A cause of chaos

is available here from the publisher website (Applied Probability Trust). A cause of chaos An abstract dynamical system consists of a collection S of points together with a transformation, or function, f which maps points of S into points of S. The points in S stand for all possible states of the system. The transformation is a “process of change ” over one time unit, that changes each state x in S into another state f(x). Then we can interpret the equation f(x)=y as meaning that if the system is in state x, over the next time unit it will change into the state y. Alternatively, x is a “cause ” of y, or x is an “antecedent ” of y. There is no reason to consider the elapse of only one time unit – the transformation can be applied over and over again. Thus, if we start off in an initial state x, the next state is f(x), then the next state is f(f(x)), and so on. The sequence of states x, f(x), f(f(x)), f(f(f(x))), ….. is called the orbit of x, and it describes the evolution of the system from an initial state x. Now the actual state x of the system may not be known, but may only be approximated, by the state y. Then, if the orbits of x,y are very different, this would mean that the behaviour of the system cannot be predicted. This inability to predict is an intrinsic feature of chaotic systems. In this paper, chaotic behaviour is linked to a property that a dynamical system may have: given a state y, it may have mor