Description:
<jats:p>A polynomial system is a real autonomous system of ordinary differential equations on the plane with polynomial nonlinearities:</jats:p><jats:p><jats:disp-formula><jats:graphic xmlns:xlink="http://www.w3.org/1999/xlink" mime-subtype="gif" mimetype="image" position="float" xlink:type="simple" xlink:href="S0305004100073412_eqn1" /></jats:disp-formula></jats:p><jats:p>with <jats:italic>a<jats:sub>ij</jats:sub></jats:italic>, <jats:italic>b</jats:italic><jats:sub>ij</jats:sub> ∈ ℝ and where <jats:italic>x</jats:italic> = <jats:italic>x</jats:italic>(<jats:italic>t</jats:italic>) and <jats:italic>y</jats:italic> = <jats:italic>y</jats:italic>(<jats:italic>t</jats:italic>) are real-valued functions.</jats:p><jats:p>The problem of analysing limit cycles (isolated periodic solutions) in polynomial systems was first discussed by Poincaré[<jats:bold>16</jats:bold>]. Then, in the famous list of 23 mathematical problems stated in 1900, Hilbert[<jats:bold>9</jats:bold>] asked in the second part of the 16th problem for an upper bound for the number of limit cycles for <jats:italic>n</jats:italic>th degree polynomial systems, in terms of <jats:italic>n</jats:italic>. Recently, it has been proved that, given a particular choice of coefficients for a system of form (1·1), the number of limit cycles is finite. This result is known as Dulac's theorem, see Ecalle[<jats:bold>8</jats:bold>] or Il'yashenko[<jats:bold>10</jats:bold>]. However, it is unknown whether or not there exists an upper bound for the number of limit cycles in system (1·1) in terms of <jats:italic>n</jats:italic>. Even for quadratic systems (i.e. polynomial systems with quadratic nonlinearities) this remains an open question.</jats:p>