A proof of the non-existence of a bounded-derivative continuous model for a discrete chaotic system

Abstract Takens' theorem states that for any dynamical system there exists a dimension in which the system can be embedded as a sampling of a continuous function. This paper presents a result which shows that a non-constant continuous model with a bounded time derivative does not exist when the system to be embedded is a discrete chaotic process. The result presented here does not contradict Takens' theorem which has no requirement of a bounded time derivative. The assumption of a bounded derivative is an important one since recurrent neural network learning rules usually implicitly assume that a bounded time derivative exists for the network dynamics equations which define the system model.