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On the Limit Cycles of the Polynomial Differential Systems with a Linear Node and Homogeneous Nonlinearities

Jaume Llibre

Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Catalonia, Spain

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Jiang Yu

Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, P. R. China

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, and

Xiang Zhang

Department of Mathematics, and MOE-LSC, Shanghai Jiao Tong University, Shanghai 200240, P. R. China

Author for correspondence.

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https://doi.org/10.1142/S0218127414500655Cited by:7 (Source: Crossref)

Abstract

We consider the class of polynomial differential equations ẋ = λx + P_n(x, y), ẏ = μy + Q_n(x, y) in ℝ² where P_n(x, y) and Q_n(x, y) are homogeneous polynomials of degree n > 1 and λ ≠ μ, i.e. the class of polynomial differential systems with a linear node with different eigenvalues and homogeneous nonlinearities. For this class of polynomial differential equations, we study the existence and nonexistence of limit cycles surrounding the node localized at the origin of coordinates.

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