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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.
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).
Tangent slopes of an asymptotically stable fixed point (left) and an unstable fixed point (right) and the state of the surrounding orbits We can see this by graphing the map: if the slope of the tangent to the curve at x f {\displaystyle x_{f}} is between −1 and 1, then x f {\displaystyle x_{f}} is stable and the orbit around it is attracted ...
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.
Then Aizerman's conjecture is that the system is stable in large (i.e. unique stationary point is global attractor) if all linear systems with f(e)=ke, k ∈(k1,k2) are asymptotically stable. There are counterexamples to Aizerman's conjecture such that nonlinearity belongs to the sector of linear stability and unique stable equilibrium coexists ...
A controversial American live-streamer is facing the prospect of prison in South Korea for his offensive antics, in a case that is shining a light on the rise of so-called “nuisance influencers ...
where : + is a Lebesgue measurable essentially bounded external input and is a Lipschitz continuous function w.r.t. the first argument uniformly w.r.t. the second one. This ensures that there exists a unique absolutely continuous solution of the system ().
One of Richardson’s more incredible plays of the game that showed what makes him different was an incomplete pass that landed out of bounds on a throwaway.