Euclidean extensions of dynamical systems

2012-02-07This thesis studies the special Euclidean group extension of discrete dynamical systems. Given a dynamical system (X, f), the extended dynamics is constructed as the pair (X×Γ, T) where Γ is the special Euclidean group and T is a map on this extended space. The special Euclidean extension is viewed as an Rⁿ extension of a SO(n) extension of the base dynamics. ❧ After a review of dynamical systems theory, dynamical systems are classified based on whether they are periodic or chaotic. Chaos is defined as exponential mixing (equivalent to decay of correlations on observables). The behavior of typical trajectories in the extended dynamics are then analyzed for these classes of dynamical system. If the base dynamics is periodic, then typical trajectories on the extended space are bounded. For chaotic dynamics these trajectories are typically unbounded. ❧ Finally these results are numerically verified for various examples of dynamical systems. The plots of typical trajectories are shown for SE(2) extensions and the growth rates for various extensions are shown and the results discussed. Despite building a more complex map on a larger space (compared to the original dynamics) extensions of dynamical systems allow us to simplify detecting behavior that is difficult to show in the original dynamics