Centers for the Kukles homogeneous systems with odd degree
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For the polynomial differential system x˙ = −y, y˙ = x+Qn(x; y), where Qn(x; y) is a homogeneous polynomial of degree n there are the following two conjectures done in 1999. (1) Is it true that the previous system for n ≥ 2 has a center at the origin if and only if its vector field is symmetric
about one of the coordinate axes? (2) Is it true that the origin is an isochronous center of the previous system with the exception of the linear center only if the system has even degree? We prove both conjectures for all n odd.
Is part ofBulletin of the London Mathematical Society, 2015, vol. 47, p. 315-324
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Giné, Jaume; Llibre, Jaume; Valls, Claudia (Shanghai Normal University & Wilmington Scientific Publisher, 2017)For the polynomial differential system x˙=−y, y˙=x+Qn(x,y), where Qn(x,y) is a homogeneous polynomial of degree n there are the following two conjectures done in 1999. (1) Is it true that the previous system for n≥2 has a ...
Giné, Jaume; Llibre, Jaume; Valls, Claudia (Texas State University, 2019)We characterize the centers of the Chiellini Hamiltonian Li´enard second-order differential equations x 0 = y, y 0 = −f(x)y − g(x) where g(x) = f(x)(k − α(1 + α) R f(x)dx) with α, k ∈ R. Moreover we study the ...
Giné, Jaume; Valls, Claudia (Taylor & Francis, 2017)In this paper, we provide necessary and sufficient conditions for the existence of local analytic first integrals for a seventh-parameter family of complex cubic systems called the complex Kukles systems.