timing event

Mathematical models for time evolution can be differential equations whose solutions represent motions developing in continuous time <math>t</math> or, often, maps whose <math>n</math>-th iterate represents motions developing at discrete integer times <math>n\ .</math> The point representing the state of the system at time <math>t</math> is denoted <math>S_tx</math> in the continuous time models or, at the <math>n</math>-th observation, <math>S^n\xi</math> in the discrete time models. Here <math>x,\xi</math> will be points on a manifold <math>X</math> or <math>\Xi</math> respectively, called the phase space, or the space of the states, of the system.

The connection between the two representations of motions is illustrated by means of the following notion of ``timing event.

Physical observations are always performed at discrete times: i.e. when some special, prefixed, timing event occurs, typically when the state of the system is in a set <math>\Xi\subset X</math> and triggers the action of a ``measurement apparatus, e.g. shooting a picture after noting the position of a clock arm. If <math>\Xi</math> comprises the collection of the timing events, i.e. of the states <math>\xi</math> of the system which induce the act of measurement, motion of the system can also be represented as a map <math>\xi\to S\xi</math> defined on <math>\Xi\ .</math>

For this reason mathematical models are often maps which associate with a timing event <math>\xi\ ,</math> i.e. a point <math>\xi</math> in the manifold <math>\Xi</math> of the measurement inducing events, the next timing event <math>S\xi\ .</math>

If the system motions also admit a continuous time representation on a space of states <math>X\supset\Xi</math> then there will be a simple relation between the evolution in continuous time <math>x\to S_tx</math> and the discrete representation <math>\xi\to S^n\xi</math> in discrete integer times <math>n\ ,</math> between successive timing events, namely <math>S\xi\equiv S_{\tau(\xi)}\xi\ ,</math> if <math>\tau(\xi)</math> is the time elapsing between the timing event <math>\xi</math> and the subsequent one <math>S\xi</math>

The discrete time representation is particularly useful mathematically in cases in which the continuous evolution shows singularities: the latter can be avoided by choosing timing events which occur when the point representing the system is not singular nor too close to a singularity (when the physical measurements become difficult or impossible).