Transient ensemble dynamics in time-independent galactic potentials

This paper summarises a numerical investigation of the short time, possibly transient, behaviour of ensembles of stochastic orbits evolving in xed nonintegrable potentials, with the aim of deriving insights into the structure and evolution of galaxies. The simulations involved three diierent two-dimensional potentials, quite diierent in appearance. However, despite these diierences ensembles in all three potentials exhibit similar behaviour. This suggests that the conclusions inferred from the simulations are robust, relying only on basic topological properties, e.g., the existence of KAM tori and cantori. Generic ensembles of initial conditions, corresponding to stochastic orbits, exhibit a rapid coarse-grained approach towards a near-invariant distribution on a timescale t H , the age of the Universe. This approach is exponential in time, with a rate, , that exhibits a direct correlation with the value of the Liapounov exponent,. However, this near-invariant distribution does not correspond to the true invariant measure: If this distribution be evolved for much longer timescales one sees systematic evolutionary eeects associated with diiusion through cantori, which on short timescales divide stochastic orbits into two distinct classes, namely connned and unconnned. For the deterministic simulations described herein, the timescale for this diiusion is t H , although various irregularities associated with external and/or internal irregularities can drastically accelerate this process. A principal tool in the analysis is the notion of a local Liapounov exponent, which provides a statistical char-acterisation of the overall instability of stochastic orbits over nite time intervals. In particular, there is a precise sense in which connned stochastic orbits are less unstable, with smaller local Liapounov exponents, than are unconnned stochastic orbits.