A dynamic system is a set of functions (rules, equations) that
specify how variables change over time.
First example ...
Alice's height diminishes by half every minute...
Second example ...
xnew = xold + yold
The second example illustrates a system with two variables, x and y. Variable x is changed by taking its old value and adding the current value of y. And y is changed by becoming x's old value. Silly system? Perhaps. We're just showing that a dynamic system is any well-specified set of rules.
discrete vs. continuous variables
stochastic vs. deterministic dynamic systems
How they differ:
Variables change in time, parameters do not.
Discrete variables are restricted to integer values, continuous variable are not.
Stochastic systems are one-to-many; deterministic systems are one-to-one
This last distinction will be made clearer as we go along ...
The current state of a dynamic system is specified by the current
value of its variables, x, y, z, ...
The process of calculating the new state of a discrete system is called iteration.
To evaluate how a system behaves, we need the functions, parameter values and initial conditions or starting state.
To illustrate...Consider a classic learning theory, the alpha model, which specifies how qn, the probability of making an error on trial n, changed from one trial to the next
qn+1 = ß qn The new error probability is diminished by ß (which is less than 1, greater than 0). For example, let the the probability of an error on trial 1 equal to 1, and ß equal .9. Now we can calculate the dynamics by iterating the function, and plot the results.
q1 = 1
q2 = ßq1 = (.9)(1) = .9
q3 = (.9)q2 = (.9)(.9) = .81
Error probabilities for the alpha model, assuming q1=1, ß =.9. This "learning curve" is referred to as a time series.
Certainly the idea that systems change in time is not new. Nor is the idea that the changes are probabilistic.
Deterministic nonlinear dynamic systems.
As we will see, these systems give us:
This last point is not pursued here.
Next Section: Nonlinear Dynamic Systems