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 consider the Hamiltonian polynomial function H of degree fourth given by either H(x,y,{p_x},{p_y}) = \frac{1}{2}(p_x^2 + p_y^2) + \frac{1}{2}({x^2} + {y^2}) + {V_3}(x,y) + {V_4}(x,y),\,\,{\text{or}}\,H(x,y,{p_x},{p_y}) = \frac{1}{2}( - p_x^2 + p_y^2) + \frac{1}{2}( - {x^2} + {y^2}) + {V_3}(x,y) + {V_4}(x,y), where V3(x,y) and V4(x,y) are homogeneous polynomials of degree three and four, respectively. Our main objective is to prove the existence and stability of periodic solutions associated to H using the classical averaging method.