System
$$\dot{x}=-x+2y^3-2y^4$$ $$\dot{y}=-x-y+xy$$
The Liapunov function of this system could be something like $V=x^m+ay^n$. I am trying to figure out the appropriate values for $m$, $n$, and $a$. I know there is no hard and fast rule to get the Liapunov functions. Any suggestions?
Looking at the linear in x terms in the right-hand side of the system, one can guess that $$ V(x,y)= \frac12 x^2+\psi(y). $$ The derivative is $$ \dot V= x(-x+2y^3-2y^4)+\psi'(y)(-x-y+xy) $$ $$ =-x^2+x(2y^3-2y^4-\psi'(y)+y\psi'(y))-y\psi'(y). $$ Let $$2y^3-2y^4-\psi'(y)+y\psi'(y)=0;$$ this gives us $$ 2y^3(1-y)-\psi'(y)(1-y)=0; $$ hence, $\psi'(y)=2y^3$, $\psi(y)=\frac12 y^4$ and $V(x,y)=\frac12 x^2+\frac12 y^4$.