Universal computation and other capabilities of hybrid and continuous dynamical systems

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Abstract

We explore the simulation and computational capabilities of hybrid and continuous dynamical systems. The continuous dynamical systems considered are ordinary differential equations (ODEs). For hybrid systems we concentrate on models that combine ODEs and discrete dynamics (e.g., finite automata). We review and compare four such models from the literature. Notions of simulation of a discrete dynamical system by a continuous one are developed. We show that hybrid systems whose equations can describe a precise binary timing pulse (exact clock) can simulate arbitrary reversible discrete dynamical systems defined on closed subsets of Rn. The simulations require continuous ODEs in R2n with the exact clock as input. All four hybrid systems models studied here can implement exact clocks. We also prove that any discrete dynamical system in Zn can be simulated by continuous ODEs in R2n + 1. We use this to show that smooth ODEs in R3 can simulate arbitrary Turing machines, and hence possess the power of universal computation. We use the famous asynchronous arbiter problem to distinguish between hybrid and continuous dynamical systems. We prove that one cannot build an arbiter with devices described by a system of Lipschitz ODEs. On the other hand, all four hybrid systems models considered can implement arbiters even if their ODEs are Lipschitz.

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Work supported by the Army Research Office and the Center for Intelligent Control Systems under grants DAAL03-92-G-0164 and DAAL03-92-G-0115.