Nonlinear Computation with Switching Map Systems

A dynamical systems based model of computation is studied. We demonstrate the computational capability of a class of dynamical systems called switching map systems. There exists a switching map system with two types of baker's map to emulate any Turing machines. The baker's maps are corresponding to the elementary operations of Turing machines such as left/right head-moving and read/write symbols. A connection between the generalized shifts by C. Moore [Moore 91] and the inputoutput mappings by L. Blum et al. [Blum, Cucker, Shub and Smale 98] is shown with our model. We present four concrete examples of switching map systems corresponding to the Chomsky hierarchy. Taking non-hyperbolic mappings as elementary operations, it is expected that the switching map systems shows a new model of computation with nonlinearity as an oracle.