Let $\mathbf{x}(t):=[x_1(t),\dots,x_n(t)]^\top$, $n>1$, $A\in\mathbb{R}^{n\times n}$ be a nonnegative matrix and $\mathbf{b}\in\mathbb{R}^n$ be a positive vector. Consider the following differential equation $$\tag{$\star$}\label{eq} \dot{\mathbf{x}}(t) = \mathbf{b}+ A\, \mathbf{\text{sin}}(\mathbf{x}(t)), \quad \mathbf{x}(0)\in\mathbb{R}^n, $$ where $\mathbf{\text{sin}}(\cdot)$ denotes the sine function applied elementwise to a vector.

A couple of questions:

By any chance, does the solution of \eqref{eq} admit an explicit form?

If the answer to my previous question is no (as I would expect), can we infer some "qualitative" properties on the behavior of the solution of \eqref{eq}? In particular, I would expect that the behavior of $\mathbf{x}(t)$ would be closer to be linear as $\mathbf{b}$ gets larger or the entries of $A$ get smaller? In particular, in the limit case $\mathbf{b}\to \infty$ the solution should tend to $\mathbf{x}(t)\to \mathbf{b}t$; am I correct?