Search results
Results from the WOW.Com Content Network
Other names for linear stability include exponential stability or stability in terms of first approximation. [ 1 ] [ 2 ] If there exists an eigenvalue with zero real part then the question about stability cannot be solved on the basis of the first approximation and we approach the so-called "centre and focus problem".
Stability diagram classifying Poincaré maps of linear autonomous system ′ =, as stable or unstable according to their features. Stability generally increases to the left of the diagram. [ 1 ] Some sink, source or node are equilibrium points .
In certain cases, von Neumann stability is necessary and sufficient for stability in the sense of Lax–Richtmyer (as used in the Lax equivalence theorem): The PDE and the finite difference scheme models are linear; the PDE is constant-coefficient with periodic boundary conditions and has only two independent variables; and the scheme uses no ...
Von Neumann stability analysis is a commonly used procedure for the stability analysis of finite difference schemes as applied to linear partial differential equations. These results do not hold for nonlinear PDEs, where a general, consistent definition of stability is complicated by many properties absent in linear equations.
The Lyapunov equation, named after the Russian mathematician Aleksandr Lyapunov, is a matrix equation used in the stability analysis of linear dynamical systems. [1] [2]In particular, the discrete-time Lyapunov equation (also known as Stein equation) for is
To determine whether the flow is stable or unstable, one often employs the method of linear stability analysis. In this type of analysis, the governing equations and boundary conditions are linearized. This is based on the fact that the concept of 'stable' or 'unstable' is based on an infinitely small disturbance.
In the control system theory, the Routh–Hurwitz stability criterion is a mathematical test that is a necessary and sufficient condition for the stability of a linear time-invariant (LTI) dynamical system or control system. A stable system is one whose output signal is bounded; the position, velocity or energy do not increase to infinity as ...
For c = 0, there is a simple proof for this statement: [8] if u 0 (x) is a stationary solution and u = u 0 (x) + ũ(x, t) is an infinitesimally perturbed solution, linear stability analysis yields the equation