On the Power of Nonlinear Mappings in Switching Map Systems

A dynamical systems based model of computation is studied, and we demonstrate the computational ability of nonlinear mappings. There exists a switching map system with two types of baker’s map to emulate any Thring machine. The baker’s maps correspond to the elementary operations of computing such as left/right shift and reading/writing of symbols in Thring machines. Taking other mappings with second-order nonlinearity (e.g. the Henon map) as elementary operations, the switching map system becomes an effective analog computer executing parallel computation similarly to MRAM. Our results show that, with bitwise Boolean AND, it has PSPACE computational power. Without bitwise Boolean AND, it is expected that its computational power is between classes RP and PSPACE. These dynamical systems execute more unstable computation than that of classical Turing machines, and this suggests a trade-off principle between stability of computation and computational power.

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