## 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**_{old
}**y**_{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.

q_{1} = 1

q_{2} = ßq_{1} = (.9)(1) = .9

q_{3} = (.9)q_{2} = (.9)(.9) = .81

etc. ...

*Error probabilities for the alpha model, assuming q*_{1}=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