Hilbert’s 16th Problem and Its Weak Form

Abstract

Consider the planar differential systems

$$ \dot x = P_n (x,y),\dot y = Q_n (x,y), $$

((1.1))

where P_n and Q_n are real polynomials in x, y and the maximum degree of P and Q is n. The second half of the famous Hilbert’s 16th problem, proposed in 1900, can be stated as follows (see [70]):

For a given integer n, what is the maximum number of limit cycles of system (1.1) for all possible P_n and Q_n ? And how about the possible relative positions of the limit cycles ?