Checking stability of a fixed point

Question

If the fixed point is hyperbolic, then it is said that linearisation gives the correct result . Is there an intuitive way of understanding why this is so ?

And for marginal cases, when the fixed points are found to be centres ot stars by linearisation, what is a way to find if they are really so for the nonlinear system too and why the marginal cases are too delicate to handle with linearisation ?

I am from physics background, hence, apart from rigorous mathematical proofs, physical arguments shall be more helpful .

Answer 1

"If the fixed point is hyperbolic, then it is said that linearisation gives the correct result . Is there an intuitive way of understanding why this is so ?"

If we have a $n-$dimensional linear system of differential equations $(\dot{x} = Ax)$ with a single fixed point at the origin we can observe several types of behaviours, such as saddle points, spiral,cycles, stars and nodes which are well understood. We classify these cases based on the eigenvalues of the matrix $A$ used to classify the system. A non linear system is difficult to analyze , fortunately we are not in dark completely because of Hartman-Grobman theorem. We can find the Jacobian matrix $J$, corresponding to the system and evaluate it at a fixed point to obtain a linear system with a characteristic coefficient matrix. The Hartman-Groban theorem tells us that, at least in the neighborhood of the fixed point, if the eigenvalues of $J$ all have non-zero real part (Hyperbolic ) then we can get a qualitative idea of the behavior of solutions in the non-linear system.

"And for marginal cases, when the fixed points are found to be centres ot stars by linearisation, what is a way to find if they are really so for the nonlinear system too and why the marginal cases are too delicate to handle with linearisation ?"

Here we can refer to this answer - Marginal cases. WIll add if I get some points!