Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-19T08:33:53.188Z Has data issue: false hasContentIssue false
  • English
  • Français

The Poincaré–Bendixson theorem for certain differential equations of higher order

Published online by Cambridge University Press:  14 November 2011

Russell A. Smith
Russell A. Smith
Affiliation:
Department of Mathematics, University of Durham
Get access Rights & Permissions [Opens in a new window]

Synopsis

By adapting its well-known proof, the Poincaré–Bendixson theorem, on the existence of periodic orbits of plane autonomous systems, is extended to vector differential equations of the form f(D)x + bφ(g(D)x) = 0. The only restrictions placed on the vector function φ(y) are that its Jacobian matrix should be continuous and lie within a suitably chosen ellintic ball.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 83 , Issue 1-2 , 1979 , pp. 63 - 79
Copyright
Copyright © Royal Society of Edinburgh 1979

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Borg, G.. A condition for the existence of orbitally stable solutions of dynamical systems. K. Tekn. Högsk. Handl. 153 (1960), 12 pp.Google Scholar
2Burkin, I. M. and Leonov, G. A.. The existence of nontrivial periodic solutions in self-induced oscillating systems (Russian). Sibirsk. Mat. Z. 18 (1977), 251–262.Google Scholar
3D'Heedene, R. N.. A third order autonomous differential equation with almost periodic solutions. J. Math. Anal. Appi 3 (1961), 344–350.CrossRefGoogle Scholar
4Grasman, W.. Periodic solutions of autonomous differential equations in higher dimensional spaces. Rocky Mountain J. Math. 7 (1977), 457–466.CrossRefGoogle Scholar
5Hirsch, M. W. and Smale, S.. Differential Equations, Dynamical Systems, and Linear Algebra (New York: Academic Press, 1974).Google Scholar
6Leonov, G. A.. Frequency conditions for the existence of nontrivial periodic solutions in autonomous systems. Siberian Math. J. 14 (1973), 884–889.CrossRefGoogle Scholar
7Mirsky, L.. An Introduction to Linear Algebra (Oxford: Clarendon Press, 1955).Google Scholar
8Newman, M. H. A.. Elements of the Topology of Plane Sets of Points (Cambridge Univ. Press, 1951).Google Scholar
9Noldus, E.. A frequency domain approach to the problem of the existence of periodic motion in autonomous nonlinear feedback systems. Z. Angew. Math. Mech. 49 (1969), 167–177.CrossRefGoogle Scholar
10Noldus, E.. A counterpart of Popov's theorem for the existence of periodic solutions. Internat. J. Control 13 (1971), 705–719.CrossRefGoogle Scholar
11Pliss, V. A.. Nonlocal Problems of the Theory of Oscillations (New York: Academic Press, 1966).Google Scholar
12Schwartz, A. J.. A generalisation of a Poincaré-Bendixson theorem to closed two-dimensional manifolds. Amer. J. Math. 85 (1963), 453–458.CrossRefGoogle Scholar
13Smith, R. A.. Absolute stability of certain differential equations. J. London Math. Soc. 7 (1973), 203–210.CrossRefGoogle Scholar
14Smith, R. A.. Forced oscillations of the feedback control equation. Proc. Roy. Soc. Edinburgh Sect. A 76 (1976), 31–42.CrossRefGoogle Scholar
15Smith, R. A.. Some elliptic balls which avoid a Nyquist set in Cn+1. Proc. Roy. Soc. Edinburgh Sect. A 79 (1977), 327–334.CrossRefGoogle Scholar
16Vaisbord, E. M.. On the existence of a periodic solution of a third-order non-linear differential equation (Russian), Mat. Sb. 56 (1962), 43–58.Google Scholar