Computational Complexity for Continuous Time Dynamics

Dissipative flows model a large variety of physical systems. In this Letter the evolution of such systems is interpreted as a process of computation; the attractor of the dynamics represents the output. A framework for an algorithmic analysis of dissipative flows is presented, enabling the comparison of the performance of discrete and continuous time analog computation models. A simple algorithm for finding the maximum of $n$ numbers is analyzed, and shown to be highly efficient. The notion of tractable (polynomial) computation in the Turing model is conjectured to correspond to computation with tractable (analytically solvable) dynamical systems having polynomial complexity.