Polynomial Planar Phase Portraits

THE PARAMETER OF INTEGRATION WINDOW

In this window you may modify the parameters that affect to the integration of separatrices and orbits through the Runge-Kutta 7/8 method and its graphical representation. They only should be modified if you are not satisfied with the results you have got, and they must be tested in order to chose the best set of values for the particular Polynomial Differential System under study.

• The Option Set  is an option which only has meaning if there is a line of singularites, i.e. if the two components of the Polynomial Differential System have a non-trivial common factor, or if the line at infinity of the Poincare Disc is filled with singular points. Then, you may choose between calculating the separatrices and orbits of the Polynomial Differential System either in the original system, or in the reduced one, that is, in the system with finitely many singularities.
  
• The Option Set  will toggle between whether you get the separatrices and orbits drawn as a sequence of dots or a set of dashes which link all these dots. Normally you will not see the difference between them unless you make a zoom. Note that you can must make this choice before you integrate the separatrices, you cannot change the type of appearance of an orbit or separatrix that is already calculated.
  
• The input   gives the starting step size that we are using when we start to integrate orbits with the Runge-Kutta 7/8 method. There is no reason for modifying it because the Runge-Kutta 7/8 method improves the size of the step continuously to better follow the orbit.
  
• The field   gives the current step that Runge-Kutta 7/8 method is using, so P4 will use this step in the case you ask to continue integrating a separatrix or an orbit further more.
  
• The input  gives the maximum step size that we will allow when we integrate orbits with the Runge-Kutta 7/8 method. Even if some orbits would accept a largest value without increasing the error, then the orbit that you would see would be a piecewise linear curve, so we suggest to leave this value unmodified as it is fast enough. On the contrary, this value may be too fast for certain systems, even producing erroneous crossings of separatrices and/or orbits. In those cases you are advised to reduce this number and evaluate the plotting.
  
• The input  gives the minimum step size that we will allow when we integrate orbits with the Runge-Kutta 7/8 method. Same as before, this value does not need be modified unless you experience strange behaviours of separatrices close to points where it is highly sensitive to small changes. Anyway, this kind of separatrices will always be conflictive ones.
  
• The input  gives the value which will make Runge-Kutta 7/8 method decide whether it must duplicate the integration step, or divide it by 2. If the difference of the values given by the two integrating functions embedded in the Runge-Kutta 7/8 method is greater than this value, it means that we must use a smaller step. If this difference is lower than a cent of this value, then we may run a little more.
  
• The integer input  gives the number of points that we will produce each time that we ask to integrate a separatrix or an orbit. This value can be increased without limit but we suggest that it should only be done to study special slow separatrices or orbits.
  

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