In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation...

Lyapunov equation Lyapunov exponent Lyapunov fractal Lyapunov function Lyapunov stability Lyapunov time Lyapunov's central limit theorem Lyapunov's condition...

Lyapunov. It is defined as the inverse of a system's largest Lyapunov exponent. The Lyapunov time mirrors the limits of the predictability of the system...

A and B. A Lyapunov fractal is constructed by mapping the regions of stability and chaotic behaviour (measured using the Lyapunov exponent λ {\displaystyle...

following are named: Lyapunov dimension Lyapunov equation Lyapunov exponent Lyapunov function Lyapunov fractal Lyapunov stability Lyapunov's central limit theorem...

the Lyapunov exponent. The rate of separation depends on the orientation of the initial separation vector, so a whole spectrum of Lyapunov exponents can...

Lyapunov exponent (related to Lyapunov's First Method of discussing stability) has received wide interest in connection with chaos theory. Lyapunov stability...

_{t_{0}}^{t_{1}}})/(t_{1}-t_{0})} , which is then precisely the finite-time Lyapunov exponent (FTLE) F T L E t 0 t 1 ( x 0 ) = 1 2 ( t 1 − t 0 ) log ⁡ λ n ( x 0...

{x}}(t)\approx {\boldsymbol {H}}\cdot \delta {\vec {x}}_{0}} The finite-time Lyapunov exponents are defined as the time average of the logarithms of the lengths of...

with the Lyapunov vector corresponding to the largest Lyapunov exponent in the system. In some cases Lyapunov vectors may not exist. Lyapunov vectors are...

fractal detail is revealed by plotting the Lyapunov exponent, as shown by the example below. The Lyapunov exponent is the error growth-rate of a given sequence...

with the positive maximal Lyapunov exponent is not as easy as stated, but even more complex (to calculate the Lyapunov exponent from an RP, the whole frequency...

mathematician Aleksandr Lyapunov because of the close connection with the Lyapunov exponents. Consider a dynamical system ( { φ t } t ≥ 0 , ( U ⊆ R n , ‖ ⋅ ‖ )...

positive Lyapunov exponents. Since on an attractor, the sum of Lyapunov exponents is non-positive, there must be at least one negative Lyapunov exponent. If...

decay rate, the fractal dimension and the Lyapunov exponents are all related. The large Lyapunov exponent, for instance, tells us how fast a trajectory...

Consequently, the equilibrium point is called "superstable". Its Lyapunov exponent is − ∞ {\displaystyle -\infty } . A similar argument shows that there...

base point x, but it requires one positive Lyapunov exponent. In addition to a positive Lyapunov exponent, boundedness is another major feature within...

all i {\displaystyle i} . This system is chaotic and has a largest Lyapunov exponent of 0.0203. From the theorems by Hirsch, it is one of the lowest-dimensional...

Floquet exponents are called Lyapunov exponents. The zero solution is asymptotically stable if all Lyapunov exponents are negative, Lyapunov stable if...

possible for billiard tracks to exhibit chaotic behavior (positive Lyapunov exponent and exponential divergence of paths) even within a convex billiard...

also associated with the presence of chaotic invariants such as the Lyapunov exponent and Kolmogorov-Sinai entropy, which quantify the rate at which nearby...

predictably lead to largely different final states. At least one Lyapunov exponent of a deterministically chaotic system is positive. Logarithms occur...

predictability Dynamical similarity index State space dissimilarity measures Lyapunov exponent Permutation methods Local flow Other univariate measures Algorithmic...

concerns the dimension of an attractor, using Lyapunov exponents. By arranging the Lyapunov exponents in order from largest to smallest λ 1 ≥ λ 2 ≥ ⋯...

\|u(0)\|e^{\lambda t}} , where λ {\displaystyle \lambda } denotes the maximum Lyapunov exponent of the isolated system. Now using the ansatz v = u e − 2 α t {\displaystyle...

conditions. This sensitivity to initial conditions can be measured with Lyapunov exponents. Markov chains and other random walks are not deterministic systems...

of the out-of-time-order correlator coincides with the classical Lyapunov exponents in the chaotic regime and at unstable points of the regular regime...

Core Attractor Bifurcation Fractal Limit set Lyapunov exponent Orbit Periodic point Phase space Anosov diffeomorphism Arnold tongue axiom A dynamical...

of exponential functions List of integrals of hyperbolic functions Lyapunov exponent Malthusian catastrophe Malthusian growth model Marshall–Olkin exponential...

Oseledets theorem provides the theoretical background for computation of Lyapunov exponents of a nonlinear dynamical system. It was proved by Valery Oseledets...