Dr. Uribe-Santibañez, Marco

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.

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.