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Lyapunov time, characteristic timescale on which a dynamical system is chaotic Probability theory , the branch of mathematics concerned with probability Dirichlet problem , the problem of finding a function which solves a specified partial differential equation in the interior of a given region that takes prescribed values on the boundary of ...
In particular, the discrete-time Lyapunov equation (also known as Stein equation) for is A X A H − X + Q = 0 {\displaystyle AXA^{H}-X+Q=0} where Q {\displaystyle Q} is a Hermitian matrix and A H {\displaystyle A^{H}} is the conjugate transpose of A {\displaystyle A} , while the continuous-time Lyapunov equation is
Among formal techniques, Lyapunov theory is one of the most powerful and widely applied frameworks for characterizing and analyzing non-asymptotic convergence behavior. For iterative methods , one common practical approach is to discuss these rates in terms of the number of iterates or the computer time required to reach close neighborhoods of ...
The mathematical theory of stability of motion, founded by A. M. Lyapunov, considerably anticipated the time for its implementation in science and technology. Moreover Lyapunov did not himself make application in this field, his own interest being in the stability of rotating fluid masses with astronomical application.
In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory .
Conley's decomposition is characterized by a function known as complete Lyapunov function. Unlike traditional Lyapunov functions that are used to assert the stability of an equilibrium point (or a fixed point) and can be defined only on the basin of attraction of the corresponding attractor, complete Lyapunov functions must be defined on the whole phase-portrait.
The Lyapunov–Malkin theorem (named for Aleksandr Lyapunov and Ioel Malkin ) is a mathematical theorem detailing stability of nonlinear systems. [ 1 ] [ 2 ] Theorem
The ordinary Lyapunov function is used to test whether a dynamical system is (Lyapunov) stable or (more restrictively) asymptotically stable. Lyapunov stability means that if the system starts in a state x ≠ 0 {\displaystyle x\neq 0} in some domain D , then the state will remain in D for all time.