While the linear stability theory addresses, in principle, the local stability of an infinitesimal region around a stationary solution, the Liapounov criteria deal with global stability of a finite region around a stationary solution.
To be more specific, consider a steady state solution to eq(1) of section C.1.
We say is stable if there exists a function such that
where is a hypercubic region of sides 2h centered at . Conditions (1-3) are called the Liapounov criteria and V a Liapounov fuction .
Thus, if the Liapounov criteria are satisfied, any trajectory that falls inside D once will forever remain in it. To see how this comes about, consider 2 more hypercubes centered at with sides 2e and 2d such that . Let be the boundaries of these hypercubes. Denoting the lower bound of on by a, we have on . Now, consider a point inside the boundary at . Let be the position of the point at time t. If is chosen such that , then, according to criterion (2), we have
so that the point must remain inside for all t.
An analogous argument can be made for the case and so that is again stable if such a function can be found.
Consider a damped oscillator described by
(a) Show that the total energy can be used as a Lyapounov function.
(b) Locate and classify the steady states for the cases and .
The equivalent set of 1st order equations can be found by introducing a new variable so that
and, with the help of (1a),
Consider now the real function
Thus, for , we have so that F is a Lyapounov function. In particular, for , the system is non-dissipative and .
A steady state to (1b,c) is . For , F is a Lyapounov function for all the space around . Hence, is a stable. To be more precise, for , it is a stable focus. For , it is a center.