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A method is L-stable if it is A-stable and () as , where is the stability function of the method (the stability function of a Runge–Kutta method is a rational function and thus the limit as + is the same as the limit as ).
The definition for discrete-time systems is almost identical to that for continuous-time systems. The definition below provides this, using an alternate language commonly used in more mathematical texts. Let (X, d) be a metric space and f : X → X a continuous function. A point x in X is said to be Lyapunov stable, if,
The stability function of implicit Runge–Kutta methods is often analyzed using order stars. The order star for a method with stability function is defined to be the set {| | | > | |}. A method is A-stable if and only if its stability function has no poles in the left-hand plane and its order star contains no purely imaginary numbers.
A Lyapunov function for an autonomous dynamical system {: ˙ = ()with an equilibrium point at = is a scalar function: that is continuous, has continuous first derivatives, is strictly positive for , and for which the time derivative ˙ = is non positive (these conditions are required on some region containing the origin).
Numerical stability, a property of numerical algorithms which describes how errors in the input data propagate through the algorithm; Stability radius, a property of continuous polynomial functions; Stable theory, concerned with the notion of stability in model theory; Stability, a property of points in geometric invariant theory
For = and =, the distribution is a Landau distribution (L) which has a specific usage in physics under this name. For α = 3 / 2 {\displaystyle \alpha =3/2} and β = 0 {\displaystyle \beta =0} the distribution reduces to a Holtsmark distribution with scale parameter c and shift parameter μ .
If ˙ is negative definite, then the global asymptotic stability of the origin is a consequence of Lyapunov's second theorem. The invariance principle gives a criterion for asymptotic stability in the case when V ˙ ( x ) {\displaystyle {\dot {V}}(\mathbf {x} )} is only negative semidefinite.
In mathematics, in the theory of differential equations and dynamical systems, a particular stationary or quasistationary solution to a nonlinear system is called linearly unstable if the linearization of the equation at this solution has the form / =, where r is the perturbation to the steady state, A is a linear operator whose spectrum contains eigenvalues with positive real part.