Prove that $V(x_{1},x_{2})$ is asymptotically stable at the origin

Prove that $V (x_ {1}, x_ {2}) = ax_ {1} ^ {2} + 2bx_ {1} x_ {2} + cx_ {2}^{2}$ is positive definite if and only if $a> 0$ and $ac> b^{2}$. Then, prove that $$\left\{\begin{array}{l} \dot{x_{1}} = x_{2} \\\dot{x_ {2}} = - x_{1} -x_{ 2} - (x_{1} + 2x_{2}) (x_{2}^2-1) \end{array} \right.$$ is asymptotically stable at the origin considering the region $| x_ {2} | <1$. Establish the domain of stability

I have problems to get the linearized system

• Please. Check the dynamics because the equilibrium set is all the axis $x_2 = 0$. – Cesareo Jun 19 '18 at 20:06
• Could you help me how to get to linearization? – Steve Jun 20 '18 at 5:06
• Discarding higher order terms, the linear version reads $\left\{\begin{array}{l} \dot{x_{1}} = x_{2} \\\dot{x_ {2}} = x_{ 2} \end{array} \right.$ – Cesareo Jun 20 '18 at 9:05
• As Cesareo above indicated, can you again verify the equation for $\dot{x_2}$ Using the equations as is, a phase portrait in MATLAB indicates that the system is not asymptotically stable for $\left| x_2 \right| \lt 1$. – Winter Soldier Jun 21 '18 at 14:42