These pages are taken from a *Workshop* presented at the annual meeting of the **Society for Chaos Theory in
Psychology and the Life Sciences**

June 28,1996 at Berkeley, California.

**© Keith Clayton**

- Exploring Chaos and Fractals from the Royal Melbourne Institute of Technology, Melbourne Australia
- Chaos Introduction from the University of Illinois.

Alice's height diminishes by half every minute...

*Another example* ...

x(new) = x(old) + y(old)

y(new)= x(old)

This last rule illustrates a system with two variables. Variable X is changed by taking its old value and adding the 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.

**variables**(dimensions) vs.**parameters****discrete**vs.**continuous**variables**stochastic**vs.**deterministic**dynamic systems

**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

The process of calculating the new state of a

To evaluate how a system behaves, we need the functions, parameter values and

q(1) = 1

q(2) = ßq(1) = (.9)(1) = .9

q(3) = (.9)q(2) = (.9)(.9) = .81

etc. ...

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

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

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*.

y = mx + b

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

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 (Figure 1 above) 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 by itself (that is, raised to some power).

We illustrate *nonlinear* systems using ...

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.

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

[r between 0 and 4.]

The [1-x(n)] term serves to inhibit growth because as x approaches 1, [1-x(n)] approaches 0.

Plotting x(n+1) vs. x(n), we see we have a nonlinear relation.

We have to

Suppose r=3, and x(1)=.1

x(2) = r x(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-x(3)]= 3(.591)(.409) = .725

It turns out that the logistic map is a very different animal, depending on its control parameter r. To see this,

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'.

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A Bifurcation

However, the system is not chaotic for all values of r greater than 3.57.

Let's zoom in a bit.

In fact, between 3.57 and 4 there is a rich interleaving of chaos and order. A small change in r can make a stable system chaotic, and vice versa.

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Suppose I have two near-by starting points.

The first 24 cycles on the left, next 24 on the right.

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- apparent randomness
- deterministic
- sensitive to 'initial' conditions

- 'bifurcation' - technically, not a dynamic change
- 'chaos' - technically, not 'overwhelming anxiety'

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When we have a system with two or more dimensions,

- its current
**state**is the current values of its variables, and - treated as a
**point**in**phase (state) space**, and - we refer to its
**trajectory**or**orbit**in time.

The number of predators is represented by y, the number of prey by x.

Ê
*The phase-space of the predator-prey system.*

The resulting trajectory (idealized)...

Points 1-4 start with about the same number of prey but with different numbers of predators.

Let's look at this system as a

- Changes in discrete variables are expressed with
**difference equations**, such as the logistic map. - Changes in continuous variables are expressed with
**differential equations**

dx/dt = (a-by)x

dy/dt = (cx-d)y

Another two-dimensional continuous dynamic system...

dx/dt = y

dy/dt = (1 - m)(ax^3 + b + cy)

With a light mass (m) and friction (c), the column eventually returns to the upright position:

With a heavy mass, the column comes to rest in one of two positions (two- point attractor).

- mutually supportive - the larger one gets, the faster the other grows
- mutually competitive - each negatively affects the other
- supportive-competitive - as in Predator-prey

dx/dt = a(y-x)

dy/dt = x(b-z) - y

dz/dt = xy-cz

The first strange attractor, the icon for chaos.

- Attractors - where the system 'settles down' to.
- Repellors - a point the system moves away from.
- Saddle points - attractor from some regions, repellor to others.

- Attractors - we've seen many
- Repellors - the value 0 in the Logistic Map
- Saddle points - the point (0,0) in the Buckling Column

2. The measures not the sum of the effects of a set of independent variables( each unaffected by the value of the dependent measure), plus random error.

3. The "determining" variables themselves influenced by the behavior being measured.

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"Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."(Mandelbrot, 1983).

- Euclidean space - 3- dimensional
- Number of variables in a dynamic system

Then

- The trajectory of a strange attractor cannot interect cross itself. (Why?)
- Nearby trajectories diverge exponentially. (Why?)
- But the attractor is bounded to the phase space.
- The trajectory does not fill the phase space.

z(n+1) = z(n)^2 + c.

where z is a complex number. z(0)=0.

For different values of c, the trajectories either: stay near the origin, or "escape".

The Mandelbrot set is the set of points that are not in the Escape Set.

- Natural objects are fractals.
- Chaotic trajectories are fractals.

To illustrate, we start with random time series.

Data generated by randomly sampling from (0,1) interval.

Here's a return map from another random time series.

This one sampled from an exponential (positively skewed) distribution.

Remember this time series?

What does its Return Map look like?

Comes from operating on a time series.

Successive n- tuples of data are treated as points in n-space.

The Return Map is an embedding dimension of 2.

Suppose, for example, that the first six data values were

4, 2, 6, 1, 5, 3,

then for an embedding dimension of 3.

P(1)= (4,2,6)

P(2)= (2,6,1)

P(3)= (6,1,5), and so forth.

**What's the point?**

Contemporary statistical analyses examine the geometric structure of obtained time series embedded with differing dimensions.

Lorenz, E. N. (1963). Deterministic non-periodic flows.

Mandelbrot, B. (1982).

van Geert, P. (1991). A dynamic systems model of cognitive and language growth.

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