Search results
Results from the WOW.Com Content Network
The tables of elements are sorted in order of decreasing number of nuclides associated with each element. (For a list sorted entirely in terms of half-lives of nuclides, with mixing of elements, see List of nuclides.) Stable and unstable (marked decays) nuclides are given, with symbols for unstable (radioactive) nuclides in italics. Note that ...
More strongly, if is Lyapunov stable and all solutions that start out near converge to , then is said to be asymptotically stable (see asymptotic analysis). The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge.
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.
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).
This solution is asymptotically stable as t → ∞ ("in the future") if and only if for all eigenvalues λ of A, Re(λ) < 0. Similarly, it is asymptotically stable as t → −∞ ("in the past") if and only if for all eigenvalues λ of A, Re(λ) > 0. If there exists an eigenvalue λ of A with Re(λ) > 0 then the solution is unstable for t → ...
If >, when ˙ hold only for in some neighborhood of the origin, and the set {˙ =}does not contain any trajectories of the system besides the trajectory () =,, then the local version of the invariance principle states that the origin is locally asymptotically stable.
The function f(n) is said to be "asymptotically equivalent to n 2, as n → ∞". This is often written symbolically as f (n) ~ n 2, which is read as "f(n) is asymptotic to n 2". An example of an important asymptotic result is the prime number theorem.
System is called globally asymptotically stable at zero (0-GAS) if the corresponding system with zero input ...