Stability theory

Part of mathematics that addresses the stability of solutions
the branch of model theory

In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial differential equation because small perturbations of initial data lead to small variations in temperature at a later time as a result of the maximum principle. In partial differential equations one may measure the distances between functions using Lp norms or the sup norm, while in differential geometry one may measure the distance between spaces using the Gromov–Hausdorff distance. In dynamical systems, an orbit is called Lyapunov stable if the forward orbit of any point is in a small enough neighborhood or it stays in a small (but perhaps, larger) neighborhood. Various criteria have been developed to prove stability or instability of an orbit. Under favorable circumstances, the question may be reduced to a well studied problem involving eigenvalues of matrices. A more general method involves Lyapunov functions. In practice, any one of a number of different stability criteria are applied.

The paradigmatic case is the stability of the origin under the linear autonomous differential equation where and is a 2 by 2 matrix. We would sometimes perform change of basis by for some invertible matrix , which gives We say is " in the new basis". Since and , we can classify the stability of origin using and , while freely using change of basis.

When , we have , so the flow is area preserving. In this case, the type of flow is classified by
If , then it is a rotation ("neutral stability") around the origin. If , then it is a shearing flow. If , then the origin is a saddle point.

The simplest kind of an orbit is a fixed point, or an equilibrium. If a mechanical system is in a stable equilibrium state then a small push will result in a localized motion, for example, small oscillations as in the case of a pendulum. In a system with damping, a stable equilibrium state is moreover asymptotically stable. On the other hand, for an unstable equilibrium, such as a ball resting on a top of a hill, certain small pushes will result in a motion with a large amplitude that may or may not converge to the original state. There are useful tests of stability for the case of a linear system. Stability of a nonlinear system can often be inferred from the stability of its linearization.

A general way to establish Lyapunov stability or asymptotic stability of a dynamical system is by means of Lyapunov functions.