Polynomial Planar Phase Portraits

HOW DOES P4 WORK?

One of the most powerful tools of P4 is that it is neither a simple numeric program nor an algebraic one, but both things altogether. Some parts of the program (like integration and Lyapunov constants) are done always in numeric mode, but all the others can be done in numeric mode or in algebraic one, or mixed, depending on the user 's options. In order to do it, P4 is written in C/C++ and REDUCE/MAPLE. The program goes running in C, but when it needs some algebraic computations, it prepares a REDUCE/MAPLE file and asks this language to execute it and produce an output in a certain format which C will read to continue working.

We are going to explain mainly what P4 does when you press the button  in the P4 window. It is assumed that you have already loaded an acceptable Polynomial Differential System in the Vector Field window and that you have selected all the default parameters.

First of all, P4 checks if there is a continuous set of finite singular points, that is, if the two polynomial components of the system introduced have a common factor. If they have, REDUCE/MAPLE will take it out and will make the study of the reduced system, giving lately the option of drawing the set of continuous singular points.

Now we must look for the finite isolated singular points. This can be done in algebraic or numeric mode. In both cases the program will ask REDUCE/MAPLE to solve the problem. REDUCE/MAPLE (like other computer algebra packages) makes use of GROEBNER basis to solve this problem (if the system is not linear). As this problem can be very complicated if the degree is high, we suggest to work in numeric mode in such cases.

After this P4 will determine the local phase portrait of each singular point. So, it will look for the Jacobian of the system evaluated at each singular point and will look for its eigenvalues. If the point is a (semi-)hyperbolic node or a strong focus, we are done. If the singular point is a focus-center, then P4 will look for its Lyapunov constants. This is done using the technique developed by Gasull and Torregrosa .You may parametrize how many Lyapunov constants you want to evaluate with the input  in the Parameters subwindow. As many as you ask, harder will be to find them. From this P4 will be able to determine if the point is a center (only in quadratic cases and a restricted cubic case (linear plus homogeneous cubic)), an unstable or stable weak focus of certain order if some of the computed Lyapunov constants is positive or negative respectively, or an undetermined weak focus if all the Lyapunov constants evaluated are zero.

If the singular point is a (semi-)hyperbolic saddle or a saddle-node, then P4 will normalize the system by moving the point to the origin and the eigenvectors to the axis. Then P4 determines the Taylor approximations of the separatrices up to a certain degree in algebraic mode (parameter  in the Parameters subwindow), and higher degrees in numeric mode (parameter  in the Parameters subwindow). Moreover, the program will test if the Taylor expansion fits properly with the real separatrix, and if not, it will evaluate a higher degree expansion up to a maximum degree of  in the Parameters subwindow.

In case of degenerate singular points, the program brings them to the origin and starts making a detailed study by using the BLOW-UP technique until it determines the exact distribution of its sectors, and the separatrices it will have. For each one of them, also a Taylor approximation of the separatrix is provided under the same conditions as explained above.

After doing the study of the finite singular points, a similar procedure is done with the infinite ones.


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