Finding Lyapunov Function Is there any Lyapunov function to investigate the stability of the following dynamical system?
$$
 dx/dt= y + x(1-x^2-y^2)
$$
$$
 dy/dt=-x + y(1-x^2-y^2)
$$
My view: I know this system has a limit cycle since the closed path $x^2+y^2=1$ is approached spirally from both the inside and outside by nonclosed paths as t tends to infinity. But I cannot find any Lyapunov function.Please help.
 A: The phase portrait a good place to start.

Candidate Lyapunov function
$$
 V(x,y) = \frac{1}{2} \left( x^{2} + y^{2} \right)
$$
Is this a strong Lyapunov function? Criteria:


*

*$V(x,y)$ is positive definite about the point $(x,y) = (0,0)$

*$\dot{V}$ is negative semidefinite about the point $(x,y) = (0,0)$


Is $V(x,y)$ positive semidefinite about the origin? Compute the discriminant
$$
 \det 
\left[ \begin{array}{cc}
  V_{x,x} & V_{x,y} \\ V_{y,x} & V_{y,y}
\end{array} \right]
=
\det
\left[ \begin{array}{cc}
  1 & 0 \\ 0 & 1
\end{array} \right]
= 1
$$
Because the discriminant is positive, the function $V(x,y)$ is positive definite.
The derivative
$$
\dot{V}(x,y)= \nabla{V} \cdot 
\left[ \begin{array}{c}
  \dot{x} \\ \dot{y}
\end{array} \right]
 = \left(x^{2}+y^{2}\right)  \left(1 - x^{2} - y^{2} \right)
$$
positive inside the unit circle, and negative outside.
So, no, $V(x,y)$ is not a strong Lyapunov function.
@Did: Red lines: nullclines where $x'=0$; yellow $y'=0$. Thanks for checking, $\dot{V}$ is corrected.
