Consider an equation, where compared to the Van der Pol oscillator equation the friction term is changed:

The equilibrium is at :formula_60

Here is a good example of an unsuccessful try to find a Lyapunov function that proves stability:

Let

so that the corresponding system is

Let us choose as a Lyapunov function

which is clearly positive definite. Its derivative is

It seems that if the parameter formula_65 is positive, stability is asymptotic for formula_66 But this is wrong, since formula_67 does not depend on formula_68, and will be 0 everywhere on the formula_68 axis. The system is Lyapunov stable.

Stability for systems with inputs
Lyapunov's second method for stability
Definition for discrete-time systems
Example