Applications in Stability Theory and Numerical Analysis
The logarithmic norm plays an important role in the stability analysis of a continuous dynamical system . Its role is analogous to that of the matrix norm for a discrete dynamical system .
In the simplest case, when is a scalar complex constant, the discrete dynamical system has stable solutions when, while the differential equation has stable solutions when . When is a matrix, the discrete system has stable solutions if . In the continuous system, the solutions are of the form . They are stable if for all, which follows from property 7 above, if . In the latter case, is a Lyapunov function for the system.
Runge-Kutta methods for the numerical solution of replace the differential equation by a discrete equation, where the rational function is characteristic of the method, and is the time step size. If whenever, then a stable differential equation, having, will always result in a stable (contractive) numerical method, as . Runge-Kutta methods having this property are called A-stable.
Retaining the same form, the results can, under additional assumptions, be extended to nonlinear systems as well as to semigroup theory, where the crucial advantage of the logarithmic norm is that it discriminates between forward and reverse time evolution and can establish whether the problem is well posed. Similar results also apply in the stability analysis in control theory, where there is a need to discriminate between positive and negative feedback.
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