### Abstract

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.

Original language | English (US) |
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Pages (from-to) | 215-239 |

Number of pages | 25 |

Journal | Proceedings of the Royal Society of Edinburgh Section A: Mathematics |

Volume | 98 |

Issue number | 3-4 |

DOIs | |

State | Published - 1984 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**The number of limit cycles of certain polynomial differential equations.** / Blows, Terence R.

Research output: Contribution to journal › Article

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TY - JOUR

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

AU - Blows, Terence R

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.

UR - http://www.scopus.com/inward/record.url?scp=84976168837&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84976168837&partnerID=8YFLogxK

U2 - 10.1017/S030821050001341X

DO - 10.1017/S030821050001341X

M3 - Article

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 -