Linear type global centers of linear systems with cubic homogeneous nonlinearities
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Linear type global centers of linear systems with cubic homogeneous nonlinearities Johanna D. García-Saldaña1
· Jaume Llibre2 · Claudia Valls3
Received: 11 October 2018 / Accepted: 19 July 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract A center p of a differential system in R2 is global if R2 \ { p} is filled of periodic orbits. It is known that a polynomial differential system of degree 2 has no global centers. Here we characterize the global centers of the differential systems x˙ = ax + by + P3 (x, y),
y˙ = cx + dy + Q 3 (x, y),
with P3 and Q 3 homogeneous polynomials of degree 3, and such that the center has purely imaginary eigenvalues, i.e. a linear type center. Keywords Center · Global center · Cubic polynomial differential system Mathematics Subject Classification 34C05 · 34C07 · 34C08
1 Introduction and statement of the main results The notion of center goes back to Poincaré and Dulac, see [6,10]. They defined a center for a vector field on the real plane as a singular point having a neighborhood filled of periodic orbits with the exception of the singular point. The problem of distinguishing when a monodromic singular point is a focus or a center, known as the focus-center problem started precisely with Poincaré and Dulac and is still active nowadays with many questions still unsolved. These
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Johanna D. García-Saldaña [email protected] Jaume Llibre [email protected] Claudia Valls [email protected]
1
Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Alonso de Ribera 2850, Concepción, Chile
2
Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Bellaterra, Barcelona, Spain
3
Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisbon, Portugal
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last years also the centers are perturbed for studying the limit cycles bifurcating from their periodic solutions, see for instance [1,3]. If an analytic system has a center, then it is known that after an affine change of variables and a rescaling of the time variable, it can be written in one of the following three forms: x˙ = −y + P(x, y),
y˙ = x + Q(x, y),
called linear type center, which has a pair of purely imaginary eigenvalues, x˙ = y + P(x, y),
y˙ = Q(x, y)
called nilpotent center x˙ = P(x, y),
y˙ = Q(x, y)
called degenerated center, where P(x, y) and Q(x, y) are real analytic functions without constant and linear terms defined in a neighborhood of the origin. We recall that a global center for a vector field on the plane is a singular point p having R2 filled of periodic orbits with the exception of the singular point. The easiest global center is the linear center x˙ = −y, y˙ = x. It is known (see [2,11]) that quadratic polynomial differential systems have no global centers. The global degenerated centers of homogeneous or quasi-homogeneous polynomial differential systems were characterized in [4] and [8], respectively. However the characterizat
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