Lyapunov Stability - Example

Example

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

The equilibrium is at :

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

\begin{align} \dot{V} &= x_{1} \dot x_{1} +x_{2} \dot x_{2}\\ &= x_{1} x_{2} - x_{1} x_{2}+\varepsilon \left(\frac{x_{2}^4}{3} -{x_{2}^2}\right)\\ &= -\varepsilon \left({x_{2}^2} - \frac{x_{2}^4}{3}\right). \end{align}

It seems that if the parameter is positive, stability is asymptotic for But this is wrong, since does not depend on, and will be 0 everywhere on the axis.

Read more about this topic:  Lyapunov Stability

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