Computation with Switching Map Systems: Nonlinearity and Computational Complexity

A dynamical-systems-based model of computation is studied. We demonstrate the computational ability of nonlinear mappings. There exists a switching map system with two types of baker's map to emulate any Turing machine. Taking non-hyperbolic mappings with second-order nonlinearity (e.g., the HŽnon map) as elementary operations, the switching map system becomes an effective analog computer executing parallel computation similar to MRAM. Our results show that, with an integer division map similar to the Gauss map, it has PSPACE computational power. Without this, we conjecture that its computational power is between class RP and PSPACE. Unstable computation with this system implies that there would be a trade-off principle between stability of computation and computational power.