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Uniformly isochronous polynomial centers
Electronic Journal of Differential Equations, 2005 We study a specific family of uniformly isochronous polynomial systems. Our results permit us to solve a problem about centers of such systems. We consider the composition conjecture for uniformly isochronous polynomial systems.
Vladimir V. Ivanov, Evgenii P. Volokitin
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A survey of isochronous centers [PDF]
Qualitative Theory of Dynamical Systems, 1999 In this survey we give an overview of the results obtained in the study of isochronous centers of vector fields in the plane. This paper consists of two parts. In the first one (sections 2--8), we review some general techniques that proved to be useful in the study of isochronicity.
Sabatini, Marco, J. Chavarriga
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On the Topology of Isochronous Centers of Hamiltonian Differential Systems [PDF]
International Journal of Bifurcation and Chaos, 2019 In this paper, we study the topology of isochronous centers of Hamiltonian differential systems with polynomial Hamiltonian functions [Formula: see text] such that the isochronous center lies on the level curve [Formula: see text]. We prove that, in the one-dimensional homology group of the Riemann surface (removing the points at infinity) of level ...
Guangfeng Dong +2 more
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Linearization of Isochronous Centers
Journal of Differential Equations, 1995 The authors propose a unified technique for the study of the isochronicity of polynomial systems. They first construct the linearizing changes of coordinates for isochronous systems having rational first integrals, namely for quadratic systems, for cubic systems symmetric with respect to a center and for reduced Kukles systems.
Mardesic, P., Rousseau, C., Toni, B.
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Quadratic Isochronous centers commute [PDF]
Applicationes Mathematicae, 1999 Summary: The author proves that every quadratic plane differential system having an isochronous center commutes with a polynomial differential system.
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Centers and Uniform Isochronous Centers of Planar Polynomial Differential Systems [PDF]
Journal of Dynamics and Differential Equations, 2018 For planar polynomial vector fields of the form (-y+X(x,y))¿¿x+(x+Y(x,y))¿¿y, where X and Y start at least with terms of second order in the variables x and y, we determine necessary and sufficient conditions under which the origin is a center or a uniform isochronous centers.
Jaume Llibre +3 more
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Centers and isochronous centers for generalized quintic systems
Journal of Computational and Applied Mathematics, 2015 In this paper we classify the centers and the isochronous centers of certain polynomial differential systems in R2 of degree d ≥ 5 odd that incomplex notation can be written as z˙ = (λ + i)z + (zz¯)d-52 (Az5 + Bz4z¯ + Cz3z¯2 + Dz2z¯3 + Ezz¯4 + Fz¯5),where λ ∈ R and A, B, C, D, E, F ∈ C.
Jaume Giné +2 more
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Center and isochronous center at infinity for differential systems
Bulletin des Sciences Mathématiques, 2004 Here, center and isochronous center conditions are investigated for differential systems at infinity. The authors give a suitable transformation by which infinity can be transferred into the origin. The properties of infinity are studied using well-known methods which are used for the origin.
Liu, Yirong, Huang, Wentao
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Phase Portraits of Uniform Isochronous Centers with Homogeneous Nonlinearities [PDF]
Journal of Dynamical and Control Systems, 2021 We classify the phase portraits in the Poincaré disc of the differential equations of the form x' = - y + xf(x, y), ẏ = x + yf(x, y) where f(x,y) is a homogeneous polynomial of degree n - 1 when n = 2, 3, 4, 5, and f has only simple zeroes. We also provide some general results on these uniform isochronous centers for all n ≥ 2.
Llibre, Jaume, Valls, Claudia
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Center conditions for a simple class of quintic systems
International Journal of Mathematics and Mathematical Sciences, 2002 We obtained center conditions for an O-symmetric system of degree 5 for which the origin is a uniformly isochronous singular point.
Evgenii P. Volokitin
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