Linearizing z'' = z - z³ about equilibrium solutions

[tasha10]

Find equilibrium solutions for the following ODE initial-value
problem and linearize the problem about those solutions
z" = z − z^3,
z(0) = z0
z'(0) = v0

i found the equilibrium solutions to be 0,1,-1. what are the steps to linearize the problem around these?

 

[jackmell]

I believe you represent it as a coupled system of first-order ODEs, then translate the equilibrium point to the origin via the transformations u=x-x_0 and v=y-y_0. Then compute the Jacobian at the equilibrium point and thus convert the system to a linear system with an equilibrium point at the origin. Try and find "Differential Equations" by Blanchard, Devaney, and Hall. It's an easy read and does a lot of work with systems.