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The Lyapunov exponents of bounded trajectory and the Lyapunov dimension of attractor are invariant under diffeomorphism of the phase space. [9] The multiplicative inverse of the largest Lyapunov exponent is sometimes referred in literature as Lyapunov time, and defines the characteristic e-folding time. For chaotic orbits, the Lyapunov time ...
The exact limit values of finite-time Lyapunov exponents, if they exist and are the same for all , are called the absolute ones [3] {+ (,)} = {()} {} and used in the Kaplan–Yorke formula. Examples of the rigorous use of the ergodic theory for the computation of the Lyapunov exponents and dimension can be found in. [ 11 ] [ 12 ] [ 13 ]
In mathematics, the Lyapunov time is the characteristic timescale on which a dynamical system is chaotic. It is named after the Russian mathematician Aleksandr Lyapunov . It is defined as the inverse of a system's largest Lyapunov exponent .
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 chaotic competitive Lotka–Volterra systems. The Kaplan–Yorke dimension, a measure of the dimensionality of the attractor, is 2.074.
In applied mathematics, the Kaplan–Yorke conjecture concerns the dimension of an attractor, using Lyapunov exponents. [ 1 ] [ 2 ] By arranging the Lyapunov exponents in order from largest to smallest λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \dots \geq \lambda _{n}} , let j be the largest index for which
Choosing the first entries of + randomly and the other entries zero, and iterating this vector back in time, the vector aligns almost surely with the Lyapunov vector () corresponding to the th largest Lyapunov exponent if and are sufficiently large. Since the iterations will exponentially blow up or shrink a vector it can be re-normalized at ...
For = and = it can be shown that almost all initial conditions inside the unit sphere generate chaotic signals with largest Lyapunov exponent. [8] Many other generalizations have been proposed in the literature.
These are the Lyapunov exponents. Further, if λ 1 > ... > λ m are the different limits then there are subspaces R n = R 1 ⊃ ... ⊃ R m ⊃ R m+1 = {0}, depending on x, such that the limit is λ i for u ∈ R i \ R i+1 and i = 1, ..., m. The values of the Lyapunov exponents are invariant with respect to a wide range of coordinate ...