Working with Discrete Dynamical Systems

The model

is an example of a discrete dynamical system. We will discuss these kinds of models in depth later on. Here we want to discuss how to compute the predictions made by this kind of model. As an example, we will assume that the constant k is 0.90, that the ambient temperature is 22 degrees Celsius, and that the initial temperature is 95 degrees Celsius. Thus, the model can be written

or, better yet,

and T₁ = 95.

Like all discrete dynamical systems this one has two pieces of information

• An initial condition (the value of T₁). This tells us how the system starts.

• A change equation of the form
  T_{i + 1} = F(T_i)

  that tells us how the system changes -- that is, how to compute each measurement from the preceding measurement.

Now we use the change equation to compute T₂ from T₁ as follows

Notice that by using the equation

twice we see that

and by using it three times we see that

and, more generally,

The formula below enables us to predict any measurement in one step.