In Arnold's "Ordinary Differential Equations-MIT Press (1978).pdf", vector flow with initial value at point $x_0$, i.e. solutions of $x'(t) = v(x(t))$. I'm mostly interested in initial value which is a singular point.
In page 13 - 14, it is said that
In 1-dim flow, $t - t_0 = \int_{x_0}^{\varphi(t)} \frac{d{\xi}}{v(\xi)} $ if $v(x_0) \neq 0$, and $\varphi(t) = x_0$ if $v(x_0) = 0 $. When $v(x_0) = 0$, just let $\varphi(t) \equiv x_0$.
In page 51, Corollary 3 says that
In 2-dim flow, the vector flow $v$ determines a local phase flow in a neighbourhood of a nonsingular point $x_0$ ($v(x_0) \neq 0$).
In page 221, corollary says that
if $v$ is continuously differentiable, given any point $x$ sufficiently close to $x_0$, there exists a neighborhood of $t_0$ in which a solution is defined. (Using Lipschitz condition, Picard approximation and contraction mapping)
My question is that:
In corollory 3, when we try to generalize 1-dim vector flow to 2-dim vector flow, why we cannot just let $\varphi(t) \equiv x_0$ for all time t and say that it is a phase flow?
In order to prove that In 2-dim flow, the vector flow $v$ determines a local phase flow in a neighbourhood of a singular point $x_0$, i.e. $v(x_0) = 0$, the book uses two chapters of direct product of 1-dim flow and eigenvalues and eigenspaces. Why don't we just use Picard's existence and uniqueness theorem with Lipschitz condition? What's the difference between these two proofs?