2.9 LINEARIZATION

In the nonlinear problem (2.8.8) considered in the previous section, even without solving the problem, we were able to conclude that all ICs u ₀ > 0 led to the steady state u _s2 = 1, while u( t) tends to increase further if u ₀ < 0. The only IC that led to u _s1 = 0 was u ₀ = 0. We thus concluded that the steady state u _s2 = 1 was stable, while u _s1 = 0 was unstable (note that the terms "stable" and "unstable" are used loosely here, and are made more precise in the next section). The problem (2.8.8) was relatively simple, and was even amenable to an analytic solution. This is typically not the case, particularly when a system of ODEs is involved. The question that arises then is whether something can be said about the character of the steady state(s) in more complex cases.

If we are interested in the behavior of the system in the immediate vicinity of the steady state, then one possibility is to consider the equations linearized about the steady state in question. For example, let us consider the problem

The steady states, u _s, are obtained by setting f( u) = 0, thus they satisfy the nonlinear equation

Let us define the perturbation variable:

which is a measure of the deviation of the solution u