I encountered the concept of the Lyapunov Function from the book Ordinary Differential Equations by Arnold. The book states the following lemma:
Let $A:\mathbb{R}^n \to \mathbb{R}^n$ be a linear operator, all of whose eigenvalues have positive real part. Then the system $$ \dot{x} = Ax, \quad x \in \mathbb{R}^n $$ is topologically equivalent to the standard system $$ \dot{x} = x, \quad x \in \mathbb{R}^n $$
The author states that the proof of this lemma is based on the construction of a special quadratic function, the so-called Lyapunov function.
However, when I searched for the term Lyapunove Function, it seems that it is close related to a theory about equilibrium point, which is not like what this book described. I then think that I need more referrences to understand this topic. Any recommendations?
The Lyapunov function for a linear system tells you the homeomorphism between the standard system (or at least some monotonically stable/unstable system, but the one that we teach is with the standard system) and the particular system being examined. Stability is preserved by homeomorphism, which gives you some insight into why Lyapunov functions work.
These notes might clarify some things about how the theory of Lyapunov functions for linear systems is basically fully understood. https://stanford.edu/class/ee363/lectures/lq-lyap.pdf