文档介绍：Section
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The Geometry of Solutions:
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 £© The Phase Plane
As mentioned in Section , we may represent a second order differential equation
x¨ = f(x, x,˙ t) ()
as a system of two first order equations
x˙= y
()
y˙= f(x, y, t).
More generally, if g and f are functions of x, y, and t, we may consider a system of
equations
x˙= g(x, y, t)
()
y˙= f(x, y, t),
of which () is a particular case when g(x, y, t) = y. In this section we shall consider
the behavior of solutions to such systems of equations, paying particular attention to those
arising in the manner of ().
Definition Suppose x(t) and y(t) are solutions of the system
x˙= g(x, y, t)
y˙= f(x, y, t)
for t in an interval [a, b]. The curve in the plane with coordinates (x(t), y(t)), a ≤ t ≤ b, is
called a phase curve of the system. The plane in which the phase curve is plotted is called
the phase plane of the system.
Note that if the system of equations arises from a second order differential equation,
then a phase curve is a plot ofx ˙(t) versus x(t). In mon cases, this is a plot of
velocity versus position.
Definition Suppose the constant functions x(t) = x0 and y(t) = y0 is a solution of the
system
x˙= g(x, y, t)
.
y˙= f(x, y, t)
Then the point (x0, y0) is called a stationary point of the system.
1
2 The Geometry of Solutions: The Phase Plane Section
If (x0, y0) is a stationary point, then the phase curve of the solution
x(t) = x0
y(t) = y0
consists of only the single point (x0, y0). That is, if (x0, y0) is a stationary point and the
system has initial conditions x(t0) = x0 and y(t0) = y0, then the system will remain at
the point (x0, y0) for all time. Moreover, note that for this solution
x˙(t) = 0
and
y˙(t) = 0
for all t. Since we must have
x˙= g(x, y, t)
y˙= f(x, y, t),
it follows that stationary points are precisely the points (x0,