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Limit cycles of a class of polynomial systems

Published online by Cambridge University Press:  14 November 2011

Marc Carbonell  and
Jaume Llibre
Marc Carbonell
Affiliation:
Departament de Matemàtiques i Informàtica, Facultat de Ciències, Universitat de les Illes Balears, 07071 Palma de Mallorca, Spain
Jaume Llibre
Affiliation:
Departament de Matemàtiques, Facultat de Ciències, Universitat Autònoma de Barcelona, 08193, Bellaterra, Barcelona, Spain
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Synopsis

We study the class of polynomial vector fields of the form = αx — y + Pn(x, y), = x + αy + Qn(x, y), where Pn and Qn are homogeneous polynomials of degree n. If we define the functions f(x, y) = xPn(x, y) + yQn(x, y) and g(x, y) = xQn(x, y)−yPn(x, y), we characterise the number of limit cycles for this class when the function g(αg − f) does not change sign.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 109 , Issue 1-2 , 1988 , pp. 187 - 199
Copyright
Copyright © Royal Society of Edinburgh 1988

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References

1Alwash, 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
2Cherkas, L. A.. Number of limit cycles of an autonomous second-order system. Differential Equations 5 (1976), 666–668.Google Scholar
3Chicone, C.. Limit cycles of a class of polynomial vector fields in the plane. J. Differential Equations 63 (1986), 68–87.CrossRefGoogle Scholar
4Coll, B., Gasull, A. and Llibre, J.. Some theorems on the existence, uniqueness and nonexistence of limit cycles for quadratic systems. J. Differential Equations 67 (1987), 372–399.CrossRefGoogle Scholar
5Gasull, A., Llibre, J. and Sotomayor, J.. Limit cycles of vector fields of the form: X(v) = Av + f(v)Bv. J. Differential Equations 67 (1987), 90–110.CrossRefGoogle Scholar
6Gonzales, E. A. V.. Generic properties of a polynomial vector fields at infinity. Trans. Amer. Math. Soc. 143 (1969) 201–222.CrossRefGoogle Scholar
7Koditschek, D. E. and Narendra, K. S.. The stability of second order quadratic differential equations. IEEE Trans. Automat. Control AC-27 (1982), 783–798.CrossRefGoogle Scholar
8Koditschek, D. E. and Narendra, K. S.. Limit cycles of planar quadratic differential equations. J. Differential Equations 54 (1984), 181–195.CrossRefGoogle Scholar
9Lins, A.. On the number of solutions of the equation dx/dt = Σ, aj(t)/xi,0≦t≦1, for which x(0) = x(l). Invent. Math. 59 (1980), 67–76.Google Scholar
10Lloyd, N. G.. The number of periodic solutions of the equation z = zN + p1(t)zN—1 + … + pN(t). Proc. London Math. Soc. (3) 27 (1973), 667–700.CrossRefGoogle Scholar
11Lloyd, 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
12Lloyd, N. G.. Small amplitude limit cycles of polynomial differential equations. Lecture Notes in Mathematics 1032, 346–357 (Berlin: Springer, 1983).Google Scholar
13Lloyd, N. G.. Limit cycles of certain polynomial systems. In Nonlinear Functional Analysis and its Applications, ed. Singh, S. P., NATO ASI Series C 173 (1986), 317–326.CrossRefGoogle Scholar
14Pliss, V. A.. Non local problems of the theory of oscillations (New York: Academic Press, 1966).Google Scholar
15Sotomayor, J.. Curvas definidas por equagoes diferenciais no piano (Rio de Janeiro: Instituto de Matematica Pura e Aplicada, 1981).Google Scholar