WorkIntro

Introduction to Dynamic Systems

What is a dynamic system?

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 ...
x_new = x_old + y_oldy_new= x_old

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.

Here are some important Distinctions:

variables (dimensions) vs. parameters
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 ...

Terms

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 q_n, the probability of making an error on trial n, changed from one trial to the next
q_n+1 = ß q_n 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

etc. ...

Error probabilities for the alpha model, assuming q₁=1, ß =.9. This "learning curve" is referred to as a time series.

So far, we have some new ideas, but much is old ...

What's not new

Dynamic Systems
Certainly the idea that systems change in time is not new. Nor is the idea that the changes are probabilistic.

What's new

Deterministic nonlinear dynamic systems.

As we will see, these systems give us:

A new meaning to the term unpredictable.
A different attitude toward the concept of variability.
Some new tools for exploring time series data and for modeling such behavior.
And, some argue, a new paradigm.

This last point is not pursued here.

Next Section: Nonlinear Dynamic Systems