This answer to a different question describes some conditions under which a time-varying discrete dynamical system can be expressed with time-invariant dynamics in a higher dimensional state space. (Full disclosure, that answer is my own. The example got me wondering more generally, but I'm not sure where to look for an answer.)
So there are some cases where a time-varying dynamical system
$$ \begin{align} \dot x = f(t, x) && x \in \mathbb R^n \end{align} \tag{1} $$
can have its state space $x \in \mathbb R^n$ augmented by some $u \in \mathbb R^m$
$$ \begin{align} r = \begin{bmatrix} x_1 \\ \vdots \\ x_n \\ u_1 \\ \vdots \\ u_m\end{bmatrix} \end{align} $$
such that there is a time-invariant dynamical system
$$ \begin{align} \dot r = g(r) && r \in \mathbb R^{n+m} \end{align} \tag{2} $$
which exactly expresses the original time-varying trajectories. That is,
- if $r(t)$ is a solution to $(\ref{2})$ then $\text{proj}_{1...n}(r(t))$ is a solution to $(\ref{1})$.
- if $x(t)$ is a solution to $(\ref{1})$, then there exists $r(t)$ a solution to $(\ref{2})$ such that $x(t) = \text{proj}_{1...n}(r(t))$.
I am wondering whether or not this is always possible. Do there exist time-varying dynamical systems which cannot be rewritten as time-invariant dynamics in a higher dimensional state space?
Another example of a tidy relationship between a time-varying system and a time-invariant one is example 8.5 from Khalil 2002, though I'm not sure it's exactly related. It shows how a time-varying linear continuous dynamical system can be written as a local linearization of some time-invariant nonlinear system. Let the system $\dot{x} = f(x), x \in \mathbb{R^2}$ be defined as
$$
\begin{align}
\dot{x_1} &= -x_2 \\
\dot{x_2} &= x_1 + (x_1^2 - 1)x_2
\end{align} \tag{1}\label{1}
$$
Clearly the system is nonlinear, but time-invariant. We can locally linearize with the Jacobian $A=\nabla f|_{x_t}$:
$$ \begin{align} \dot{x}|_{x_t}&=A(x_t)x_t \\ A(x)&=\begin{bmatrix}0 & 1 \\ 1+2x_1x_2 & x_1^2 - 1\end{bmatrix} \end{align} \tag{2}\label{2} $$
Hence, along any particular solution trajectory of the time-invariant nonlinear system $(\ref{1})$, we obtain time-varying linear system $(\ref{2})$, since $x_1, x_2$ are functions of time.