Resumen de Dynamics and Periodic Solutions in Cubic Polynomial Hamiltonian Systems

Dante Carrasco Olivera, Claudio Vidal

• We consider the Hamiltonian function defined by the cubic polynomial H=12(y12+y22)+V(x1,x2) where the potential V(x)=δV2(x1,x2)+V3(x1,x2), with V2(x1,x2)=12(x12+x22) and V3(x1,x2)=13x13+fx1x22+gx23, with f and g are real parameters such that f≠0 and δ is 0 or 1. Our objective is to study the number and bifurcations of the equilibria and its type of stability. Moreover, we obtain the existence of periodic solutions close to some equilibrium points and an isolated symmetric periodic solution distant of the equilibria for some convenient region of the parameters. We point out the role of the parameters and the difference between the homogeneous potential case (δ=0) and the general case (δ=1).