I am looking for any reference on the regularity of flows of non-autonomous ODEs. Any suggestions on books or articles are very much welcome. I have had little luck finding any usable results.

Let us assume that

$$ f: [0,1]\times \mathbb{R}^N \to \mathbb{R}^N $$

is continuous, satisfies the Lipschitz condition (i.e., it is Lipschitz continuous in the second argument), and is continuously differentiable in the second argument. Furthermore, assume that its derivative in the second argument, $\frac{\partial f}{\partial x}$, also satisfies the Lipschitz condition.

Consider the flow $\varphi : [0, 1] \times \mathbb{R}^N \to \mathbb{R}^N$ of the ODE

$$ \dot x = f(t, x) $$

Meaning that $\varphi$ is defined via:

\begin{equation} \begin{cases} \frac{d}{dt}\varphi(t, x_0) = f(t, \varphi(t, x_0)) \\ \varphi(0, x_0) = x_0 \end{cases} \end{equation}

For fixed $t\in[0,1]$, I expect that $\varphi(t, \cdot):\mathbb{R}^N \to \mathbb{R}^N$ is differentiable. In fact, I would expect the derivative $\frac{\partial \varphi}{\partial x}(\cdot, x_0)$ to satisfy the usual variational equation

\begin{equation} \begin{cases} \frac{d}{dt}X(t, x_0) = \frac{\partial f}{\partial x}(t, \varphi(t, x_0))X(t, x_0) \\ X(0, x_0) = I \end{cases} \end{equation}

I have been searching through dozens of books for any result like this. Most books, e.g., "Differential Equations, Dynamical Systems & An Introduction to Chaos", only consider autonomous ODEs with a right-hand side which is $C^1$. Oftentimes, non-autonomous systems are dismissed at the very beginning with the note that it is possible to reduce to the autonomous setting. In my case, this will not be possible since the autonomized right-hand side will not be in $C^1$.

I would be grateful if somebody could point me towards helpful literature. Thanks!

  • 1
    $\begingroup$ Maybe not exactly what you're looking for, but Section 9 in Amann's Ordinary Differential Equations may be helpful. He assumes that $f$ is continuous as a function of all variables, but doesn't require differentiability with respect to $t$. $\endgroup$ Jul 5 at 10:52
  • $\begingroup$ This looks promising! Thank you very much! $\endgroup$
    – Markus K
    Jul 5 at 11:13

Consider the following system:

$$\begin{cases} \dot{\tau} = \omega\\ \dot{\omega} = 0 \end{cases}$$

with initial conditions $\tau(0) = 0$ and $\omega(0) = 1.$

The solution is $\tau(t) = t$ and $\omega(t) = 1.$

Now, consider the following system:

$$\begin{cases} \dot{x}_1 = f_1(t, x_1, \ldots, x_n)\\ \ldots \\ \dot{x}_n = f_n(t, x_1, \ldots, x_n)\\ \end{cases}.$$

You can add the equations of $\tau$ and $\omega$, and replace the explicit $t$ with $\tau$:

$$\begin{cases} \dot{x}_1 = f_1(\tau, x_1, \ldots, x_n)\\ \ldots \\ \dot{x}_n = f_n(\tau, x_1, \ldots, x_n)\\ \dot{\tau} = \omega\\ \dot{\omega} = 0 \end{cases}.$$

In this way, you are able to transform a non-autonomous system into an autonomous one.

  • 1
    $\begingroup$ I am aware that we can transform non-autonomous systems into autonomous ones. However, this autonomous system does not have a right-hand side in C^1, does it? I have no assumption regarding the differentiability of f w.r.t. the first argument. Thus, I cannot apply the result about autonomous ODEs. $\endgroup$
    – Markus K
    Jul 5 at 10:35
  • $\begingroup$ Well, The right hand side of $\tau$ and $\omega$ are $C^1$ for sure. You just need that all $f_i$ are $C^1$, right? $\endgroup$ Jul 5 at 10:39
  • $\begingroup$ Yes, but that's not really the case. The $f_i$ are continuously differentiable in space, but not in time. $\endgroup$
    – Markus K
    Jul 5 at 10:44
  • $\begingroup$ Sorry, I misread the question. $\endgroup$ Jul 5 at 11:32

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