# American Institute of Mathematical Sciences

July  2000, 6(3): 609-624. doi: 10.3934/dcds.2000.6.609

## Center-focus and isochronous center problems for discontinuous differential equations

 1 Dept. de Matemàtiques i Informàtica, Universitat de les Illes Balears, Facultat de ciències, 07071, Palma de Mallorca, Spain, Spain 2 Dept. de Matemàtiques, Universitat Autónoma de Barcelona, Edifici C, 08193 Bel-laterra, Barcelona, Spain

Received  October 1999 Revised  April 2000 Published  April 2000

The study of the center focus problem and the isochronicity problem for differential equations with a line of discontinuities is usually done by computing the whole return map as the composition of the two maps associated to the two smooth differential equations. This leads to large formulas which usually are treated with algebraic manipulators. In this paper we approach to this problem from a more theoretical point of view. The results that we obtain relate the order of degeneracy of the critical point of the discontinuous differential equations with the order of degeneracy of the two smooth component differential equations. Finally we apply them to some families of examples.
Citation: B. Coll, A. Gasull, R. Prohens. Center-focus and isochronous center problems for discontinuous differential equations. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 609-624. doi: 10.3934/dcds.2000.6.609
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