I suggest you follow https://www.cds.caltech.edu/~murray/courses/cds101/fa02/caltech/pph02-ch19-23.pdf

https://www.sciencedirect.com/topics/engineering/equilibrium-point

https://books.google.com.np/books?id=OsfFCgAAQBAJ&pg=PA43&lpg=PA43&dq=How+could+I+take+into+account+the+change+of+the+equilibrium+point+during+the+simulation+of+a+Jacobian+linearized+system?&source=bl&ots=_Joul07D8i&sig=ACfU3U1mYCZFll3vMcROu8T861unqLVUuQ&hl=en&sa=X&ved=2ahUKEwiGjqvewY_qAhVJXn0KHdp9BmIQ6AEwCnoECAQQAQ#v=onepage&q=How%20could%20I%20take%20into%20account%20the%20change%20of%20the%20equilibrium%20point%20during%20the%20simulation%20of%20a%20Jacobian%20linearized%20system%3F&f=false

You have to change the A matrix at every operating point with the Jacobian.

Linearisation is valid only for a small region around the operating point.

If you want to use one single model over a range of x ,possibly an average model of A obtained over the range might help. Stability inherent as well as numerical (condition of the A matrix) is another aspect to be considered.

Good luck

Cheers

The following reference may be helpful

An Introduction to Differential Equation (Order and Chaos)

by Florin Diacu

Dear Ádám Domina,

First of all, you need to make sure that the Jacobian matrix evaluated at the equilibrium state is hyperbolic. ( The eigenvalues has non zero real parts) Then you can apply the Hartman-Grobman theorem. That is, the linear system and nonlinear system behave the same in the neighborhood of the equilibrium point. So, the main point is to determine the eigenvalues of the Jacobian matrix which essential to study the stability of your system.

Best regards