TANGENTIAL VERSION OF HILBERT 16th PROBLEM FOR THE ABEL EQUATION
نویسنده
چکیده
Two classical problems on plane polynomial vector fields, Hilbert’s 16th problem about the maximal number of limit cycles in such a system and Poincaré’s center-focus problem about conditions for all trajectories around a critical point to be closed, can be naturally reformulated for the Abel differential equation y′ = p(x)y + q(x)y. Recently, the center conditions for the Abel equation have been related to the composition factorization of P = R p and Q = R q and to the vanishing conditions for the moments mi,j = R P Qq. On the basis of these results we start in the present paper the investigation of the “Hilbert’s tangential problem” for the Abel equation, which is to find a bound for the number of zeroes of I(t) = R b
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Around Hilbert –Arnol′d Problem
H(n) = uniform bound for the number of limit cycles of (1) . One way to formulate the Hilbert 16th problem is the following: Hilbert 16th Problem (HP). Estimate H(n) for any n ∈ Z+. To prove that H(1) = 0 is an exercise, but to find H(2) is already a difficult unsolved problem (see [DRR,DMR] for work in this direction). Below we discuss two of the most significant branches of research HP has ge...
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