ABSTRACT
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
y
dy dt
x x y
=
= − − −
⎧
⎨
⎪
⎪
⎩
⎪
⎪
ε( )1 2 (4.5)
In this section, we introduce a few important theorems.
