So, I wanted to introduce all of you

in the class

to Professor James Meiss who is in the Department

of applied mathematics at the University of Colorado

and my good colleague and downstairs neighbor.

Jim, I know you are particularly interested in

the standard map.

And we've been talking lately

in the...that I'm teaching about maps.

So can you tell us a little bit

about what the standard map is

and what is interesting to you about it.

Sure, I'm happy to talk about my favorite

dynamical system--the standard map.

So, my background is in physics

and the standard map is a good

model for some very simple physical systems

and one of them is

imagine that you have a pendulum

and ordinarily be rocking back and forth.

If you don't do anything to the pendulum

and there is no friction then it just rocks back and forth

the same amplitude forever and ever.

But if someone over here has a hammer.

and occasionally in this situation

gives it a kick periodically

So, every time the ball, wherever it happens to be

at t=1, the person hits it

and at t=2 the person will hit it again

and wherever it happens to be

the amplitude of the kick is fixed

but the hammer is more effective for different angles

than other angles

The kick will be more effective if say

the pendulum bob is horizontal

then you get a big kick.

And basically ineffective if it's straight down.

And give a negative kick if it's over here.

The physics of this tells you

just from Newton's equation

that you can reduce this dynamics to a map

a very simple map.

Here, the angle is represented by this variable x.

And, the angular momentum is represented

by the variable y.

And, so you get a two dimensional map that I've written

down totally wrong here.

There it is.

And, the dynamics of this system

is amazingly complicated

So if the kick is 0,

if parameter k is 0

then the angular momentum is constant.

And that's reflection of

the constant amplitude that I was telling you about.

But, if the k is nonzero than you have complicated

dynamics.

which sometimes is regular

and sometimes is chaotic

and I think the interested thing about this kind of system

which isn't dissipated is that there isn't just one.

typical attractor for this system

But is either strange or not strange

But, an inter mixture of some orbits

that's being chaotic and some orbits being regular

all through the same fixed parameter.

So, I have written a little program

that if you have a macintosh computer

you can download from my homepage.

If you click on the programs link here

There are several programs that I've written about

dynamical systems

One of them is the standard map program.

And, just download it by clicking on the link here.

It runs on most of the recent operating systems

for mac osx.

So, when you open the standard map program

you see two windows.

One is the phase space window

and the other is a text window

In the phase space based window

the horizontal variable is x

which is I showed you earlier is

the angle of the pendulum

and the vertical variable is y

the angular momentum.

And there is one parameter in the system

we can access by this map parameters menu called k

And, here I have just set parameter k to zero

as the initial trial.

If you click anywhere in the window

you see immediately horizontal lines in this situation.

These horizontal lines are actually consist

of many, many points.

So, if I instead of iterating continuously

go to single step and go to initial condition

you see a square where the initial condition is

Then hitting the space bar, shows you

one iterative of the map.

So, the analytical formula says that the

new value of x is the old value of x

plus whatever y we happen to be at.

Here the y is in the order of .3

So we move over to .3 and hit the space bar again

The colors change but the new point is again

the same distance horizontally over.

Since, this is an angle,

we wrap x around periodically.

Now we are over here some place.

Now we are over here some place.

Opps, I think I clicked some place incorrectly.

And if we keep iterating with the space bar.

It begins filling in the horizontal line

and if I go off to the find menu

and do continuously it just fills in a horizontal line

that indicates that initial condition

So changing the value of this parameter

k to something that is not zero

Something that is positive where we have a kick

We are kicking the system.

Now, if I click somewhere in the window.

The dynamical property of the orbit is quite different

So when you click, Jim, you are setting

an initial condition.

-Yes, Exactly. Setting an initial condition.

So, at different initial conditions have different behaviors.

So here the initial conditions is near x = 0,

which is where the pendulum would be oscillating

So, the range of x is bounded

and the momentum is positive and negative

So these are oscillatory orbits.

So up here, if I click with initial condition

the pendulum is going over the top.

And the interesting thing,

is there are orbits that have a hard time of deciding

if they are going to over the top

or oscillate.

And that's exactly where chaos starts to appear

in this dynamical system

So these are orbits on what we think of seperatrix

in the case of an integrable pendulum

this is an orbit where the pendulum starts with the bob

at the top. It falls down

crosses the down position

and goes back up vertically to the top

and the energy is preserved and it just asymptotic

to the top position and stays there

but when we are kicking it

the kick isn't very large here

but we are kicking it

and that results in orbit that seems to fit phase space

it seems to be chaotic

If I hit the up arrow key

I can just increment the value of K.

Here, the size of the chaotic region seems to grow.

Now, I mentioned before

is what is really interested about these conservative

dynamical systems

is I think there is an intermixture

of chaotic orbits and regular orbits

So the regular orbits that you see here

are these islands

And there are lots of islands in the system

and they are caused by

what we think of as resonances

in the Hamitonian dynamics

So there is a resonance here for example

that corresponds to three kicks per period

of the oscilation

-- And this kind is the resonance that causes planets

to have interesting behavior as well.

Yeah, so if we had a similar resonance

between the period of Jupiter and the period of an asteroid

Then, the asteroid, is effectively getting kicked

every time it goes near Jupiter and gets

kicked out of its orbit.

Another interesting thing about the standard map

is that there is a progression

of bifurcations that cause this chaotic region

to grow larger and larger

And if I make the parameter k,

and in this case let me type it in

on the order of say 5.

Then, the system becomes almost completely

chaotic

except for some small and irregular regions.

If we continue to increase the parameter value k

Those regions seem to be going away.

--Do they ever completely go away?

--No one knows.

It's been proved for any particular value of k

that's pretty large there are arbitrary close values

of k for which there are elliptic orbits.

Which are these islands.

But, it seems like if I set k on the order 10 say

the system as far as the computer is concerned

is completely chaotic.

We are filling one orbit here

and every pixel on the screen is gone.

It seems like the orbit fills the entire area.

But if i increase k to be 12 in this situation.

Suddenly an island--a visible island appears.

And there are visible islands that appear

for arbitrary large values of k.

--So a whole chain of bifurcations

but nothing like the period doubling

in the logistic map. This is much more complicated

--Yes, this is a different sequence of bifurcations.

--Another thing you might be interested in

on Jim's home page, is a link to the wonderful book

that he has written about dynamical systems.

--That's nice of you to say.

So, I wrote this book

for a class I teach for first year graduate students

in applied math and anybody in engineering or science

is interested in dynamics.

--Thanks Jim

--You are more welcomed.

In his lecture, Jim used a couple of terms

that we will talk about a little bit later into the semester

So, I just wanted to give you a forward pointer

and a quite explaination about those terms

First of all, Phase Space this is a synonym for state space.

State space is the very powerful representation

that we use to describe the evolution of trajectories

of dynamical systems.

It's the space of axis of the state variables.

For, the standard map, the state space

looks like this

And each point on that state space

The ones that jim was putting down by clicking on his app

represents a pair of values for x and y.

and the standard map which was invoking with the

click is a box of mathematics

that takes a value for x and a value for y

and tells you where the next point will be.

So, the action of the standard map is to evolve

the map one step at a time

as scanitized by that red arrow.

Of course the point doesn't actually follow

the path of the red arrow. It just hops from one dot

to the other.

A separatrix is what it sounds like

its a curve or surface in phase space

that separates two separate regions.

Kind of like the continental divide.

Integrable is a loaded word.

It basically means that the system is not chaotic

that the equations cannot be solved in closed form.

We will get back to this in the next unit.

This is another nest of words

that we will talk about later on

dissipation is a more general term

for friction.

Dissipation of energy, lost of energy

or technically a gain of energy.

In systems that are conservative

energy is conserved.

Those are non dissipative systems

Hamiltonian is a synonym for conservative

So, if I put a non in front of dissipative

those three words are synonyms.

Dissipation is a necessary condition for contractors.

You have to have dissipation

for an attractor in your system

because something has to eat up the energy

and cause the energy to relax down to the fixed point

where the chaotic cycle or limited attractor

So in the logistical map

where the trajectory is relaxing down to a fixed point

that behavior lets you know the system is dissipative

because there is an attractor there.

The standard map that Jim was talking about

is non dissipative, so there aren't any attractors.

But, there is still chaos

So just to sum that up.

Dissipation, friction, is a neccesary condition

for attractors

but not for the existence of chaos.

So, you can have chaos

if you don't have chaotic attractors

I'm not going to do a whole lot

of hamiltonian dynamics in this course

because we don't have time.

But it is a rich field

as Jim, I hope, suggested to you.