Study, Princeton, N. J. MR 0442980
S. Smale and R. F. Williams, The qualitative analysis of a difference equation of population growth, J. Math. Biol. 3 (1976), no. 1, 1–4. MR 414147, DOI 10.1007/BF00307853
David Ruelle, Sensitive dependence on initial condition and turbulent behavior of dynamical systems, Bifurcation theory and applications in scientific disciplines (Papers, Conf., New York, 1977) Ann. New York Acad. Sci., vol. 316, New York Acad. Sci., New York, 1979, pp. 408–416. MR 556846