### Abstract

In this paper we deal with ordinary differential equations of the form dy/dx = P(x, y) where P(x, y) is a real polynomial in the variables x and y, of degree n in the variable y. If y = φ(x) is a solution of this equation defined for x ∈ [0, 1] and which satisfies φ(0) = φ(1), we say that it is a periodic orbit. A limit cycle is an isolated periodic orbit in the set of all periodic orbits. If φ(x) is a polynomial, then φ(x) is called a polynomial solution. We study the maximum number and the multiplicity of the polynomial limit cycles of dy/dx = P(x, y) with respect to n. We prove that this differential equation has at most n polynomial limit cycles and that this bound is sharp. If n = 1 (linear equation) or n = 2 (Riccati equation), we prove that the differential equation dy/dx = P(x, y) has at most n polynomial limit cycles counted with their multiplicities. For n = 3 (Abel equation), we show that at most three polynomial limit cycles can exist whereas the multiplicity of a polynomial limit cycle can be unbounded. © 2011 Hebrew University Magnes Press.

Original language | English |
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Pages (from-to) | 461-475 |

Journal | Israel Journal of Mathematics |

Volume | 181 |

DOIs | |

Publication status | Published - 1 Mar 2011 |

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## Cite this

Giné, J., Grau, M., & Llibre, J. (2011). On the polynomial limit cycles of polynomial differential equations.

*Israel Journal of Mathematics*,*181*, 461-475. https://doi.org/10.1007/s11856-011-0019-3