Linear approximations around a given fixed point.

[andyb177]

Saw this mentioned, didn't understand what it was or how it would be done.
Given the continuous system given by x'₁,x'₂,x'₃
Find the linear approximation for each x* (fixed point)

Guessing first of course to find the fixed points.
Then find the Jacobian Df for the solved system.
What to do then?

Idea was given in an example of course, hoping some one will be able to explain what to do for a general case.

Thanks in advance.

 

[HallsofIvy]

Yes, find the Jacobian and evaluate it at each of the fixed points (equilibrium points in terms of dynamics). Since the entries in the Jacobian matrix are now constants rather than functions of t, you will have the linearized equation about that point.

 

[andyb177]

Yes, have evaluated it. So the Jacobian for each point, a 3x3 matrix filled with constants. Is this the linear approximation? or do I take the det? There was a HINT to find eigenvectors and values and find for a non fixed point and permute the indices?. Still confused as what to do/what I am looking for?