I am studying the equilibrium of smooth ODEs $y'(t)=f(t,y(t))$ and I read in wikipedia that an equilibrium is stable if all the eigenvalues of the Jacobian matrix of $f$ have modulus strictly inferior to 1. I can't find a proof of this property and I can't find a name or a keyword on this topic.
Could you please point me to any reading related to this relation between the jacobian matrix and the stability of an equilibrium?
Many thanks
First of all, I'm afraid that you mistook discrete-time dynamical systems for systems of ODEs: it is for the former that the Principle of Linearized Stability has the form that the moduli of all eigenvalues being less than $1$ implies the asymptotic stability of the fixed point. For autonomous systems of ODEs the sufficient condition for the asymptotic stability of an equilibrium is that the moduli of all eigenvalues have negative real parts. You can search Google, for example see BYU lecture notes.
Further, even after correcting that mistake, the property as you formulated it here, that is for nonautonomous systems of ODEs, need not hold. Consider the following two-dimensional time-periodic system of linear ODEs, due to Markus and Yamabe: $$ y'(t) = \begin{bmatrix} -1 + \frac{3}{2} \cos^2(t) & 1 - \frac{3}{2} \cos(t) \sin(t) \\ -1 - \frac{3}{2} \cos(t) \sin(t) & - 1 + \frac{3}{2} \sin^2(t) \end{bmatrix} y(t). $$ For all $t \in \mathbb{R}$ the eigenvalues of the matrices are just $(-1 \pm \sqrt{7} i)/4$, nevertheless the function $$ \exp(t/2) \begin{bmatrix} -\cos(t) \\ \sin(t) \end{bmatrix} $$ is a solution that is unbounded as $t \to \infty$.