Renato C. Calleja

Renato C. Calleja, Eusebius J. Doedel, Antony R. Humphries et al.

We demonstrate the remarkable effectiveness of boundary value formulations coupled to numerical continuation for the computation of stable and unstable manifolds in systems of ordinary differential equations. Specifically, we consider the Circular Restricted Three-Body Problem (CR3BP), which models the motion of a satellite in an Earth- Moon-like system. The CR3BP has many well-known families of periodic orbits, such as the planar Lyapunov orbits and the non-planar Vertical and Halo orbits. We compute the unstable manifolds of selected Vertical and Halo orbits, which in several cases leads to the detection of heteroclinic connections from such a periodic orbit to invariant tori. Subsequent continuation of these connecting orbits with a suitable end point condition and allowing the energy level to vary, leads to the further detection of apparent homoclinic connections from the base periodic orbit to itself, or the detection of heteroclinic connections from the base periodic orbit to other periodic orbits. Some of these connecting orbits could be of potential interest in space-mission design.

Adrian P. Bustamante, Renato C. Calleja

Conformally symplectic systems include mechanical systems with a friction proportional to the velocity. Geometrically, these systems transform a symplectic form into a multiple of itself making the systems dissipative or expanding. In the present work we consider the limit of small dissipation. The example we study is a family of conformally symplectic standard maps of the cylinder for which the conformal factor, $b(\varepsilon)$, is a function of a small complex parameter, $\varepsilon$. We assume that for $\varepsilon=0$ the map preserves the symplectic form and the dependence on $\varepsilon$ is cubic, i.e., $b(\varepsilon) = 1 - \varepsilon^3$. We compute perturbative expansions formally in $\varepsilon$ and use them to estimate the shape of the domains of analyticity of invariant circles as functions of $\varepsilon$. We also give evidence that the functions might belong to a Gevrey class at $\varepsilon = 0$. We also perform numerical continuation of the solutions as they pass through the boundary of the domain to illustrate that the monodromy of the solutions is trivial. The numerical computations we perform support conjectures on the shape of the domains of analyticity.