On the (qualitative) behavior of a coupled differential equation

Question

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:

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

2. 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?

Answer 1

Regarding your second question, consider the flow on the quotient space $\mathbb{T}^n = (\mathbb{R}/(2 \pi \mathbb{Z}))^n$.

When $n = 2$, in the unperturbed case, the behavior depends on whether $b_1/b_2$, where $\mathbf{b} = (b_1, b_2)$, is rational or not. In the former case, the torus $\mathbb{T}^2$ is foliated by periodic orbits, in the latter it is a minimal set.

M. Peixoto in the 1960s initiated the investigation of the structural stability of flows. He gave a characterization of structurally stable systems on two-dimensional manifolds. Unfortunately, according to Peixoto's theorem, your unperturbed systems are not structurally stable.

But this means only that arbitrarily close there is some perturbed system whose phase portrait is different from the unperturbed one (a priori, perturbing by adding the terms as in your question need not change the picture). However, at least in the case of rational $b_1/b_2$ the phase portrait (periodic orbits alone) seems extremely precarious, and it is highly improbable (to me) that it would remain the same under perturbation. Perhaps for irrational $b_1/b_2$ there is some spark of hope?