Linear estimate of the number of zeros of Abelian integrals for a class of integrable non-Hamiltonian systems

We consider the class of all polynomial systems having a conic center (i.e. all periodic orbits round the center are conics). It is proved that all such systems have an isochronous center, and up to affine transformations of the coordinates, each such system coincides with linear isochronous center (S-0), quadratic isochronous centers (S-1),(S-2) due to Loud's classification [1] or a special kind of a cubic isochronous system (S-*). Moreover, it is proved that the upper bounds of the number of zeros of Abelian integrals of (S-1), (S-2) and (S-*) in case of time reversible, are linearly dependent on n, when such systems are Perturbed inside the class of all polynomial systems of degree n.Mathematics, AppliedMathematicsSCI(E)CPCI-S(ISTP)