No abstract

We investigate cubic systems which can be transformed to an equation of Abel form. The conditions for the origin to be a centre and, in particular, an isochronous centre are obtained. The maximum number of limit cycles which can bifurcate from a fine focus is determined and some information is obtained about the global phase portrait.1998 Academic Press

We consider conditions under which the second-order differential equationx þ f ðxÞ 'x þ gðxÞ ¼ 0 has a family of periodic orbits with constant period. This condition is equivalent to seeking conditions under which the two-dimensional autonomous system 'x ¼ y; ' y ¼ ÀgðxÞ À f ðxÞy has a center with constant period: i.e., an isochronous center. In turn, this is equivalent to the latter system being locally linearizable. We give a simple necessary and sufficient condition for this when f and g are analytic functions. In the case where when f and g are polynomials, we show that this reduces to a finitely determinable system of equations. A complete classification is given of all such systems of degree 34 or less. r 2004 Published by Elsevier Inc.