1
$\begingroup$

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

$\endgroup$
  • 2
    $\begingroup$ Please. Check the dynamics because the equilibrium set is all the axis $x_2 = 0$. $\endgroup$ – Cesareo Jun 19 '18 at 20:06
  • $\begingroup$ Could you help me how to get to linearization? $\endgroup$ – Steve Jun 20 '18 at 5:06
  • $\begingroup$ 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. $ $\endgroup$ – Cesareo Jun 20 '18 at 9:05
  • $\begingroup$ 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 $\left| x_2 \right| \lt 1$. $\endgroup$ – Winter Soldier Jun 21 '18 at 14:42

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.