A question on dynamical system's focus values on its center manifold

Question

I have recently come across this problem involving the center focus of dynamical systems of a parameter vector field related to center manifold:

We define a vector field on $ R^3 $ given by:

$ {x_1}' = -x_1+{x_3}^2 $

$ {x_2}' = x_3+{x_1}^2 $

$ {x_3}' = -x_2 +a{x_2}^3+b{x_2}^2x_3 + cx_2{x_3}^2+d{x_3}^3$

Where a,b,c,d are real parameters

We are asked to find a necessary and sufficient condition on the parameters a,b,c,d for which the first focus/Lyapunov value restricted to the center manifold is not zero

The fact that I am dealing with $ R^3 $ is puzzling me and my problem here is I cannot really find the restriction of this system to its center manifold so this i where I am stuck and where I need the help. Thanks all helpers

Answer 1

In general, it is impossible to get the center manifold of a vector field, even some simple vector fields. But you can get the system restricted to a patch of the center manifold around the center equibirium point, and you can refer to chapter "center manfiold for vector fields", in the book "introduction to applied nonlinear dynamical systems and chaos"-by Stephen Wiggins. Maybe it can help you.