Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-24T16:19:16.206Z Has data issue: false hasContentIssue false
  • English
  • Français

Word problems related to periodic solutions of a non-autonomous system

Published online by Cambridge University Press:  24 October 2008

James Devlin
James Devlin
Affiliation:
Department of Mathematics, The University College of Wales, Aberystwyth, Dyfed
Get access Rights & Permissions [Opens in a new window]

Extract

In this paper, we develop an abstract formulation of a problem which arises in the investigation of the number of limit cycles of systems of the form

where p and q are homogeneous polynomials. This is part of the much wider study of Hilbert's sixteenth problem, in which information is sought about the number of limit cycles of systems of the form

where P and Q are polynomials, and their possible configurations.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 108 , Issue 1 , July 1990 , pp. 127 - 151
Copyright
Copyright © Cambridge Philosophical Society 1990

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

REFERENCES

[1]Alwash, M. A. M. and Lloyd, N. G.. Non-autonomous equations related to polynomial two-dimensional systems. Proc. Roy. Soc. Edinburgh Sect. A 105 (1987), 129–152.CrossRefGoogle Scholar
[2]Alwash, M. A. M. and Lloyd, N. G.. Periodic solutions of a quartic non-autonomous system. Nonlinear Anal. 11 (1987), 809–820.CrossRefGoogle Scholar
[3]Bautin, N. N.. On the number of limit cycles which appear with the variation of coefficients from an equilibrium position of focus or centre type. Math. Sb. 30 (1952), 181–196.(in Russian)Google Scholar
Amer. Math. Soc. Transl. 100 (1954), 1–19.Google Scholar
[4]Blows, T. R. and Lloyd, N. G.. The number of limit cycles of certain polynomial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 98 (1984), 215–239.CrossRefGoogle Scholar
[5]Blows, T. R. and Lloyd, N. G.. The number of small-amplitude limit cycles of Liénard equations. Math. Proc. Cambridge Philos. Soc. 95 (1984), 359–366.CrossRefGoogle Scholar
[6]Lansun, Chen and Mingshu, Wang. The relative position and number of limit cycles of quadratic differential equations. Acta Math. Sinica 22 (1979), 751–758.Google Scholar
[7]Fliess, M.. Fonctionelles causales non linéaires et indéterminées non commutatives. Bull. Soc. Math. France 109 (1981), 3–40.CrossRefGoogle Scholar
[8]James, E. M. and Yasmin, N.. Limit cycles of a cubic system. (Preprint, The University College of Wales, Aberystwyth, 1989).Google Scholar
[9]Neto, A. Lins. On the number of solutions of the equation , 0 ≤ t ≤ 1, for which x(0) = x(1). Invent. Math. 59 (1980), 67–76.CrossRefGoogle Scholar
[10]Lloyd, N. G.. The number of periodic solutions of the equation ż = zN + p1(t)zN − 1 +…+ pN(t). Proc. London Math. Soc. (3) 27 (1973), 667–700.CrossRefGoogle Scholar
[11]Lloyd, N. G.. Small amplitude limit cycles of polynomial differential equations. In Ordinary Differential Equations and Operators (eds. Everitt, W. N. and Lewis, R. T.), Lecture Notes in Math. vol. 1032 (Springer-Verlag, 1982). pp. 357–364.Google Scholar
[12]Lloyd, N. G.. Limit cycles of polynomial systems – some recent developments. In New Directions in Dynamical Systems (eds. Bedford, T. and Swift, J.), London Math. Soc. Lecture Notes Series no. 127 (Cambridge University Press, 1988). pp. 192–234.CrossRefGoogle Scholar
[13]Lloyd, N. G., Blows, T. R. and Kalenge, M. C.. Some cubic systems with several limit cycles. Nonlinearity 1 (1988), 653–669.CrossRefGoogle Scholar
[14]Lloyd, N. G. and Lynch, S.. Small-amplitude limit cycles of certain Liénard systems. Proc. Roy. Soc. London Ser. A 418 (1988), 199–208.Google Scholar
[15]Lothaire, M.. Combinatorics on Words. Encyclopedia of Mathematics and its Applications, vol. 17 (Cambridge University Press, 1984).Google Scholar
[16]Melançon, G. and Reutenauer, C.. Lyndon words, free algebras and shuffles. (Preprint, Universités Paris 7 et P. et M. Curie, 1987.)Google Scholar
[17]Songling, Shi. A concrete example of the existence of four limit cycles for plane quadratic systems. Sci. Sinica Ser. A 23 (1980), 153–158.Google Scholar