Abstract:
In this paper we consider the system
˙u=P(u),
where u=(u0,u1,u2)∈C3, P=(P0,P1,P2) and the Pi are homogeneous polynomials of degree 2n (n⩾1) with complex coefficients. Let An be the space of coefficients of the right-hand sides of the system (1). Any point α∈An defines a system of the form (1).
Our aim in this paper is to show that the property of the solutions of the system (1) being dense in CP2 is locally characteristic, i.e. we prove that in An there exists an open set U such that the solutions of the system (1) with right-hand side α∈U are everywhere dense in CP2.
This result can be extended without difficulty to the case in which the degree of the homogeneous polynomials appearing in the right-hand side of the system (1) is odd.
Bibliography: 4 titles.
A. A. Shcherbakov, “Dynamics of Local Groups of Conformal Mappings and Generic Properties of Differential Equations on C2”, Proc. Steklov Inst. Math., 254 (2006), 103–120
Ilyashenko Y., “Centennial History of Hilbert's 16th Problem”, Bull. Amer. Math. Soc., 39:3 (2002), 301–354