Phase portraits of a quadratic system of differential equations occurring frequently in applications

Reyn, J.W.

Nieuw Arch. Wisk. (4), 5, no. 2 (1987), 107--151.

Abstract

The phase portraits of the quadratic system of differential equations, occurring frequently in applications (*), are represented in 178 figures, among which are 101 topologicaliy different ones. The presentation is aimed at giving information that can be used directly in applications. If (*) has 4 finite singular points there are 72 figures of which 20 are topoiogically different; for 3 finite singular points 53 figures of which 38 are topoiogically different, for 2 finite singular points 37 figures of which 29 are topologicaliy different, and for 1 finite singular point 16 figures of which 14 are topologically different.

Phase portraits (98)

• \(AAD01\). \(AAD02\). \(ADD01\). \(ADD04\). \(ADD06\). \(ADD07\). \(BCDD1\). \(BCDD2\). \(BDD3\). \(BDD4\). \(CCD1\). \(CDD03\). \(CDD04\). \(DD02\). \(DD03\).
• \(QS1^{(0)}_1\). \(QS1^{(1)}_1\). \(QS1^{(2)}_1\).
• \(QS2_3\).
• \(QS5^{(0)}_3\). \(QS5^{(1)}_4\). \(QS5^{(1)}_6\).
• \(QS6_2\).
• \(QS7^{(1)}_3\). \(QS7^{(2)}_7\). \(QS7^{(2)}_8\).
• \(QS10^{(0)}_{16}\). \(QS10^{(1)}_5\). \(QS10^{(1)}_{21}\).
• \(QS11^{(1)}_{20}\). \(QS11^{(2)}_{25}\).
• \(QS28^{(2)}_2\).
• \(QS31^{(1)}_6\). \(QS31^{(2)}_4\). \(QS31^{(2)}_5\). \(QS31^{(2)}_8\).
• \(QS32^{(3)}_5\). \(QS32^{(3)}_{20}\).
• \(QS33_1\).
• \(QS38^{(1)}_{33}\). \(QS38^{(2)}_{10}\). \(QS38^{(2)}_{32}\). \(QS38^{(2)}_{33}\). \(QS38^{(2)}_{34}\).
• \(QS39^{(3)}_3\). \(QS39^{(3)}_{43}\). \(QS39^{(3)}_{46}\). \(QS39^{(4)}_1\). \(QS39^{(4)}_6\).
• \(QS52^{(5)}_1\).
• \(QS68^{(4)}_1\).
• \(QS69^{(4)}_1\).
• \(QS70^{(5)}_1\).
• \(QS71^{(4)}_1\).
• \(QS72^{(5)}_1\).
• \(QS74^{(1)}_5\). \(QS74^{(2)}_2\). \(QS74^{(2)}_5\). \(QS74^{(2)}_6\).
• \(QS75^{(2)}_4\).
• \(QS76^{(1)}_{10}\). \(QS76^{(2)}_2\). \(QS76^{(2)}_3\).
• \(QS83^{(4)}_1\).
• \(QS99^{(2)}_{23}\). \(QS99^{(3)}_2\). \(QS99^{(3)}_3\). \(QS99^{(3)}_4\). \(QS99^{(3)}_5\). \(QS99^{(4)}_1\).
• \(QS100^{(4)}_{16}\). \(QS100^{(5)}_2\).
• \(QS104^{(3)}_1\). \(QS104^{(3)}_4\). \(QS104^{(4)}_1\).
• \(QS112^{(5)}_1\).
• \(QS123^{(4)}_1\).
• \(QS124^{(3)}_1\).
• \(QS125^{(3)}_1\).
• \(QS126^{(3)}_1\).
• \(QS128^{(5)}_1\).
• \(QS129^{(4)}_1\).
• \(QS131^{(5)}_1\).
• \(QS133_2\).
• \(QS134^{(2)}_4\).
• \(QS144^{(4)}_1\).
• \(QS147^{(6)}_1\).
• \(QS149^{(4)}_1\).
• \(QS160^{(5)}_1\).
• \(QS161^{(4)}_1\).
• \(QS177^{(5)}_1\).
• \(QS179^{(5)}_1\).
• \(QS185^{(5)}_1\).
• \(QS187^{(5)}_1\).
• \(QS189^{(5)}_1\).
• \(QS198^{(6)}_1\).
• \(QS207^{(7)}_1\).
• \(QS208^{(8)}_1\).

Comments

• Type of paper: Bifurcation diagram with possible islands.

• This paper produces the complete set of phase portraits for quadratic systems having two real invariant intersecting straight lines.

• There is no systematic way to compare if two phase portraits are topologically equivalent or not. The author claims to have 178 phase portraits of which 101 are different. However, under our account, we have obtained 185 portraits of which only 98 are different. If someone can point out and report an error in our database, we will be greatly thankful.

• Moreover the author also details the number of phase portraits having a certain number of finite singularities is given in the abstract. However, there are 8 portraits with an infinite number of finite singularities, and these are not mentioned in the abstract. With the configurations of singularities, it is easy to check these numbers also. Under our recount we have: 8 phase portraits with an infinite number of finite singularities, all them are different. 76 portraits with 4 finite singularities of which 20 are different. 55 portraits with 3 finite singularities of which 35 are different. 34 portraits with 2 finite singularities of which 25 are different. 12 portraits with 1 finite singularity of which 10 are different. Moreover, we have find that there are several misses when compared with other articles. We point them out below.

• There are some missed arrows.

• The goal of this paper is very similar to the goal of Global topological classification of Lotka--Volterra quadratic differential systems and they should have to contain the same phase portraits. But there are some few portraits which are not in common. We detail each difference.

• Phase portrait \(QS145^{(5)}_1\) is not present here but it is in Global topological classification of Lotka--Volterra quadratic differential systems. Morever, its existence is confirmed by other papers. It seems to be an error in this paper.

• There are several degenerate phase portraits which are present in the other paper but not here. Concretely \(QS176^{(4)}_1\), \(QS178^{(4)}_1\), \(QS180^{(5)}_1\), \(QS182^{(6)}_1\), \(QS183^{(5)}_1\), \(QS184^{(6)}_1\), \(QS186^{(6)}_1\), \(QS188^{(6)}_1\), \(QS194^{(7)}_1\), \(QS196^{(8)}_1\), \(QS198^{(6)}_1\) and \(QS200^{(7)}_1\). Possibly the normal forms considerend by Reyn did not cover all the possibilities of degenerate systems.