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.
Mathematics Subject Classification: Primary: 37M99; Secondary: 70F07.