timing event

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Giovanni Gallavotti (2008), Scholarpedia, 3(1):5906.

revision #27611 [link to/cite this article]

Curator: Dr. Giovanni Gallavotti, Physics, University di Roma, Italy

Mathematical models for time evolution can be differential equations whose solutions represent motions developing in continuous time $t$ or, often, maps whose $n$ -th iterate represents motions developing at discrete integer times $n$ . The point representing the state of the system at time $t$ is denoted $S_tx$ in the continuous time models or, at the $n$ -th observation, $S^n\xi$ in the discrete time models. Here $x,\xi$ will be points on a manifold $X$ or $\Xi$ 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 $\Xi\subset X$ and triggers the action of a ``measurement apparatus, e.g. shooting a picture after noting the position of a clock arm. If $\Xi$ comprises the collection of the timing events, i.e. of the states $\xi$ of the system which induce the act of measurement, motion of the system can also be represented as a map $\xi\to S\xi$ defined on $\Xi$ .

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

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

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).

Invited by:	Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia
Action editor:	Dr. Eugene M. Izhikevich, Editor-in-Chief of Scholarpedia, the peer-reviewed open-access encyclopedia

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timing event

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