Stability for a infinite dimensional dynamical system Suppose I have a infinite dimensional dynamical system as
$\dot{x_n}=Ax$
where $A$ is an infinite-dimensional matrix and $\{x_n\}$ is a sequence of states. I was wondering if you can help me find about the stability of this system based on the properties of matrix $A$ and sequence $\{x_n\}$. 
More specifically, I was searching for stability properties of dynamical system that is arranged as:
$\dot{x_n}=a_n x_{n+1}$
where $a_n$ is a constant that means each state integrates the next step, but the process does not stop any where. 
Thank You!
 A: To address your more specific question, we consider the parameterized finite dimensional case with $n+1$ elements in the sequence and then take the limit as $n \rightarrow \infty$. Note that without any boundary condition specified for $x_n$, the problem is ill posed. In the following, we assume a periodic boundary where
$$ \dot{x}_n = a_n x_0 $$
Solving for $x_k$;
$$ \frac{d^{n + 1}}{dt^{n + 1}} x_k  = \left(\prod_{i = 0}^{n} a_i \right) x_k,$$
which holds $\forall k$ due to the symmetry of the ring and the commutativity of multiplication.
The eigenvalues of your system are the solutions to
$$ \lambda_i = \sqrt[n + 1]{- \prod_{i = 0}^{n} a_i} = \sqrt[n + 1] {R e^{\imath\Theta}}, $$ 
which has $n + 1$ different roots for the finite dimensional case. These roots are of the form $r e^{\imath\theta_k}$; where
$$\theta_k = \frac{\Theta + 2\pi k}{n + 1},$$ $$ r = \sqrt[n + 1]{R}, k \in \{0 \dots n\}$$
 For $n > 1$; i.e., for dimensions $3$ and above, we end up with at least one root with positive real part. Thus, proving that the system is generally unstable. 
