{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,17]],"date-time":"2026-07-17T03:17:09Z","timestamp":1784258229807,"version":"3.55.0"},"reference-count":28,"publisher":"World Scientific Pub Co Pte Lt","issue":"04","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Int. J. Bifurcation Chaos"],"published-print":{"date-parts":[[2002,4]]},"abstract":"<jats:p> The center problem for degenerate singular points of planar systems (the degenerate-center problem) is a poorly-understood problem in the qualitative theory of ordinary differential equations. It may be broken down into two problems: the monodromy problem, to decide if the singular point is of focus-center type, and the stability problem, to decide whether it is a focus or a center. <\/jats:p><jats:p> We present an outline on the status of the center problem for degenerate singular points, explaining the main techniques and obstructions arising in the study of the problem. We also present some new results. Our new results are the characterization of a family of vector fields having a degenerate monodromic singular point at the origin, and the computation of the generalized first focal value for this family V<jats:sub>1<\/jats:sub>. This gives the solution of the stability problem in the monodromic case, except when V<jats:sub>1<\/jats:sub> = 1. Our approach relies on the use of the blow-up technique and the study of the blow-up geometry of singular points. The knowledge of the blow-up geometry is used to generate a bifurcation of a limit cycle. <\/jats:p>","DOI":"10.1142\/s0218127402004693","type":"journal-article","created":{"date-parts":[[2002,7,27]],"date-time":"2002-07-27T07:05:54Z","timestamp":1027753554000},"page":"687-707","source":"Crossref","is-referenced-by-count":38,"title":["ON THE CENTER PROBLEM FOR DEGENERATE SINGULAR POINTS OF PLANAR VECTOR FIELDS"],"prefix":"10.1142","volume":"12","author":[{"given":"V\u00cdCTOR","family":"MA\u00d1OSA","sequence":"first","affiliation":[{"name":"Departament de Matem\u00e0tica Aplicada III, Universitat Polit\u00e8cnica de Catalunya, Colom 1, 08222  Terrassa, Spain"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"219","published-online":{"date-parts":[[2011,11,20]]},"reference":[{"key":"rf1","doi-asserted-by":"publisher","DOI":"10.1017\/S0308210500021971"},{"key":"rf2","first-page":"187","volume":"8","author":"Andreev A.","journal-title":"Transl. Amer. Math. Soc."},{"key":"rf3","volume-title":"Theory of Bifurcation of Dynamic Systems on a Plane","author":"Andronov A. A.","year":"1973"},{"key":"rf4","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-61551-1"},{"key":"rf6","volume-title":"Geometric Theory of Differential Equations and Dynamical Systems","author":"Arnol'd V. I.","year":"1988"},{"key":"rf7","first-page":"1","volume":"13","author":"Berezovskaya F. S.","journal-title":"Sel. Math."},{"key":"rf8","first-page":"268","volume":"211","author":"Chavarriga J.","journal-title":"J. Math. Anal. Appl."},{"key":"rf10","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(77)90136-X"},{"key":"rf11","volume-title":"Singularities of Vector Fields","author":"Dumortier F.","year":"1978"},{"key":"rf12","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1017\/S0143385700008725","volume":"16","author":"Fran\u00e7oise J. P.","journal-title":"Ergod. Th. Dyn. Syst."},{"key":"rf13","doi-asserted-by":"publisher","DOI":"10.1088\/0951-7715\/13\/3\/311"},{"key":"rf14","author":"Gasull A.","journal-title":"J. Diff. Eqs."},{"key":"rf15","doi-asserted-by":"crossref","first-page":"4479","DOI":"10.1016\/S0362-546X(01)00561-2","volume":"47","author":"Gasull A.","journal-title":"Nonlin. Anal.: Proc. IIIrd World Congress on Nonlinear Analysis"},{"key":"rf16","first-page":"149","volume":"20","author":"Gasull A.","journal-title":"Comput. Appl. Math."},{"key":"rf18","volume-title":"Table of Integrals Series and Products","author":"Grandshteyn I. S.","year":"1980"},{"key":"rf19","series-title":"Translations of Math. Monographs","doi-asserted-by":"crossref","DOI":"10.1090\/mmono\/094","volume-title":"Finiteness Theorems for Limit Cycles","volume":"94","author":"Il'Yashenko Y. S.","year":"1991"},{"key":"rf20","first-page":"277","volume":"20","author":"Lloyd N. G.","journal-title":"J. London Math. Soc."},{"key":"rf21","doi-asserted-by":"publisher","DOI":"10.1007\/BF00971099"},{"key":"rf22","first-page":"212","volume":"213","author":"Medvedeva N. B.","journal-title":"Proc. Steklov Inst. Math."},{"key":"rf23","doi-asserted-by":"publisher","DOI":"10.1007\/BF02674907"},{"key":"rf24","first-page":"130","volume":"13","author":"Medvedeva N. B.","journal-title":"Algebra i Analiz"},{"key":"rf25","first-page":"1","volume":"19","author":"Medvedeva N. B.","journal-title":"Electron. J. Qual. Th. Diff. Eqs."},{"key":"rf26","doi-asserted-by":"crossref","first-page":"241","DOI":"10.1017\/S0143385700001553","volume":"2","author":"Moussu R.","journal-title":"Ergod. Th. Dyn. Syst."},{"key":"rf27","volume-title":"Qualitative Theory of Differential Equations","author":"Nemytskii V. V.","year":"1989"},{"key":"rf28","doi-asserted-by":"publisher","DOI":"10.1137\/S0036144595283575"},{"key":"rf29","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4684-0249-0"},{"key":"rf30","doi-asserted-by":"publisher","DOI":"10.1016\/0022-0396(73)90034-X"},{"key":"rf31","doi-asserted-by":"publisher","DOI":"10.1006\/jdeq.1994.1049"}],"container-title":["International Journal of Bifurcation and Chaos"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.worldscientific.com\/doi\/pdf\/10.1142\/S0218127402004693","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,8,6]],"date-time":"2019-08-06T20:12:20Z","timestamp":1565122340000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.worldscientific.com\/doi\/abs\/10.1142\/S0218127402004693"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2002,4]]},"references-count":28,"journal-issue":{"issue":"04","published-online":{"date-parts":[[2011,11,20]]},"published-print":{"date-parts":[[2002,4]]}},"alternative-id":["10.1142\/S0218127402004693"],"URL":"https:\/\/doi.org\/10.1142\/s0218127402004693","relation":{},"ISSN":["0218-1274","1793-6551"],"issn-type":[{"value":"0218-1274","type":"print"},{"value":"1793-6551","type":"electronic"}],"subject":[],"published":{"date-parts":[[2002,4]]}}}