Global analysis of Riccati quadratic differential systems (Q6538973)

The article considers a system of two ordinary differential equations \N\[\N\dot{X}=F(X),\tag{1}\N\]\Nwhere \(F(X)=(P_1(x,y), Q_1(x,y))^T\), \(P_1(x,y)=a+cx+dy+gx^2+2hxy+ky^2\), \(Q_1(x,y)=b+ex+fy+lx^2+2mxy+ny^2\), \(X=(x,y)^T\), \(x,y\) are real variables, \(a, c, d, g, h, k, b, e, f, l, m, n\) are real constant (parameters). Along with the system (1), the Riccati system is considered: \N\[\N\dot{X}=G(X),\tag{2}\N\]\Nwhere \(G(X)=(P_2(x,y), Q_1(x,y))^T\), \(P_2(x,y)= a+cx+gx^2\). If system (1) can be reduced to system (2) by some affine transformation, then system (1) is called a \(QSL^{2p}\) class system. An essential part of the article is an overview and contains previously known results of the authors obtained for systems of the \(QSL^{2p}\) class. The article summarizes, modifies and systematizes previously known results, as well as adds new results. All possible relations between the values of the system parameters (1) are considered. Depending on these relations, the following results are obtained: (i) The algorithm by which it is possible to determine whether the system (1) belongs to the \(QSL^{2p}\) class. (ii) Various topological portraits of \(QSL^{2p}\) class systems (a total of 119 different phase portraits were obtained). (iii) Bifurcation diagrams of \(QSL^{2p}\) class systems.