Let $f$ denote a continuous-time dynamical system in $\mathbb{R}^{n}$. That is, $\dot{x} = f(x)$.
Let $\phi_t : \mathbb{R}^{n} \to \mathbb{R}^{n}$ be the flow function of $f$; that is, $\frac{\mathrm{d}}{\mathrm{d}t}\phi_t(x) = f(x)$.
I am concerned with sufficient conditions for:
- $\phi_t$ to be smooth (that is, continuously differentiable, $C^1$)
- $\phi_t$ to have non-vanishing Jacobian
- $\phi_t$ to be injective (that is, one-to-one)
I think that Hirsch et al. 3rd, Ch 7.4, pp. 154, "Smoothness of Flows" Theorem implies that if $f$ is smooth ($C^1$), then $\phi_t$ is also smooth ($C^1$). Is this true?
However, I am not sure how to find a sufficient condition for 2) and/or 3 to hold. I have been told that, if the vector field $f$ has bounded first derivatives, then 2) and 3) are always true. Could you please point to a specific source (or sources) that will show this result?
Bonus: if you would like to prove the conditions for which 2) and 3) hold for completeness of this entry for the sake of future Math.SE users, that would be great.