Bifurcations of links of periodic orbits in Mathieu systems

We consider the nonlinear dissipative Mathieu equation x + delta(x) over dot + omega(2)(1 + epsilon cos t)x + gx(3) = 0, g, delta > 0 We prove that orbits escape from infinity, and that therefore the sphere S-3 can be considered as its phase space. If the parameter delta is large enough, the system is non-singular Morse-Smale, and its periodic orbits define a Hogf link. As delta decreases, the system undergoes some bifurcations that we describe geometrically. We relate the bifurcation orbits to periodic orbits continued from the linear Mathieu equation.