As for the phase space, which is the Euclidean plane R2, some important results were established by H. Poincaré and I. Bendixson. For example, the given differential equations

dx dt

P x y

dy dt

Q x y



( , )

( , ) (4.3)

where P(x, y), Q(x, y) are continuous in a connected region D⊂R2 and must satisfy some conditions in order to keep any initial-value problem have a unique solution. This kind of system appears in the studies of physics and finance frequently. For example, there is the following differential equation with damping in the oscillating financial market:

d x dt

x dx dt

x 2

2 21 0+ − + =ε( ) (4.4)

where ε > 0. Let y = (dx/dt). Then, (4.4) becomes the following quadratic system:

dx dt


dy dt

x x y


= − − −

ε( )1 2 (4.5)

In this section, we introduce a few important theorems.