• Anisotropic Kepler Problem
• Chaos
• Conley-Moser conditions
• Lyapunov characteristic number.
• Smale horseshoe
• Symbolic Dynamics

The Anisotropic Kepler Problem (AKP) is a Hamiltonian System which posseses chaotic dynamics. Following two different approaches, one by Devaney and the other by Contopoulos, we show the existence of chaotic behaviour in the AKP for all the cases in which there is anistropy. In the first approach we use the Conley-Moser conditions which provide a generalization of the situation of the Smale horseshoe. We start by making an accurate study of the flow of the problem inside the Hamiltonian energy level sets, by using many techniques from the theory of dynamical systems such as the regularization of the singularities and the use of Poincaré maps; particular attention is given to the bi-collision orbits, which begin and start with a collision. We then isolate a collection of "windows" for the flow and we define a Poincaré map of a two dimensional square in phase space, considering only orbits which meet those windows in a prescribed order. We define horizontal and vertical strips as in the Smale horseshoe and we discuss a way to label these strips, so that we can associate a bi-infinite sequence of integers to each admissible orbit and we discuss the behaviour of such orbits in the configuration space. Following Smale and Moser we then find an invariant set on which the dynamics is topologically conjugate to a shift on a suitable set of sequences.
The second approach uses the concept of Lyapunov characteristic number as a way to measure the sensitive dependence on initial conditions (and then chaos) of a system. This number is zero in the proximity of a stable periodic orbit and it is a positive number in a chaotic domain. We describe some tests and their results by Contopoulos.