Well, gee Mikey, it's one that can be written in the form of a straight
line. Remember the formula ...

y = mx + b

where m is the slope and b is the y-intercept?

Any function that ain't linear!

Is the Alpha model a linear model?

Yes, because q_{n+1} is a linear function of q_{n}

But wait! Its **output**, the plot of its behavior over time (its
time series shown earlier) is not a straight line. Doesn't that make it
a nonlinear system?

No, what makes a dynamic system *nonlinear *....

is whether the function specifying the change is nonlinear. Not whether
its behavior is nonlinear.

And y is a nonlinear function of x **if**
x is multiplied by another (non-constant) variable, or multiplied by itself
(i. e., raised to some power).

We illustrate *nonlinear* systems using ...

... a model often used to introduce chaos. The Logistic Difference Equation,
or *Logistic Map*, though simple, displays the major chaotic concepts.

We start, generally, with a model of growth.

**x**_{new} = **r x**_{old}

We prefer to write this in terms of n:

**x**_{n+1} = **r** x_{n}.

This says x changes from one time period, n, to the next, n+1, according
to r.* If r is larger than one*, x gets larger with successive iterations
*If r is less than one*, x diminishes. (In the "Alice" example
at the beginning, r is .5).

Let's set r to be larger than one...

We start, year 1 (n=1), with a population of 16 [x_{1}=16], and
since r=1.5, each year x is increased by 50%. So years 2, 3, 4, 5, ...
have magnitudes 24, 36, 54, ...

Our population is growing exponentially. By year 25 we have over a quarter
million.

*Iterations of Growth model with r = 1.5*

So far, notice, we have a *linear* model that produces unlimited
growth.

The Logistic Map prevents unlimited growth by inhibiting growth whenever
it achieves a high level. This is achieved with an additional term, [1
- x_{n}].

The growth measure (x) is also rescaled so that the maximum value x can
achieve is transformed to 1. (So if the maximum size is 25 million, say,
x is expressed as a proportion of that maximum.)

Our new model is

**x _{n+1} = r x_{n} [1 - x_{n}]**

[r between 0 and 4.]

The** [1-x _{n}] **term serves to inhibit growth because as

Plotting x_{n+1} vs. x_{n}, we see we have a nonlinear
relation.

*Limited growth (Verhulst) model. X _{n+1} vs. x_{n},
r = 3.*

We have to **iterate this function **to see how it will behave ...

Suppose r=3, and x_{1}=.1

x_{2} = rx_{1}[1-x_{1}] = 3(.1)(.9) = .27

x_{3}= r x_{2}[1-x_{2}]= 3(.27)(.73) = .591

x_{4}= r x_{3}[1-_{3}]= 3(.591)(.409) = .725

*Behavior of the Logistic map for r = 3, x _{1} = .1, iterated
to give x_{2}, x_{3}, and x_{4
}*It turns out that the logistic map is a very different animal,
depending on its control parameter r. To see this,

*Behavior of the Logistic map for r=.25, .50, and .75. In all cases
x _{1}=.5. *

The same fates awaits any starting value. So long as r is less than
1, x goes toward 0. This illustrates a **one-point attractor.**

*Behavior of the Logistic map for r=1.25, 2.00, and 2.75. In all cases
x _{1}=.5.*

Now, regardless, of the starting value, we have non-zero one-point attractors.

*Behavior of the Logistic map for r=3.2.*

Moving just beyond r=3, the system settles down to alternating between
two points. We have a *two-point attractor*. We have illustrated a
**bifurcation**, or **period doubling**,

*Behavior of the Logistic map for r= 3.54. Four-point attractor*

Another bifurcation. The concept: an **N-point attractor**.

**Chaotic** behavior of the Logistic map at r= 3.99.

So, what is an **attractor**? Whatever the system "settles down
to".

Here is a very important concept from nonlinear dynamics: A system eventually
"settles down". But what it settles down to, its attractor, need
not have 'stability'; it can be very 'strange'.