Exemplo de dinâmica caótica em mecânica celeste

Abstract

In this work we will study the Sitnikov restricted three body problem. We will show that the solutions of the second order ordinary di erential equation which describes the movement of the null mass body are de ned for all time and that a non-null solution is necessarily oscillatory, parabolic or hyperbolic when the time goes to + or -. We shall characterize in the plane of initial conditions what are the values that corresponds to each of this solutions. We will show that the mapping S which takes one zero of one solution to the next zero of the same solution is a di eomorphism and, near a solution that has only one zero and is parabolic when the time goes to + and to -, the dynamical system given by the map S-1 : I > I (where I is the completely invariant set by S) has chaotic behavior.