Question on Linearised system

Question

I have this question :

study the nature of the critical point for the linearized system of : $x''+x'^3+x=0$

please how we find the linearised system of $x''+x'^3+x=0$.

Please help me ,

Thank you

Answer 1

Given:

$$x''+(x')^3+x=0$$

We want to study the nature of the critical point for the linearized system.

We start off by writing this as a system of equations (they happen to be nonlinear) as follows. Let $x_1 = x$, so

• $x_1' = x' = x_2$
• $x_2' = x'' = -(x')^3 - x = -x_2^{3} - x_1$

So we can write this as a system:

• $x_1' = x_2$
• $x_2' = -x_1 -x_2^{3}$

The next steps:

• $(1)$ Find the critical points
• $(2)$ Linearize the system using those critical points
• $(3)$ Classify the linearization of those critical points
• $(4)$ Draw the phase portrait to validate or invalidate the above
• $(5)$ Analyze that result using the theory for correctness based on all of the above

Here are phase portraits, for example (the second one gets closer into the origin).