We investigate the Dirichlet solution for a semianalytic continuous function on the boundary of a semianalytic bounded domain in the plane. We show that the germ of the Dirichlet solution at a boundary point with angle greater than 0 lies in a certain quasianalytic class used by Ilyashenko in his work on Hilbert’s 16th problem. With this result we can prove that the Dirichlet solution is defina...

We apply the averaging method in a class of planar systems given by linear center perturbed sum continuous homogeneous vector fields, to study lower bounds for their number limit cycles. Our results can be applied models where smoothness is lost on set Σ = { x y 0 } . They also motivate consider variant Hilbert 16th problem, goal bound cycles terms monomials family polynomial instead doing this...

we present a collocation method to obtain the approximate solutionof troesch's problem which arises in the confinement of a plasmacolumn by radiation pressure and applied physics. by using thechristov rational functions and collocation points, this methodtransforms troesch's problem into a system of nonlinear algebraicequations. the rate of convergence is shown to be exponential. then...

This paper is concerned with bifurcation of limit cycles in a fourth-order near-Hamiltonian system with quartic perturbations. By bifurcation theory, proper perturbations are given to show that the system may have 20, 21 or 23 limit cycles with different distributions. This shows thatH(4) ≥ 20, whereH(n) is the Hilbert number for the second part of Hilbert’s 16th problem. It is well known that ...

One of the main problems in the qualitative theory of real planar differential systems is the determination of number and relative positions of limit cycles. The problem concerns “the most elusive” second part of Hilbert’s 16th problem (see [Smale, 1998; Lloyd, 1988]). In 1983, Jibin Li (see [Li, 2003; Li & Li, 1985; Li & Liu, 1991, 1992]) posed a method of detection functions to investigate po...

Given a C ? family of planar vector fields { X ? ˆ } ? W having hyperbolic saddle, we study the Dulac map D ( s ; ) and time T between two transverse sections located in separatrices at arbitrary distance from saddle. We show (Theorems A B, respectively) that, for any 0 L > , functions have an asymptotic expansion = ? with remainder being uniformly -flat respect to parameters. The principal par...