lyapunov function for my system

Question

I need to find a Lyapunov function for (0,0) in the these two systems:

\begin{cases} x' = -x + y^6 \\ y' = x^6 + y^3 \end{cases}

and

\begin{cases} x' = -x^3 + y \\ y' = x^6 -y^3 \end{cases}

regards,

Answer 1

Since the systems have polynomial vector fields you can use SOS to find a Lyapunov function.

However, the first system seems to be unstable. Even with extremly small initial conditions like $x_0 = \begin{bmatrix}0 & 10^{-16}\end{bmatrix}$ the system doesn't return to the equilibrium. Also no polynomial Lyapunov function is found up to a degree of 16.

For the second system a Lyapunov function is for example

$V_2(x, y) = 0.000353x^{10} - 8.207\cdot 10^{-11}x^9y - 0.005261x^9 + 3.683\cdot 10^{-5}x^8y^2 - 0.007904x^8y + 0.3201x^8 + 0.0001484x^7y^3 - 0.01358x^7y^2 + 0.03524x^7y - 1.408x^7 - 8.886\cdot 10^{-5}x^6y^4 - 0.007686x^6y^3 + 0.07872x^6y^2 - 0.9889x^6y + 14.11x^6 - 0.000665x^5y^5 + 0.02739x^5y^4 - 0.1359x^5y^3 + 0.5374x^5y^2 - 2.881x^5y - 15.28x^5 - 8.886\cdot 10^{-5}x^4y^6 + 0.02739x^4y^5 - 0.277x^4y^4 + 0.04691x^4y^3 - 1.388x^4y^2 - 15.68x^4y + 40.07x^4 + 0.0001484x^3y^7 - 0.007686x^3y^6 - 0.1359x^3y^5 + 0.04691x^3y^4 + 3.485x^3y^3 - 12.69x^3y^2 + 34.87x^3y - 41.74x^3 + 3.683\cdot 10^{-5}x^2y^8 - 0.01358x^2y^7 + 0.07872x^2y^6 + 0.5373x^2y^5 - 1.388x^2y^4 - 12.69x^2y^3 + 44.05x^2y^2 - 77.45x^2y + 49.54x^2 + 4.339\cdot 10^{-11}xy^9 - 0.007904xy^8 + 0.03524xy^7 - 0.9889xy^6 - 2.881xy^5 - 15.68xy^4 + 34.87xy^3 - 77.45xy^2 + 30.93xy + 0.000353y^10 - 0.005261y^9 + 0.3201y^8 - 1.408y^7 + 14.11y^6 - 15.28y^5 + 40.07y^4 - 41.74y^3 + 49.54y^2.$