Discrete time and continuous timeFrom Wikipedia, the free encyclopedia

In mathematics and in particular mathematical dynamics, discrete time and continuous time are two alternativeframeworks within which to model variables that evolve over time.

Contents

1 Discrete time

2 Continuous time

3 Relevant contexts

4 Types of equations

4.1 Discrete time

4.2 Continuous time

5 Graphical depiction

6 See also

Discrete time

Discrete time views values of variables as occurring at distinct, separate "points in time", or equivalently as beingunchanged throughout each non-zero region of time ("time period"). Thus a variable jumps from one value toanother as time moves from time period to the next. This view of time corresponds to a digital clock that gives afixed reading of 10:37 for a while, and then jumps to a new fixed reading of 10:38, etc. In this framework, eachvariable of interest is measured once at each time period. The number of measurements between any two timeperiods is finite. Measurements are typically made at sequential integer values of the variable "time".

Continuous time

In contrast, continuous time views variables as having a particular value for potentially only an infinitesimally shortamount of time. Between any two points in time there are an infinite number of other points in time. The variable"time" ranges over the entire real number line, or depending on the context, over some subset of it such as the non-negative reals.

Relevant contexts

Discrete time is often employed when empirical measurements are involved, because normally it is only possible tomeasure variables sequentially. For example, while economic activity actually occurs continuously, there being nomoment when the economy is totally in a pause, it is nevertheless possible to measure economic activity onlydiscretely. Hence published data on, for example, gross domestic product will show a sequence of quarterly values.

When one attempts to empirically explain such variables in terms of other variables and/or their own prior values,one uses time series or regression methods in which variables are indexed with a subscript indicating the time periodin which the observation occurred. For example, yt might refer to the value of income observed in unspecified time

period t, y3 to the value of income observed in the third time period, etc.

Moreover, when a researcher attempts to develop a theory to explain what is observed in discrete time, often thetheory itself is expressed in discrete time in order to facilitate the development of a time series or regression model.

On the other hand, it is often more mathematically tractable to construct theoretical models in continuous time, andoften in areas such as physics an exact description requires the use of continuous time. In a continuous-time contextthe value of a variable y at an unspecified point in time is denoted as y(t) or, when the meaning is clear, simply as y.

Types of equations

Discrete time

Discrete time makes use of difference equations, also known as recurrence relations. An example, known as thelogistic map or logistic equation, is

in which r is a parameter in the range from 2 to 4 inclusive, and x is a variable in the range from 0 to 1 inclusivewhose value in period t nonlinearly affects its value in the next period, t+1. For example, if and

, then for t=1 we have , and for t=2 we have

.

Another example models the adjustment of a price P in response to non-zero excess demand for a product as

where is the positive speed-of-adjustment parameter which is less than or equal to 1, and where is the excess

demand function.

Continuous time

Continuous time makes use of differential equations. For example, the adjustment of a price P in response to non-zero excess demand for a product can be modeled in continuous time as

where the left side is the first derivative of the price with respect to time (that is, the rate of change of the price), isthe speed-of-adjustment parameter which can be any positive finite number, and is again the excess demand

function.

Graphical depiction

The values of a variable measured in discrete time can be plotted as a step function, in which each time period isgiven a region on the horizontal axis of the same length as every other time period, and the measured variable isplotted as a height that stays constant throughout the region of the time period. In this graphical technique, the graphappears as a sequence of horizontal steps. Alternatively, each time period can be viewed as a detached point intime, usually at an integer value on the horizontal axis, and the measured variable is plotted as a height above thattime-axis point. In this technique, the graph appears as a set of dots.

The values of a variable measured in continuous time are plotted as a continuous function, since the domain of timeis considered to be the entire real axis or at least some connected portion of it.

See also

Discrete calculus

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Categories: Time Dynamical systems

This page was last modified on 14 August 2013 at 16:25.

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