Periodic orbits associated to Hamiltonian functions of degree four

Abstract

We consider the Hamiltonian polynomial function H of degree fourth given by either H (x, y, p_x, p_y) = (+) + (x² +y²) + V₃(x;y) + V₄(x,y), or H (x, y, p_x, p_y) = (- + ) + (-x² +y²) + V₃ (x, y) + V₄ (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.

Keywords:

• Hamiltonian systems
• Periodic orbits
• Stability
• Averaging theory

2000 Mathematics Subject Classification:

• Primary 34C29
• 37J25
• 34C25
• Secondary 85A05