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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 ...
Squire's theorem allows many simplifications to be made in stability theory. If we want to decide whether a flow is unstable or not, it suffices to look at two-dimensional perturbations. These are governed by the Orr–Sommerfeld equation for viscous flow, and by Rayleigh's equation for inviscid flow.
In the mathematical field of model theory, a theory is called stable if it satisfies certain combinatorial restrictions on its complexity. Stable theories are rooted in the proof of Morley's categoricity theorem and were extensively studied as part of Saharon Shelah's classification theory, which showed a dichotomy that either the models of a theory admit a nice classification or the models ...
In the theory of dynamical systems, Peixoto's theorem, proved by Maurício Peixoto, states that among all smooth flows on surfaces, i.e. compact two-dimensional manifolds, structurally stable systems may be characterized by the following properties: The set of non-wandering points consists only of periodic orbits and fixed points.
The sub-class of Lur'e systems studied by Popov is described by: ˙ = + ˙ = = + = where x ∈ R n, ξ,u,y are scalars, and A,b,c and d have commensurate dimensions. The nonlinear element Φ: R → R is a time-invariant nonlinearity belonging to open sector (0, ∞), that is, Φ(0) = 0 and yΦ(y) > 0 for all y not equal to 0.
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For higher degree polynomials the extra computation involved in this mapping can be avoided by testing the Schur stability by the Schur-Cohn test, the Jury test or the Bistritz test. Necessary condition: a Hurwitz stable polynomial (with real coefficients ) has coefficients of the same sign (either all positive or all negative).
Ideas of structural stability were taken up by Stephen Smale and his school in the 1960s in the context of hyperbolic dynamics. Earlier, Marston Morse and Hassler Whitney initiated and René Thom developed a parallel theory of stability for differentiable maps, which forms a key part of singularity theory. Thom envisaged applications of this ...