The number of limit cycles of certain polynomial differential equations

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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 languageEnglish (US)
Pages (from-to)215-239
Number of pages25
JournalProceedings of the Royal Society of Edinburgh Section A: Mathematics
Volume98
Issue number3-4
DOIs
StatePublished - 1984

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Polynomial equation
Limit Cycle
Differential equation
Quadratic Systems
Two-dimensional Systems
Differential System
Polynomial
Estimate

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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title = "The number of limit cycles of certain polynomial differential equations",
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.",
author = "Blows, {Terence R}",
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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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