TY - JOUR

T1 - The number of limit cycles of certain polynomial differential equations

AU - Blows, T. R.

N1 - Funding Information:
The authors gratefully acknowledge the support of the Science and Engineering Research Council (GR/B 57156). * Present address: Department of Mathematics, Northern Arizona University, Flagstaff, Arizona 86011, U.S.A.

PY - 1984

Y1 - 1984

N2 - Two-dimensional differential systems x=P(x, y), y= Q(x, y) are considered, where P and Q are polynomials. The question of interest is the maximum possible number of limit cycles of such systems in terms of the degree of P and Q. An algorithm is described for determining a so-called focal basis; this can be implemented on a computer. Estimates can then be obtained for the number of small-amplitude limit cycles. The technique is applied to certain cubic systems; a class of examples with exactly five small-amplitude limit cycles is constructed. Quadratic systems are also considered.

AB - Two-dimensional differential systems x=P(x, y), y= Q(x, y) are considered, where P and Q are polynomials. The question of interest is the maximum possible number of limit cycles of such systems in terms of the degree of P and Q. An algorithm is described for determining a so-called focal basis; this can be implemented on a computer. Estimates can then be obtained for the number of small-amplitude limit cycles. The technique is applied to certain cubic systems; a class of examples with exactly five small-amplitude limit cycles is constructed. Quadratic systems are also considered.

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U2 - 10.1017/S030821050001341X

DO - 10.1017/S030821050001341X

M3 - Article

AN - SCOPUS:84976168837

VL - 98

SP - 215

EP - 239

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

SN - 0308-2105

IS - 3-4

ER -