Dr. Uribe-Santibañez, Marco

In this paper we consider the autonomous Hamiltonian system with two degrees of freedom associated to the function H = ½ (x2 + y2) + ½ (p2/x + p2/y) + V5(x, y), where V5(x, y) = (A/5x5 + Bx3y2 + C/5 xy4) which is related to a homogeneous potential of degree five. We prove the existence of different families of periodic orbits and the type of stability is analyzed through the averaging theory which guarantee the existence of such orbits on adequate sets defined by the parameters A, B, C.

In this work, we show the existence of zero-Hopf periodic orbits in a 10-parametric differential equation of third order x′′′ + (a1x′ + b1x + c1)x′′ + (a2x′ + b2x + c2)x′ + (a3x′ + b3x + c3)x + k = 0, where ai, bi, ci, k ∈ R for i = 1, 2, 3. This family is based on a generalization of the equation associated to the Hiemenz flow, when the boundary conditions are neglected, and it will be named as generalized Hiemenz equation. Our approach relies in the use of averaging method. Moreover, the kind of stability of the periodic orbits is determined according to the parameters.

We provide explicit lower and upper bounds for the maximum number of isolated zeros of the complete Abelian integral associated with a rational polynomial, with non-trivial global monodromy, and a polynomial 1-form of degree n. Moreover, we obtain the explicit form of the relative cohomology of the polynomial 1-forms with respect to the rational polynomial.

It is known that the principal Poincaré Pontryagin function is generically an Abelian integral. We give a sufficient condition on monodromy to ensure that it is also an Abelian integral in non-generic cases. In non-generic cases it is an iterated integral. Uribe (2006 J. Dyn. Control. Syst. 12 109–34, 2009 J. Diff. Eqns 246 1313–41) gives in a special case a precise description of the principal Poincaré Pontryagin function, an iterated integral of length at most 2, involving logarithmic functions with only 1 ramification at a point at infinity. We extend this result to some non-isomonodromic families of real Morse polynomials.