Quadratic ODE systems

You are right, these systems are not completely understood even in the simple case of dimension $n=2$ (i.e., on the plane). You can check out this book on planar quadratic differential equations. In general you can consider systems of the form $$ \dot x_i=x_i(c_i+(Ax)_i), $$ which is slightly less general than you are asking for (the notation $(Ax)_i$ means the $i$th element of the vector $Ax$). These are the so-called Lotka--Volterra systems in mathematical ecology. There is an equivalent system defined on the simplex $$ \dot p_i=p_i((Ap)_i-p\cdot Ap), $$ which is called the replicator equation. There are a lot of open questions about these equations. A good book to look for some existing theory is Evolutionary games and popualtion dynamics (or just google the names).

For quadratic systems there is the famous paper by Larry Markus "Quadratic Differential equations and non-associative algebras" in Contributions to the Theory of Nonlinear Oscillations Vol V (1960) pages 185 - 213.

The PhD thesis: "Quadratic differential Equations a Study in Nonlinear Systems Theory" by M. Frayman Univ. of Maryland, 1974.

Also, "Extensions of Linear Quadratic Control, Optimization and Matrix Theory" by David H. Jacobson 1977.

Also, "Bilinear Control Systems" by David Elliot, Springer, 2009.

Quadratic systems have very interesting properties and include some chaotic systems; Lorenz's butterfly effect is a quadratic system. Also, these systems can be "super unstable" or explosive in the sense that a solution can go to infinity in finite time; whereas in a linear system a solution that goes to infinity takes infinite time to get there.

All sorts of other good stuff.