arXiv subjects
Publications and source records attributed to Bassem Ben Hamed.
We find an upper bound for the number of limit cycles, bifurcating from the 8-loop of the Duffing oscillator $x"= x-x^{3}$ under the special cubic perturbation $$ x"= x-x^{3}+λ_{1}y+λ_{2}x^{2}+λ_{3}xy+λ_{4}x^{2}y . $$
We show that the center set of reversible cubic systems, close to the symmetric Hamiltonian system $x'=y, y'= x-x^3$ has two irreducible components of co-dimension two in the parameter space. One of them corresponds to the Hamiltonian stratum, the other to systems which are polynomial pull back of an appropriate linear system
We consider arbitrary one-parameter cubic deformations of the Duffing oscillator $x"=x-x^3$. In the case when the first Melnikov function $M_1$ vanishes, but $M_2\neq 0$ we compute the general form of $M_2$ and study its zeros in a suitable complex domain.
The cyclicity of the exterior period annulus of the asymmetrically perturbed Duffing oscillator is a well known problem extensively studied in the literature. In the present paper we provide a complete bifurcation diagram for the number of the zeros of the associated Melnikov function in a suitable complex domain.
We prove that the holonomy group at infinity of the Painleve VI equation is virtually commutative.
We classify all functions satisfying non-trivial families of PVI equations. It turns out that, up to an Okamoto equivalence, there are exactly four families parameterized by affine planes or lines. Each affine space is generated by points of "geometric origin", associated either to deformations of elliptic surfaces with four singular fibers, or to deformations of three-sheeted covers of the projective line with branching locus consisting of four points.
We describe two algebraic solutions of the sixth Painlevé equation which are related to (isomonodromic) deformations of Picard-Fuchs equations of order two.