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I came across the following question in one of my professor's past exams:

Find a Lyapunov function for $(0,0)$ in the system: $$ \left\{ \begin{array}{ll} \dot{x} = -x -2y + x^2\\ \dot{y} = x - 4y + xy \end{array}\right. $$

I know there is no formula for finding Lyapunov functions for a system, so how do I start solving such problems?

Thanks!

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up vote 2 down vote accepted

For many ODEs, a good bet is to try a polynomial as a Lyapunov function candidate. In fact, in practice, a common method is to search for a sum-of-squares polynomial, i.e., a polynomial that can be given as $\sum_{i=1}^k p_i(x)^2$, with $p_1, \dots, p_k$ polynomial.

I haven't tried this example, but a good approach might be to try something of the form $V(x,y) = (x+ay+b)^2 + cy^2 + d$, with parameters $a,b,c,d$.

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