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Archived 2022-06-28 at the Wayback Machine Dr. Ronald Joe Record's mathematical recreations software laboratory includes an X11 graphical client, lyap, for graphically exploring the Lyapunov exponents of a forced logistic map and other maps of the unit interval.
The logistic map is a discrete dynamical system defined by the quadratic difference equation: ... For a one-dimensional map, the Lyapunov exponent λ can be ...
A Lyapunov fractal is constructed by mapping the regions of stability and chaotic behaviour (measured using the Lyapunov exponent) in the a−b plane for given periodic sequences of a and b. In the images, yellow corresponds to λ < 0 {\displaystyle \lambda <0} (stability), and blue corresponds to λ > 0 {\displaystyle \lambda >0} (chaos).
Bifurcation diagram for the Logistic map. RQA measures of the logistic map for various setting of the control parameter a. The measures RR and DET exhibit maxima at chaos-order/ order-chaos transitions. The measure DIV has a similar trend as the maximal Lyapunov exponent (but it is not the same!).
In mathematics, a chaotic map is a map (an evolution function) that exhibits some sort of chaotic behavior. Maps may be parameterized by a discrete-time or a continuous-time parameter. Discrete maps usually take the form of iterated functions. Chaotic maps often occur in the study of dynamical systems.
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
The tent map with parameter μ = 2 and the logistic map with parameter r = 4 are topologically conjugate, [1] and thus the behaviours of the two maps are in this sense identical under iteration. Depending on the value of μ, the tent map demonstrates a range of dynamical behaviour ranging from predictable to chaotic.
Feigenbaum originally related the first constant to the period-doubling bifurcations in the logistic map, but also showed it to hold for all one-dimensional maps with a single quadratic maximum. As a consequence of this generality, every chaotic system that corresponds to this description will bifurcate at the same rate.