In this section we introduce the important idea of
stability of an equilibrium point of a system of ordinary
differential equations. An equilibrium point
is stable if all nearby solutions stay nearby. It is
asymptotically stable if all nearby solutions not only stay
nearby, but also tend to
.
Consider the nonlinear system
Definition 3.5.1: Let
,
,
be an equilibrium point of
the system (3.19) and
be the flow defined
by (3.19). Then
is said to be
In 2-d, the above concepts can be described by the following phase diagram:
It follows from Theorems 3.1.1, 3.1.3 and 3.4.1 that the
equilibrium point
of (3.19) is
asymptotically stable if it is a sink, unstable if it is a
source or a saddle. In other words, a hyperbolic
equilibrium point of (3.19) is either asymptotically
stable or unstable. This also implies that stable
equilibrium points which are not asymptotically stable can
only occur at nonhyperbolic equilibrium points. But the
question as to whether a nonhyperbolic equilibrium point is
stable, asymptotically stable or unstable is a delicate one.
To motivate a method, which is very useful in answering this question, we discuss a simple example below.
Example 3.5.1: Consider the nonlinear system
Thus
E(x1(t), x2(t)) = E(x1(0), x2 (0)),
,
for any solution of (3.20), i.e.
The following method, given by A.M. Lyapunov, a Russian mathematician and engineer, in his doctoral thesis in 1892, is a generalization of the (energy) idea contained in the preceding example.
Let
be a C1
function. Then the derivative of V along
solutions of (3.19) is defined as
Theorem 3.5.1: Let
be an
equilibrium point of (3.19),
be an open set containing
,
and
be continuously differentiable such that
is stable,
is asymptotically stable,
is unstable.
Before proving this theorem, we discuss some examples and try to get some geometric intuition.
Example 3.5.2: Consider the nonlinear system
Since
and
,
i.e.
,
it follows that
no conclusion can be made by linearization.
To apply the Lyapunov method, we set
V (x,y) = x2 +
y2. Then
Thus by Theorem 3.5.1, (0,0) is asymptotically stable.
Example 3.5.3: Consider the nonlinear system
But let
V(x,y) = x2 + 2y2. Then
V(x,y) > 0 if
,
V(0,0) = 0 and
if
.
Thus by Theorem 3.5.1, (0,0) is unstable.
The function V in Theorem 3.5.1 is often called Lyapunov function. It is called positive definite if a) is satisfied, negative definite if -V is positive definite, and ``strict'' Lyapunov function if a) and c) hold.
To get an intuitive idea about Theorem 3.5.1, we note that
a) implies that
is a strict local minimum
point of V. Thus we expect the level sets of V,
in a neighbourhood of
,
to be concentric
curves (n=2) or surfaces (n=3). Recall that
if
,
then
is
orthogonal to the level set
V(x) - V(a) through
a and points outward, and
,
where
is the angle between
and f. Thus if
Proof of Theorem 3.5.1: (omitted)
The beauty of the Lyapunov function approach lies in the fact that no knowledge of solutions is required. Thus it exhibits great power in applications. It should be noted, that there is no rigid scheme to construct Lyapunov functions, especially, when the systems are written in a general setting. Maybe this is one of the reasons why Lyapunov's stability theory has lasted so long and is still a very active area of research. However, in the case of equations which model physical systems, e.g. mechanical systems, there will be some natural candidates, which can be seen in the next section.
It should be noted that when
is a sink of
(3.19), one can construct, in view of Theorem 3.1.2,
V(x) in a neighbourhood of
.
Namely
V(x) = xT Qx, where Q is a positive
definite matrix and satisfies
You might question why we would even bother to consider the Lyapunov's method when linearization works. The purpose was to really ``tie up loose ends'' and provide a complete picture of the successes and failures of both methods. Again, the ``triumph'' of the Lyapunov's method is to boldly go where linearization could not take us, i.e. eigenvalues with zero real parts. It should be noted that information obtained by linearization can be very misleading in real world problems. The extent of the stability, e.g. the size of the region of attraction and the behaviour of transients as they approach the equilibrium is determined by the nonlinearity of the system. Lyapunov's method takes nonlinearity into account. When one can find Lyapunov functions, they make quantitative information available.