How Do We Define Stability in a Dynamically Growing System?

[nacadaryo]

Suppose we have a dynamical system x_{t+1} = Ax_{t} where A is matrix, x is vector. We suppose that $x$ always grow as time goes on.

If we treat equilibrium as the whole time evolution(path) of x given x_0 = a and no disturbance to the value of x - that is $x$ follows from the initial condition, how would we be able to define stability of the system? What would be the equation?

 

[pasmith]

Suppose we have a dynamical system x_{t+1} = Ax_{t} where A is matrix, x is vector. We suppose that $x$ always grow as time goes on.

I think you mean that \|x_t\| grows.

If we treat equilibrium as the whole time evolution(path) of x given x_0 = a and no disturbance to the value of x - that is $x$ follows from the initial condition, how would we be able to define stability of the system? What would be the equation?

What do you mean by stability? Do you mean that a small change in the initial condition tends to 0 as t \to \infty? The formal expression of that is that the solution starting at x_0 is stable if and only if there exists some \epsilon > 0 such that for any solution y_t starting at y_0, if \|y_0 - x_0\| < \epsilon then \|y_t - x_t\| \to 0 as t \to \infty.

If so your system is unstable, since for \|x_t\| to grow there must exist an eigenvalue \lambda of A such that |\lambda| > 1. This eigenvalue has a corresponding eigenvector v, and we can assume that \|v\| = 1. Let \epsilon > 0, and y_0 = x_0 + \frac12\epsilon v. Then \|y_0 - x_0\| = \frac12 \epsilon < \epsilon, and
<br /> \|y_t - x_t\| = \left\|\frac12 \epsilon A^t v \right\| = \frac12 \epsilon |\lambda^t|<br />
which tends to infinity as t \to \infty for any strictly positive \epsilon.

Alternatively you could define stability to mean that the solution starting at x_0 is stable if and only if, for all finite t and all \epsilon &gt; 0, there exists \delta &gt; 0 such that for any solution y_t starting at y_0, if \|x_0 - y_0\| &lt; \delta then \|x_t - y_t\| &lt; \epsilon. That's equivalent to requiring that x_t - y_t is a continuous function of x_0 - y_0, which in this case it is.