The term stability has been used in the literature for a large variety of different concepts. The basic, qualitative idea is that a model that produces a solution (output) for given data (input) should possess the property that if the input is perturbed by a small amount then the output should also be perturbed by only a small amount. But the precise application of this idea to initial value ordinary differential equations (ODEs), to boundary value ODEs, and to numerical methods has given rise to a multitude of definitions. The reader should therefore be careful, when speaking of stability, to distinguish between stability of problems and of numerical methods, and between stability of initial value problems (IVPs) and boundary value problems (BVPs).
In this chapter we briefly discuss the stability of IVPs. No numerical solutions or methods are discussed yet; that will start only in the next chapter. Matrix eigenvalues play a central role here, so we also include a quick review below.
Consider at first the simple scalar ODE, often referred to later as the test equation
where ? is a constant. We allow ? to be complex, because it represents an eigenvalue of a system's matrix. The solution for t ? 0 is
If y( t) and ?( t) are two solutions of the test equation, then their difference for any t depends on their difference at the initial time:
Review: Matrix eigenvalues.
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