Sergey Yakovenko - Tangential Hilbert's 16-th problem and differential equations

We discuss some recent achievements related to bifurcations of limit cycles from periodic orbits of planar Hamiltonian systems. In a somewhat inexpected way, all progress in counting zeros of Abelian integrals (the qualitative property of a completely algebraic object) was achieved by applying purely transcendental methods of ``quantitative'' theory of ordinary differential equations.