Polynomial Planar Phase Portraits
THE PLOT WINDOW
In this window you will be able to produce the phase portrait of the Polynomial
Differential System you are studying. It may appear in different versions depending on the options you are running.
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The most usual will be the Poincare Disc version. In this one you will get a circle representing the
infinity and some symbols representing the finite and infinite singular points of the Polynomial Differential
System.
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If you had modified the values in the
Parameters
subwindow of the main P4 window. for the good natural numbers which allow the
Poincare-Lyapunov Compactification of your Polynomial Differential System, you will get the
Poincare-Lyapunov Disc of degree (p,q).
You will see
two circles, in the inner circle you have all the finite singular points with modulus lower than one. If
the modulus of a singular point is greater than 1, you will see it in the annulus limited by the circle of
radius 1 and the infinity circle. Then P4 makes a transformation of x=cos(a)/r^p and
y=sin(a)/r^q where a is the polar angular coordinate. The values p and q have to be
introduced by the user in the Parameters window. By default they are 1 and
that is equivalent to the standard Poincare Disc. You will be interested in this representation if you
have one non-elementary singular point at infinity because then, with appropiate values of p and q
you will see it splitted in elementary points. You may see orbits crossing the circle of radius 1, and for
such orbits it may appear as if they had non-continuous derivative. This is due to the fact that we are making
two different transformations which cannot be connected in a differentiable way on such a circle.
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In case you decided to study only one singular point, you will see a planar representation of
the neighbourhood of such a point.
Buttons:
Zooming:
Even there is no special button, you may make a ZOOM of any rectangle of the Plot window.
In order to do it you must press the left button of your mouse together with the Control key of
your keyboard on the left-up corner. Then you release the Control key and move your mouse
towards the right-down corner. You will see as a rectangle is formed. Then you click the left button of
your mouse in the right-down corner and you will get the ZOOM window:
In this window you also have the buttons ,
and which work exactly in the same way as described above.
You are not limited to do only one ZOOM, you may do as many as you want.
Changing the size of the plot/zoom windows:
The plot and zoom windows are resizable, just like any other window on your computer. While resizing,
you will see that the "aspect ratio" is displayed on the bottom bar:
If this aspect ratio is 1.0000, then you have rescaled your window in an euclidean way. In the picture
above, the aspect ratio is 0.6333 because only the width of the window was decreased and the height
remained the same.
LET US FINISH BY GIVING THE DIFFERENT WAYS OF SELECTING POINTS IN ANY PLOT OR ZOOM WINDOW:
- If you select a point by simply clicking it with the
left button of your mouse, then you enter in the Orbits
window in order to draw the orbit passing through this point.
- If you select a point by clicking it with the left button
of your mouse while mantaining pressed the Shift key of your keyboard,
then you are selecting the closest singular point having separatrices and
you enter automatically in the Plot
Separatrices window which will allow you to draw (or to continue) the
separatrices of this singular point one by one.
- If you select a point by clicking it with the left button
of your mouse while mantaining pressed the Control key of your keyboard,
then you are selecting the left-up corner of the rectangle which you want
to ZOOM.
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