Examples

Abstract

In this section we study the decay to zero of solutions of the equation

$$ \ddot r + \dot r + f\left( r \right) = 0 $$

(3.1.1)

where f is a smooth function with

$$ f\left( r \right) = {r^3} + a{r^5} + 0\left( {{r^7}} \right)\,as\,r \to 0, $$

(3.1.2)

where a is a constant. By using a suitable Liapunov function it is easy to show that the zero solution of (3.1.1) is asymptotically stable. However, because f′(0) = 0, the rate of decay cannot be determined by linearization.