---
title: "Discrete time and continuous time"
chunk: 2/2
source: "https://en.wikipedia.org/wiki/Discrete_time_and_continuous_time"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T03:51:24.635876+00:00"
instance: "kb-cron"
---

== Relevant contexts ==
Discrete time is often employed when empirical measurements are involved, because normally it is only possible to measure variables sequentially. For example, while economic activity actually occurs continuously, there being no moment when the economy is totally in a pause, it is only possible to measure economic activity discretely. For this reason, 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 period in 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 the theory 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, and often in areas such as physics an exact description requires the use of continuous time. In a continuous time context, the 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 the logistic map or logistic equation, is

  
    
      
        
          x
          
            t
            +
            1
          
        
        =
        r
        
          x
          
            t
          
        
        (
        1
        −
        
          x
          
            t
          
        
        )
        ,
      
    
    {\displaystyle x_{t+1}=rx_{t}(1-x_{t}),}
  

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 inclusive whose value in period t nonlinearly affects its value in the next period, t + 1. For example, if 
  
    
      
        r
        =
        4
      
    
    {\displaystyle r=4}
  
 and 
  
    
      
        
          x
          
            1
          
        
        =
        1
        
          /
        
        3
      
    
    {\displaystyle x_{1}=1/3}
  
, then for t = 1 we have 
  
    
      
        
          x
          
            2
          
        
        =
        4
        (
        1
        
          /
        
        3
        )
        (
        2
        
          /
        
        3
        )
        =
        8
        
          /
        
        9
      
    
    {\displaystyle x_{2}=4(1/3)(2/3)=8/9}
  
, and for t = 2 we have 
  
    
      
        
          x
          
            3
          
        
        =
        4
        (
        8
        
          /
        
        9
        )
        (
        1
        
          /
        
        9
        )
        =
        32
        
          /
        
        81
      
    
    {\displaystyle x_{3}=4(8/9)(1/9)=32/81}
  
.
Another example models the adjustment of a price P in response to non-zero excess demand for a product as

  
    
      
        
          P
          
            t
            +
            1
          
        
        =
        
          P
          
            t
          
        
        +
        δ
        ⋅
        f
        (
        
          P
          
            t
          
        
        ,
        .
        .
        .
        )
        ,
      
    
    {\displaystyle P_{t+1}=P_{t}+\delta \cdot f(P_{t},...),}
  

where 
  
    
      
        δ
      
    
    {\displaystyle \delta }
  
 is the positive speed-of-adjustment parameter which is less than or equal to 1, and where 
  
    
      
        f
      
    
    {\displaystyle f}
  
 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

  
    
      
        
          
            
              d
              P
            
            
              d
              t
            
          
        
        =
        λ
        ⋅
        f
        (
        P
        ,
        .
        .
        .
        )
        ,
      
    
    {\displaystyle {\frac {dP}{dt}}=\lambda \cdot f(P,...),}
  

where the left side is the first derivative of the price with respect to time (that is, the rate of change of the price), 
  
    
      
        λ
      
    
    {\displaystyle \lambda }
  
 is the speed-of-adjustment parameter which can be any positive finite number, and 
  
    
      
        f
      
    
    {\displaystyle f}
  
 is again the excess demand function.

== Graphical depiction ==
A variable measured in discrete time can be plotted as a step function, in which each time period is given a region on the horizontal axis of the same length as every other time period, and the measured variable is plotted as a height that stays constant throughout the region of the time period. In this graphical technique, the graph appears as a sequence of horizontal steps. Alternatively, each time period can be viewed as a detached point in time, usually at an integer value on the horizontal axis, and the measured variable is plotted as a height above that time-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 time is considered to be the entire real axis or at least some connected portion of it.

== See also ==

== References ==

Gershenfeld, Neil A. (1999). The Nature of mathematical Modeling. Cambridge University Press. ISBN 0-521-57095-6.
Wagner, Thomas Charles Gordon (1959). Analytical transients. Wiley.