The rate of separation of infinitesimally close trajectories
Explanations of the Lyapunov exponent
In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectories. Quantitatively, two trajectories in phase space with initial separation vector diverge (provided that the divergence can be treated within the linearized approximation) at a rate given by
where is the Lyapunov exponent.
The rate of separation can be different for different orientations of initial separation vector. Thus, there is a spectrum of Lyapunov exponents—equal in number to the dimensionality of the phase space. It is common to refer to the largest one as the maximal Lyapunov exponent (MLE), because it determines a notion of predictability for a dynamical system. A positive MLE is usually taken as an indication that the system is chaotic (provided some other conditions are met, e.g., phase space compactness). Note that an arbitrary initial separation vector will typically contain some component in the direction associated with the MLE, and because of the exponential growth rate, the effect of the other exponents will be obliterated over time.
In mathematics, the Lyapunovexponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation...
_{t_{0}}^{t_{1}}})/(t_{1}-t_{0})} , which is then precisely the finite-time Lyapunovexponent (FTLE) F T L E t 0 t 1 ( x 0 ) = 1 2 ( t 1 − t 0 ) log λ n ( x 0...
Lyapunov. It is defined as the inverse of a system's largest Lyapunovexponent. The Lyapunov time mirrors the limits of the predictability of the system...
following are named: Lyapunov dimension Lyapunov equation LyapunovexponentLyapunov function Lyapunov fractal Lyapunov stability Lyapunov's central limit theorem...
the Lyapunovexponent. The rate of separation depends on the orientation of the initial separation vector, so a whole spectrum of Lyapunovexponents can...
Lyapunovexponent (related to Lyapunov's First Method of discussing stability) has received wide interest in connection with chaos theory. Lyapunov stability...
A and B. A Lyapunov fractal is constructed by mapping the regions of stability and chaotic behaviour (measured using the Lyapunovexponent λ {\displaystyle...
{x}}(t)\approx {\boldsymbol {H}}\cdot \delta {\vec {x}}_{0}} The finite-time Lyapunovexponents are defined as the time average of the logarithms of the lengths of...
fractal detail is revealed by plotting the Lyapunovexponent, as shown by the example below. The Lyapunovexponent is the error growth-rate of a given sequence...
Consequently, the equilibrium point is called "superstable". Its Lyapunovexponent is − ∞ {\displaystyle -\infty } . A similar argument shows that there...
mathematician Aleksandr Lyapunov because of the close connection with the Lyapunovexponents. Consider a dynamical system ( { φ t } t ≥ 0 , ( U ⊆ R n , ‖ ⋅ ‖ )...
Floquet exponents are called Lyapunovexponents. The zero solution is asymptotically stable if all Lyapunovexponents are negative, Lyapunov stable if...
with the positive maximal Lyapunovexponent is not as easy as stated, but even more complex (to calculate the Lyapunovexponent from an RP, the whole frequency...
with the Lyapunov vector corresponding to the largest Lyapunovexponent in the system. In some cases Lyapunov vectors may not exist. Lyapunov vectors are...
base point x, but it requires one positive Lyapunovexponent. In addition to a positive Lyapunovexponent, boundedness is another major feature within...
predictably lead to largely different final states. At least one Lyapunovexponent of a deterministically chaotic system is positive. Logarithms occur...
positive Lyapunovexponents. Since on an attractor, the sum of Lyapunovexponents is non-positive, there must be at least one negative Lyapunovexponent. If...
also associated with the presence of chaotic invariants such as the Lyapunovexponent and Kolmogorov-Sinai entropy, which quantify the rate at which nearby...
predictability Dynamical similarity index State space dissimilarity measures Lyapunovexponent Permutation methods Local flow Other univariate measures Algorithmic...
decay rate, the fractal dimension and the Lyapunovexponents are all related. The large Lyapunovexponent, for instance, tells us how fast a trajectory...
characteristic p > 0 Lyapunov characteristic exponent, a quantity that characterizes the rate of separation Characteristic exponent of Stable distribution...
conditions. This sensitivity to initial conditions can be measured with Lyapunovexponents. Markov chains and other random walks are not deterministic systems...
of exponential functions List of integrals of hyperbolic functions Lyapunovexponent Malthusian catastrophe Malthusian growth model Marshall–Olkin exponential...
theorem Lyapunovexponent Shub, Michael (1987). Global Stability of Dynamical Systems. Springer. pp. 65–66. Pesin, Ya B (1977). "Characteristic Lyapunov Exponents...
hypothesis, he proposed a Market Instability Indicator based on the first Lyapunovexponent of flows of funds evolution. In collaboration with Nassim Nicholas...
Core Attractor Bifurcation Fractal Limit set Lyapunovexponent Orbit Periodic point Phase space Anosov diffeomorphism Arnold tongue axiom A dynamical...
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