On the high order approximation of the centre manifold for ODEs

Many times in dynamical systems one wants to understand the bounded motion around an equilibrium point. From a numerical point of view, we can take arbitrary initial conditions close to the equilibrium points, integrate the trajectories and plot them to have a rough idea of motion. If the dimension of the phase space is high, we can take suitable Poincaré sections and/or projections to visualise the dynamics. Of course, if the linear behaviour around the equilibrium point has an unstable direction, this procedure is useless as the trajectories will escape quickly. We need to get rid, in some way, of the instability of the system.
   Here we focus on equilibrium points whose linear dynamics is a cross product of one hyperbolic directions and several elliptic ones. We will compute a high order approximation of the centre manifold around the equilibrium point and use it to describe the behaviour of the system in an extended neighbourhood of this point. Our approach is based on the graph transform method. To derive an efficient algorithm we use recurrent expressions for the expansion of the non - linear terms on the equations of motion.
   Although this method does not require the system to be Hamiltonian, we have taken a Hamiltonian system as an example. We have compared its efficiency with a more classical approach for this type of systems, the Lie series method. It turns out that in this example the graph transform method is more efficient than the Lie series method. Finally, we have used this high order approximation of the centre manifold to describe the bounded motion of the system around and unstable equilibrium point.