I am working on a project which essentially involves me generating high dimensional vectors which evolve smoothly (Lipschitz) over a finite time interval. I am trying to come up with 'interesting' vector valued functions $f:\mathbb{R}^1\rightarrow\mathbb{R}^p$, in a similar flavor to the way that $$g(t)=\begin{bmatrix}\sin t\\\cos t\end{bmatrix}$$
is an interesting function.
My trouble is that it's not hard to come up with cases where $p=2,3$ but for something like $p=100$ it is less clear.
One thing I have considered is something along the lines of
$$g(t)=\mathbf{A}\mathbf{X}_t$$
where $\mathbf{A}\in \mathbb{R}^{p\times p}$ is a constant coefficient matrix and $\mathbf{X}_t$ is a vector of fourier basis functions (e.g. $\mathbf{X}_t=[\sin(t),\cos(t),\sin(2t),\cos(2t),\dots]^T$), although I wonder if another basis, or another form of $g(t)$ may do the trick better? Any thoughts/comments would be greatly appreciated.