P4 Plot Help Page

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.

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.
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.
In case you decided to study only one singular point, you will see a planar representation of the neighbourhood of such a point.

Buttons:

The button will close this window but it will not stop the program.
The button will clear the window and it will redraw everything evaluated up to now, so it may avoid any problem with the screen which may have appeared. It is useful once you have got all separatrices to bring up the singular points which may have been shadowed by the lines, or after you have deleted orbits.
The button will link you with the Legend window.
The button will link you with the Orbits window. You may also access to this window by simply selecting one point with the left button of your mouse, to be the initial conditions of the orbit you want to draw.
The button will link you with the Parameters of Integration window.
The button will link you with the GCF window. This button is disabled if the vector field that you entered did not have a nontrivial greatest common factor.
The button will link you with the Plot Separatrices window. You may also access to this window by simply selecting one singular point (or close to it) with the left button of your mouse while maintaining pressed the Shift key of your keyboard.
The button will start the execution of the integration program and it will draw every separatrix available. It is possible that some separatrices appear too short or even do not appear at all. This is due to the fact that we are integrating a fixed number of steps of the separatrix. This number is given in the option of the Parameters of Integration window. You may either modify this number, or to go to the Plot Separatrices window to deal with these slow separatrices. We highly recommend the latter.
The button will link you with the Limit Cycles window.
The button will link you with the Plot View window. This will allow you to view the phase portrait in one of the charts at infinity, such as the U1 chart:
The circle at infinity is in this chart shown as a line of infinity. To the right of this line, one finds the U1 chart; to the left of this line the V1' chart is shown. More information on the charts at infinity is found here.
The button will link you with the Print window.

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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