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
SN - 0308-2105
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
IS - 3-4
ER -