MHB Determining Stability of Equilibrium Point for x'=y-x3-xy2, y'=-x-x2y-y3

[onie mti]

i have this system

x'=y-x³-xy²
y'=-x-x²y-y³

i worked it out and found the equilibrium point to be 0.

how do i determine whether it is stable, assymp stable or not stable

 

[chisigma]

i have this system

x'=y-x³-xy²
y'=-x-x²y-y³

i worked it out and found the equilibrium point to be 0.

how do i determine whether it is stable, assymp stable or not stable

Let's define $x_{1}$ and $x_{2}$ the state variables, so that the system becomes...

$\displaystyle x_{1}^{\ '} = x_{2} - x_{1}^{3} - x_{1}\ x_{2}^{2}$

$\displaystyle x_{2}^{\ '} = - x_{1} - x_{1}^{2}\ x_{2} - x_{2}^{3}\ (1)$

You can choose as Lyapunov function $\displaystyle V(\overrightarrow x)= \frac{x_{1}^{2} + x_{2}^{2}}{2}$, so that is...

$\displaystyle V^{\ '} (\overrightarrow x) = x_{1}\ x_{2} - x_{1}^{4} - x_{1}^{2}\ x_{2}^{2} - x_{1}\ x_{2} - x_{1}^{2}\ x_{2}^{2} - x_{2}^{4} = - (x_{1}^{2} + x_{2}^{2})^{2} \le 0\ \forall \overrightarrow x\ (2)$

... so that the system is stable...

Kind regards

$\chi$ $\sigma$