Verification of chaotic behaviour in the planar restricted three body problem Original Research Article

We present a computer-assisted method to verify the existence of chaotic behaviour in discrete dynamical systems. Our approach is closely related to the procedure introduced by Stoffer and Palmer [Nonlinearity 12 (1999) 1683]. Yet the scheme of Stoffer and Palmer is substantially improved leading to a much more efficient procedure with a wider range of potential applications. Moreover, the application to the planar restricted three body problem with two primaries of equal masses is offered. As is well known, this problem is described by a four-dimensional system of differential equations. Restricting the flow to an energy surface and defining an appropriate Poincaré section the problem is reduced to a map in View the MathML source to which our method is shown to apply. Strictly speaking we do not rigorously establish chaotic behaviour. This is due to the fact that we replace certain rigorous error bounds by what we call realistic estimated upper bounds.