OS/2 Professional/OS2PRO194.ISO/os2/wps/progs/chaos/pmchaos.hlp

OS/2 Help File | 1993-12-11 | 17KB | 460 lines

ΓòÉΓòÉΓòÉ 1. PM Chaos: Extended Help ΓòÉΓòÉΓòÉ

PM Chaos is a simulation of a simple chaotic system: the standard mapping.
This is a simple function that takes a point (J,theta) into another point
(J',theta').  If you keep applying this function again and again, you get a
sequence of points.  For certain initial conditions this sequence is a smooth
trajectory, and for other initial conditions it is chaotic.

To start a calculation, click the left mouse button anywhere within the box of
the graph.  PM Chaos will begin calculating the sequence of points starting
with the initial point you have chosen.  The calculation continues until you
stop it, either by clicking the left mouse button somewhere else (this will
stop the first calculation and start a new one), by selecting Stop! from the
Control menu, or by clearing the screen.

Related information:

o  The standard mapping

o  Using the program


ΓòÉΓòÉΓòÉ 2. Physics of the standard mapping ΓòÉΓòÉΓòÉ

The standard mapping is a simplified set of equations that has many of the same
qualitative topological properties as do actual physical systems in classical
mechanics.

The evolution of a classical mechanical system is given by a set of
differential equations determined by the Hamiltonian. It is numerically easier,
however, to work with a difference equation, that is, one that doesn't evolve
continuously, but where, given a point, there is a function that just gives you
the next point in the sequence.

If you don't know much classical mechanics, you may want to skip the next three
paragraphs.  You don't need to understand them in order to use the program.

Consider a time-independent Hamiltonian with two degrees of freedom, i.e., two
canonical coordinates and two canonical momenta.  The state of the system is
represented by a point in the four-dimensional phase space, and the time
evolution of the system from some initial state is a trajectory in phase space.
We can choose some arbitrary two-dimensional surface in this phase space, a
surface of section, and study the successive intersections of a trajectory with
that surface.  This gives us a set of difference equations on a two-dimensional
reduced phase space.

A time-independent Hamiltonian leads to a conserved energy; this means that a
trajectory must lie in some three-dimensional subspace of the original
four-dimensional phase space.  It also means that the mapping given by the
surface of section (a canonical transformation) is area-preserving, that is,
that any region of the two-dimensional reduced phase space gets mapped to a
region of equal area.  If the Hamiltonian is integrable (i.e., if there exists
a constant of the motion other than the total energy), then motion is actually
in a two-dimensional subspace of phase space.  In that case, the successive
points in the surface of section lie on closed curves in the reduced phase
space.

This is most easily seen in angle-action variables.  In that case, the mapping
equation for an integrable system is

     J'   = J
     theta' = theta + f J.
The standard mapping is very similar to this: it is a near-integrable
area-preserving mapping in angle-action coordinates.

o  Equations of the standard mapping

o  Where to learn more


ΓòÉΓòÉΓòÉ 3. The standard mapping ΓòÉΓòÉΓòÉ

The standard mapping is given by the equations

     J'   = J + K sin(theta)
     theta' = theta + J', mod(2 pi).

It is defined for J and theta in the range (0, 2 pi).

o  Significance of K


ΓòÉΓòÉΓòÉ 4. The K factor ΓòÉΓòÉΓòÉ

K is the stochasticity parameter.  For K=0, the standard mapping is integrable,
and the action variable, J, is a constant of the motion. For K>0 there are some
regions of stochasticity, but they are isolated when K is small, and J is
nearly constant for many trajectories.  For any K<1 there are separatrices
dividing the screen into at least two regions.  That is: if the motion begins
near J=0, it will never end up near J=2 pi.

Motion becomes globally stochastic at K=1. At this point, the last separatrix
disappears, and there is nothing separating the J<pi and J>pi regions.

For any value of K there are still some islets of stability, but they become
more and more isolated, and motion becomes more and more chaotic, as K
increases.  Note that there are always at least two fixed points: (J,theta) =
(0,pi), and (J,theta) = (2pi,pi).  For K<4, these fixed points are stable.

This program limits K to the range 0<K<3, because motion with larger values of
K is too disordered to be very interesting.

Related information:

o  Setting K

o  Equations of the standard mapping

o  Discussion of the physics


ΓòÉΓòÉΓòÉ 5. References ΓòÉΓòÉΓòÉ

The standard mapping is Equation 4.1.3 of

        A. J. Lichtenberg and M. A. Lieberman,  Regular and Stochastic Motion,
        New York: Springer-Verlag, 1983.

 Section 4.1b describes the standard mapping in detail; chapter 3 discusses
 mappings in general.  This book is somewhat advanced, and assumes that you
 already know about Hamiltonian dynamics, angle-action variables, and so on.
 If you don't, the standard reference is

        H. Goldstein,  Classical Mechanics (second edition),  Reading:
        Addison-Wesley, 1980.


ΓòÉΓòÉΓòÉ 6. Using PM Chaos ΓòÉΓòÉΓòÉ

At the most basic level, using PM Chaos is very simple.  Just click the left
mouse button anywhere within the graph on the main screen. PM Chaos will use
the point you have selected as the initial conditions for a calculation.  It
will then begin calculating a sequence of points and displaying them on the
screen; each point is determined from the last by a simple equation. If you
click the left mouse button in the graph while a calculation is in progress, it
will stop the old calculation and start a new one.

The Control menu contains a command to stop a calculation that is in progress,
a command to clear the screen (if there is a calculation in progress, clearing
the screen will stop it), and a command to quit the program. You can also quit
the program by selecting Close from the system menu.

Normally, the points get displayed very quickly.  In  slow motion mode,
however, the points are display slowly enough so that you can see the progress
of the calculation explicitly.

You can resize PM Chaos's window in any of the normal ways that you can resize
any Presentation Manager** window: either by using the sizing border, or by
using the minimize and maximize buttons at the upper right-hand corner of the
window, or by selecting Size from the system menu.

That's really all you need to know to use PM Chaos.  There are a number of
additional options that you can select, though.

 o  Choosing the color

 o  Setting K

 o  Changing the range of the graph


ΓòÉΓòÉΓòÉ 7. Choosing the color ΓòÉΓòÉΓòÉ

You can change the color that PM Chaos will use to display results.  You can
mix different colors on the same plot; that is, you can display the results of
a calculation in one color, and the results of the next calculation in a
different color.  This is a convenient way to distinguish trajectories that lie
close to each other.

You choose the color by using the color dialog, which you get to by selecting
Colors from the Options menu.

The color dialog displays sixteen different colors, each of which has a button
next to it.  Click on the color that you would like to use for the next
calculation.

When you click on the OK button, the color for the next calculation will be
changed.  If there is already a calculation in progress, it will stop.  If you
click on the Cancel button, then the current color will not be changed, and the
calculation in progress (if any) won't be stopped.


ΓòÉΓòÉΓòÉ 8. Setting K ΓòÉΓòÉΓòÉ

The equation for the standard mapping depends on the K factor, which deterines
the degree of stochasticity.  Essentially, the larger K is, the more chaotic is
the behavior of the system. If K=0, the system is completely regular.  If K is
much larger than 1, the system is so chaotic that it is no longer very
interesting.

You can change the value of K by using the K factor dialog, which you get to by
selecting Set K from the Options menu.

This dialog box uses a slider.  You can change the value of K by dragging the
slider arm; you can also use the buttons to the left of the slider, or you can
use the arrow keys on the keyboard.  The value you have selected is displayed
just above the slider control.

When you click on the OK button, the value of K will be changed, and the screen
will be cleared.  If you click on the Cancel button, then the current value of
K will remain unchanged, and the calculation in progress (if any) won't be
stopped.


ΓòÉΓòÉΓòÉ 9. Setting the range ΓòÉΓòÉΓòÉ

You can change the range of the graph, so that you can look more closely at
some particular part of it.  There are three ways to change the range. No
matter which method you use, however, changing the range will stop any
calculation that may be in progress, and will clear the screen.

 o Zooming out

 o Zooming in

 o Using the range dialog


ΓòÉΓòÉΓòÉ 10. Zooming out ΓòÉΓòÉΓòÉ

If you select Zoom Out from the Range menu, then the graph will be set to the
maximum possible range: both axes will go from 0 to 2 pi.

Doing this will clear the screen; if there is a calculation in progress, it
will be stopped.


ΓòÉΓòÉΓòÉ 11. Zooming in ΓòÉΓòÉΓòÉ

To zoom in to a particular region of the graph, you must first define a zoom
rectangle.  You do this with the right mouse button.

Move the mouse pointer to somewhere within the graph, and then move the mouse
while holding down the right mouse button.  You will see a rectangle on the
screen: one corner of it will be at the point where you first began to hold
down the right mouse button, and the other will be at the current posigion of
the mouse.  The rectangle will continue to move around as long as you're
holding down the right mouse button.

When you release the right mouse button, the zoom rectangle will remain in
position.  If you select Zoom In  from the Range menu, then the range of the
graph will be set to be the region outlined by the zoom rectangle.  Doing this
will clear the screen; if there is a calculation in progress, it will be
stopped.

The zoom rectangle will be cleared if you start a calculation, if you clear the
screen, or if you press the right mouse button to define a different zoom
rectangle.


ΓòÉΓòÉΓòÉ 12. Using the range dialog ΓòÉΓòÉΓòÉ

The simplest way to set the range is usually with the Zoom Out  or the Zoom In
options.  If you want to type the ranges in explicitly, though, the range
dialog allows you to do that.  You get to the range dialog by selecting Set
Ranges  from the Range menu.

The range dialog has four entry fields: the minimum and maximum values of the
range for each of the two coordinate axes.  You can type in a number in each of
those entry fields.

For each axis, the minimum value must be at least 0 and the maximum must be no
greater than 2pi.  Also, of course, the maximum must be greater than the
minimum...

When you click on the OK button, the range will be changed to the value you
have typed in, and the screen will be cleared.  If you click on the Cancel
button, then the range will remain unchanged, and the calculation in progress
(if any) won't be stopped.


ΓòÉΓòÉΓòÉ 13. Slow motion mode ΓòÉΓòÉΓòÉ

In slow motion mode, a calculation is done slowly enough so that you can see
the points calculated one at a time.  This lets you see, explicitly, how the
mapping function takes one point into another.

When a calculation is in progress in slow motion mode, the current point is
displayed prominently in the graph window, and its coordinates are displayed in
a separate window.

You can set the speed of slow motion mode by selecting the Delay Time item from
the Options menu.

You can turn slow motion mode on or off by selecting the Slow Motion item from
the Options menu.  If a calculation is in progress when you turn slow motion
mode on or off, it will be stopped.


ΓòÉΓòÉΓòÉ 14. Setting the slow motion speed ΓòÉΓòÉΓòÉ

In slow motion mode, PM Chaos pauses for a specified length of time after
displaying each new point.  Selecting the Delay Time item from the Options menu
will display a dialog box that allows you to set that length of time. This
delay time has no effect unless you are in slow motion mode.

The delay time is specified in units of 1/10 seconds.  So, for example, if you
want a delay of 1 second, you type in a delay time of 10. The delay time must
be a non-negative integer.  You may specify a delay time of 0, but doing that
isn't very useful.

When you click on the OK button, the delay time will be changed.  If there is a
calculation in progress, and if it is using slow motion mode, it will be
stopped.  If you click on the Cancel button, then the delay time will remain
unchanged.


ΓòÉΓòÉΓòÉ 15. The Control menu ΓòÉΓòÉΓòÉ

The Control menu contains the following options:

 o Stop!

 o Clear Screen

 o Exit


ΓòÉΓòÉΓòÉ 16. Stop! ΓòÉΓòÉΓòÉ

This menu option stops the calculation that is in progress.  If there isn't any
calculation in progress, this option is disabled.

Another way to stop a calculation in progress, without using the menu, is to
click the mouse somewhere outside the borders of the graph.


ΓòÉΓòÉΓòÉ 17. Clear Screen ΓòÉΓòÉΓòÉ

This option clears the screen.  If a calculation is in progress, it is stopped.


ΓòÉΓòÉΓòÉ 18. Exit ΓòÉΓòÉΓòÉ

Use this option to quit PM Chaos.  When you select it, the program will display
a dialog box asking if you really want to quit; click on the Yes button if you
do.

You can also quit PM Chaos by selecting Close from the system menu.


ΓòÉΓòÉΓòÉ 19. The Options menu ΓòÉΓòÉΓòÉ

The Options menu contains the following choices:

 o Slow Motion

 o Delay Time

 o Colors

 o Set K


ΓòÉΓòÉΓòÉ 20. Slow Motion ΓòÉΓòÉΓòÉ

Use this option to turn slow motion mode on or off.

If there is a check mark next to this menu item, that means that slow motion
mode has been selected.  If there is no check mark, then slow motion mode has
not been selected.

If a calculation is in progress, turning slow motion mode on or off will end
it.


ΓòÉΓòÉΓòÉ 21. The Range menu ΓòÉΓòÉΓòÉ

The Range menu contains the following choices:

 o Zoom In

 o Zoom Out

 o Set Ranges

 Related information:

  General comments about the range.


ΓòÉΓòÉΓòÉ 22. Key assignments ΓòÉΓòÉΓòÉ

For on-line help, and for tasks like switching between windows, PM Chaos uses
the same keys as any other Presentation Manager** program.  Additionally, PM
Chaos defines some accelerator keys of its own for commonly used tasks.

 o PM Chaos keys

 o Help keys

 o System and window keys


ΓòÉΓòÉΓòÉ 23. PM Chaos key assignments ΓòÉΓòÉΓòÉ

 Key     Action
  Esc     Stop!
  Alt+C   Clear Screen
  Alt+X   Exit
  C       Colors
  K       Set K
  <       Zoom In
  >       Zoom Out
  R       Set Ranges


ΓòÉΓòÉΓòÉ 24. On-line help key assignments ΓòÉΓòÉΓòÉ

 Key         Action
  F1          Get help
  F2          Extended help (from within help)
  F9          Keys help (from within help)
  F11         Help index (from within help)
  Shift+F10   Using help (from within help)


ΓòÉΓòÉΓòÉ 25. System and window key assignments ΓòÉΓòÉΓòÉ

  Alt+F9
       Minimize the window

  Alt+F10
       Maximize the window

  Alt+Esc
       Switch to the next program

  Ctrl+Esc
       Switch to the Task List

  Shift+Esc or Alt+Spacebar
       Go to or from the system menu

  F10 or Alt
       Go to or from the action bar

  Underlined letter
       Move among the choices on the action bar and pull-down menus

  Arrow keys
       Move among the choices on the action bar and pull-down menus


ΓòÉΓòÉΓòÉ <hidden>  ΓòÉΓòÉΓòÉ

Actually, the transition isn't exactly at K=1.  The approximate value is
K=0.9716.  See A.J. Lichtenberg and M.A. Lieberman,  Regular and Stochastic
Motion,  for more details.


ΓòÉΓòÉΓòÉ <hidden>  ΓòÉΓòÉΓòÉ

Presentation Manager is a trademark of the IBM Corporation.


ΓòÉΓòÉΓòÉ <hidden>  ΓòÉΓòÉΓòÉ

Presentation Manager is a trademark of the IBM Corporation.