In the following links we show a page for every phase portrait of quadratic systems without limit cycles that has been cataloged and given a name. Most of the configurations of singularities have already been proved to produce a given number of phase portraits, but some are still open. For those which are still open, and specially in codimensions 2, 3 and 4, we cannot assure that the name given here will be definitive. Not all the parts and object of the Quadripedia are still complete. We are working on it.
Each page of the Quadripedia contains the following information:
2) A description of the topological configuration of singularities according to TCSP.
3) A topologically equivalent image of the phase portrait done with Corel Draw. Please note that the real numerical image that we may see in 6) is sometimes very difficult to grasp. The image presented here may be not numerically or geometrically accurate but it is an image easier to understand and topologically correct.
4) The main topological invariants that identify the phase portrait.
5) An example of a differential equation producing the phase portrait.
6) The exact image of that previous example done with P4. In some cases where the interesting part of the picture is too concentrated to understand it, we have produced a ZOOM version to clarify it. However, some portraits need two or more zones to be zoomed and we provide just one of them. Anyway, these zooms are better understood when done by oneself in P4. So we advise the user to test the example given in P4 and make the zooms required up to confirm that the phase portrait is correct. Note that some portraits with a separatrix connection are impossible to be provided with an exact set of parameters, and the values given are just approximated. We have tried to draw them the most accurate possible and trying to show the separatrix connection by integrating both separatrices up to a point where they are very close. Also some phase portraits must be studied with a very specific set of integration parameters so to minimize numerical errors. Anyway, there are some phase portraits like QS010^0_4 or QS010^0_8 for which the numeric as so forced, that no image is possible to numerically confirm its existence. In those cases the reader may rely on the proof given in the corresponding paper where they appear.
There are some examples obtained from perturbation of more degenerated cases which produce a topologically equivalent modulo limit cycles phase portrait. That is, the portrait contains limit cycles (normally one). We have left that example with the limit cycle since trying to force its collapse could make the portrait even more difficult to grasp.
There are other examples taken from normal forms with weak foci that still maintain the weak focus. We could have forced the weak focus to become strong but this would complicate the system for no good reason. Even more, in those cases one may easily modify the system to produce one or more limit cycles.
Most examples seem quite strange and with many features concentrated in a very short space. This is because they were systematically obtained from perturbations of more degenerate systems in order to prove their existence and we have taken the examples we already had produced. Most probably, many of them may also have real images which look not so forced. For some very special portraits that live in very narrow regions of the parameter space, may really be only possible in those narrow views.
In case some inconsistency is found, please report it to us.
7) A list of the papers where this phase portrait has appeared. We add a link to a page for every article where we give the abstract, the list of portraits found there, and some comments on the articles. We have tried to identify every phase portrait with the name given by their authors, or the Figure where they appear. We have also added comments in some of them if we have detected some inconsistency as an arrow missed or capsized which may be just typos. However, sometimes we cannot be sure on how to repair some misses.
8) We have indicated which other phase portraits of one more codimension live in the border regions of the given phase portrait, and which other phase portrait of the same codimension of the original one, lives beyond that border.
9) We have indicated in which phase portraits of one less codimension this phase portrait may bifurcate. Sometimes one (or even both) bifurcating phase portrait has limit cycles.
10) We indicate if the given phase portrait is missed in a paper where it was supposed to be present according to the goal of that paper.
11) Sometimes a comment on that phase portrait may be included.
Please, report to the authors any inconsistency detected and we will check it and repair.