Linearization of a Second Order Nonlinear Differential Equation

Question

I am in need of help with the linearization of this equation around the initial condition u=pi

My main confusion lies in the middle term .5xdotx. I cannot for the life of me figure out what to do with this term when linearizing.

Answer 1

Hint.

This system can be arranged as

$$ \cases{ \dot x_1 = x_2\\ \dot x_2 = -\frac 12 x_1x_2-\arctan(x_1)+u } $$

or $\dot X = f(X)+U$ with $X=(x_1,x_2)^T$ and $U = (0,u)^T$ . Now assuming an operating point at equilibrium $(\dot x_1=\dot x_2 = 0)$ and $p_0 = \{X_0,U_0\}$ such that $f(X_0)+U_0=0 $ we have

$$ f(X) = f(X_0) + \nabla f(X_0)\delta X+O(\|\delta X\|^2) $$

or

$$ \delta f(X) = \nabla f(X_0)\delta X+O(\|\delta X\|^2) $$

and finally the linearized system around $p_0$

$$ \delta X' = \delta f(X) +\delta U = \nabla f(X_0)\delta X+\delta U $$

Here $\delta X = X-X_0$ and $\delta U = U-U_0$