Polynomial Planar Phase Portraits

THE PARAMETERS (SUB)WINDOW

• The Option Set  will toggle between the two options showed here. In Algebraic mode some computation will be done in algebraic mode. These computations are the search of singular points (finite and infinite), and the first terms of the Taylor approximations of the separatrices. In Numeric mode, everything is done in numerics. You will need to use the Numeric mode if the degree of the Polynomial Differential System is high and it has many coefficients. This is somehow subjective, and the fact is that you will simply need to toggle to Numeric when you have already tried a system in the Algebraic mode and the program has collapsed.
  
• The Option Set  will decide whether P4 tests every Taylor approximation it has found, so it enters in a self-controlled procedure for improving it or not. Normally it is recommended to use this option, but if you see a concrete system where a separatrix is hard to deal with and you are not specially interested in it, you may deactivate this option.
  
• The input  gives the value (negative power of 10) such that we will consider any number below it (in module) as 0. In any numeric program you need to define such a number so to avoid that a rounding error may produce a variable being different from 0 when it should be null. Of course this may mean that you take as zero something that it should not, and so we allow this value to be modified. Anyway, if a problem depends greatly on such small values, its numerical study may be hazardous.
  
• The input  gives the distance we will move from the singular point in order to chose the first point of the integration for each separatrix. This value is the default one we will use for every separatrix. When we choose the initial point we are making an error which is directly proportional to this number elevated to the power of the Taylor approximation that we have done (greater or equal than input  explained in this same window). You may modify this number but it may be more interesting to modify it for individual slow separatrices and this can be done in the Plot Separatrices window.
  
• The integer input  will determine the degree up to which the coefficients of the Taylor approximations of the separatrices will be done in any case. The computations will be done in the mode defined by the Option Set . If the Option Set  is activated, then P4 will test if the Taylor approximation fits properly with the vector field. In case it does not, it will look for one degree more and will repeat the test until the maximum degree (see below) is reached or the test passes.
  
• The procedure described above will continue being done in algebraic mode until we reach the value defined in the integer input . From now on, the computation will be done in numeric mode. We remark that  must be larger than .
  
• As this process needs to have an end, the integer input  is the maximum value we will allow for the separatrix. If the test fails up to this level, this will be explained in the report that P4 will produce.
  
• The integer input  will determine the number of Lyapunov constants that we wish to calculate for every focus-center that P4 may meet. We know that you may be very interested in getting a large number of them, but you must be careful as the time for computing them increases exponentialy. This can be a good reason to use the option  in the P4 window and then leave the REDUCE/MAPLE program running in batch mode.
  
• In the case you have selected to study all the singular points (or just the infinite ones) in the P4 window, you will be asked to enter the values of the powers  and  which will define the Poincare-Lyapunov Compactification. By default these two numbers are 1, and this leads to the standard Poincare Compactification.
  
• In the case you have selected to study only one singular point in the P4 window, you will be asked to enter the coordinates of that point in the inputs  and . Be sure to introduce a real singular point. If you introduce parameters in the Vector Field window, you may use these parameters in the (x0,y0) fields as well.
  

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