An inverse approach to the center-focus problem for polynomial differential system with homogenous nonlinearities
We consider polynomial vector fields of the form \[ \X=(-y X_m) x (x Y_m) y, \] where X_m=X_m(x,y) and Y_m=Y_m(x,y) are homogenous polynomials of degree m. It is well--known that \X has a center at the origin if and only if \X has an analytic first integral of the form \[ H=12(x^2 y^2) _j=3^ H_j, \]...
| Authors: | , , |
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| Format: | article |
| Publication Date: | 2017 |
| Country: | España |
| Institution: | Universitat Autònoma de Barcelona |
| Repository: | Dipòsit Digital de Documents de la UAB |
| Language: | English |
| OAI Identifier: | oai:ddd.uab.cat:182507 |
| Online Access: | https://ddd.uab.cat/record/182507 https://dx.doi.org/urn:doi:10.1016/j.jde.2017.04.030 |
| Access Level: | Open access |
| Keyword: | Analytic planar differential system Center-foci problem Darboux's first integral Holomorphic isochronous center Isochronous center Liapunov's constants Uniform isochronous center Weak condition for a center |
| Summary: | We consider polynomial vector fields of the form \[ \X=(-y X_m) x (x Y_m) y, \] where X_m=X_m(x,y) and Y_m=Y_m(x,y) are homogenous polynomials of degree m. It is well--known that \X has a center at the origin if and only if \X has an analytic first integral of the form \[ H=12(x^2 y^2) _j=3^ H_j, \] where H_j=H_j(x,y) is a homogenous polynomial of degree j. The classical center-focus problem already studied by H. Poincar\'e consists in distinguishing when the origin of \X is either a center or a focus. In this paper we study the inverse center-focus problem. In particular for a given analytic function H defined in a neighborhood of the origin we want to determine the homogenous polynomials X_m and Y_m in such a way that H is a first integral of \X and consequently the origin of \X will be a center. Moreover, we study the case when \[H=12(x^2 y^2)(1 _j=1^ \Upsilon_j),\] where \Upsilon_j is a convenient homogenous polynomial of degree j for j 1. The solution of the inverse center problem for polynomial differential systems with homogenous nonlinearities, provides a new mechanism to study the center problem, which is equivalent to Liapunov's Theorem and Reeb's criterion. |
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