Hostname: page-component-586b7cd67f-t7fkt Total loading time: 0 Render date: 2024-11-26T14:27:00.191Z Has data issue: false hasContentIssue false
  • English
  • Français

Word problems related to derivatives of the displacement map

Published online by Cambridge University Press:  24 October 2008

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

Extract

In [6], we considered the equation

where z ∈ ℂ and the pi are real-valued functions; abstract word-problem concepts and techniques were applied to the local problem of the bifurcation of periodic solutions out of the solution Z ≡ 0. This paper is a sequel to [6]; we present an extension of certain concepts given in that paper, and give a global version of some of our word-problem results.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 3 , November 1991 , pp. 569 - 579
Copyright
Copyright © Cambridge Philosophical Society 1991

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]Carbonell, M. and Llibre, J.. Limit cycles of a class of polynomial systems. Proc. Roy. Soc. Edinburgh Sect. A 109 (1988), 187–199.CrossRefGoogle Scholar
[4]Coppel, W. A.. A simple class of quadratic systems. J. Differential Equations 64 (1986), 275–282.CrossRefGoogle Scholar
[5]Coll, B., Gasull, A. and Llibre, J.. Some theorems on the existence, uniqueness and nonexistence of limit cycles of quadratic systems. J. Differential Equations 76 (1987), 372–399.CrossRefGoogle Scholar
[6]Devlin, J.. Word problems related to periodic solutions of a non-autonomous system. Math. Proc. Cambridge Philos. Soc. 108 (1990), 127–151.CrossRefGoogle Scholar
[7]Devlin, J.. Periodic solutions of certain non-autonomous differential equations. Ph.D. thesis, The University College of Wales (1990).Google Scholar
[8]Lins Neto, A.. On the number of solutions of the equation dx/dt 0 ≤ x ≤ 1, for which x(0) = x(1). Invent. Math. 59 (1980), 67–76.Google Scholar
[9]Lloyd, N. G.. The number of periodic solutions of the equation ż = zN +p 1(t)z N-1 + … +pN(t). Proc. London Math. Soc. (3) 27 (1973), 667–700.CrossRefGoogle Scholar
[10]Lloyd, N. G.. A note on the number of limit cycles in certain two-dimensional systems. J. London Math. Soc. (2) 20 (1979), 277–286.CrossRefGoogle Scholar
[11]Lloyd, N. G.. Limit cycles of certain polynomial systems. In Nonlinear Functional Analysis and Applications (ed. Singh, S. P.), NATO Adv. Sci. Inst. Ser. C no. 173 (Reidel, 1986), pp. 317–326.CrossRefGoogle Scholar
[12]Pliss, V. A.. Nonlocal Problems in the Theory of Oscillations (Academic Press, 1966).Google Scholar
[13]Shahshahani, S.. Periodic solutions of polynomial first order differential equations. Nonlinear Anal. 5 (1981), 157–165.CrossRefGoogle Scholar