State Prediction for Chaotic 1-D-Maps

Abstract

In order to predict future states of a dynamical system which is assumed to be well described by a deterministic law of motion we must first measure the initial state and then look to the development in time of this initial state under the action of the law of motion which is usually done by using an analytical mathematical expression of the solution of the equations of motions or, if such expression is not known, by simulating the dynamics on a computer. Beside the fact that in general any equation of motion can approximate only within a certain finite precision the motion of a real system there remains the problem that initial states cannot be measured exactly. Nowadays it is widely known that alone this missing knowledge of the exact initial state can make the above procedure of state prediction questionable or even without support. This is the case if we have a chaotic system which is characterized by an exponential grow of small errors of initial states in the time mean.