P4 Limit Cycles Help Page

Polynomial Planar Phase Portraits

THE LIMIT CYCLES WINDOW

Once you enter in this window, you will be allowed to search for non semi-stable limit cycles up to a certain degree of precision. You need to introduce two points forming a transverse segment such that you suspect the existence of one (or several) limit cycle(s) cutting it. You may do it with your mouse by simply clicking with the left button on one of the endpoints of the segment that you one to select, move with the mouse while keeping the mouse button clicked and releasing the mouse at the other endpoint.

The input fields
will allow you to enter the two endpoints of the segment of coordinates (x0,y0) and (x1,y1) from your keyboard (or modify the values selected with the mouse).
The input determines the precision up to which the limit cycles will be determined. That is, if two consecutive limit cycles (one stable and another unstable) cut the selected segment in two points separated more than this value, then the program will detect them. Otherwise, it may fail in detecting one, or even both.
The integer input determines the number of integration steps that the Runge-Kutta 7/8 method will calculate at every orbit starting with initial conditions in a point of the segment. If the orbit has not crossed the line defined by this segment at this time, the program will consider that the orbit does not cut again the segment. You may note that this default number is bigger than in the Parameters of Integration window. We suggest to keep it around this value or even bigger since you may find very slow limit cycles which would remain undetected with low values. In fact you may get an approximate idea of which number you should enter by studying the integration of an orbit close to the limit cycle (in the Orbits window) and estimating how many steps you have needed to make a complete turn.
The button will begin the search of the limit cycles. It may take a bit long, specially if you have selected a segment close to infinity which will be much longer than it may seem. During the search the dialog box
In case there are some limit cycles found P4 will draw it with the colour described in the Legend window. In case that it does not find any limit cycle under the stated assumptions, the control of the program will return to you once P4 has ended the exploration. Press the button to cancel the search and return control of the program immediately.
The button will clear the (x0,y0) and (x1,y1) fields.
The button will remove the last limit cycle that was found.
The button will remove all limit cycles that were previously drawn on the phase plane.

Note:After you finish searching limit cycles, close the limit cycles window. (Because its presence causes the mouse click to have a different meaning than the usual meaning of selecting initial points for orbits.)

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