TY - JOUR

T1 - On the polynomial limit cycles of polynomial differential equations

AU - Giné, Jaume

AU - Grau, Maite

AU - Llibre, Jaume

PY - 2011/3/1

Y1 - 2011/3/1

N2 - 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.

AB - 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.

U2 - https://doi.org/10.1007/s11856-011-0019-3

DO - https://doi.org/10.1007/s11856-011-0019-3

M3 - Article

VL - 181

SP - 461

EP - 475

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

ER -