[2.11] What are Lyapunov exponents? (Thanks to Ronnie Mainieri & Fred Klingener for contributing to this answer) The hardest thing to get right about Lyapunov exponents is the spelling of L yapunov, which you will variously find as Liapunov, Lyapunof and even Liapun off. Of course Lyapunov is really spelled in the Cyrillic alphabet: (Lambda) (backwards r)(pi)(Y)(H)(0)(B). Now that there is an ANSI standard of transli teration for Cyrillic, we expect all references to converge on the version L yapunov. Lyapunov was born in Russia in 6 June 1857. He was greatly influenced by Che byshev and was a student with Markov. He was also a passionate man: Lyapunov shot himself the day his wife died. He died 3 Nov. 1918, three days later. According to the request on a note he left, Lyapunov was buried with his wif e. [biographical data from a biography by A. T. Grigorian]. Lyapunov left us with more than just a simple note. He left a collection of papers on the equilibrium shape of rotating liquids, on probability, and on the stability of low-dimensional dynamical systems. It was from his disserta tion that the notion of Lyapunov exponent emerged. Lyapunov was interested i n showing how to discover if a solution to a dynamical system is stable or n ot for all times. The usual method of studying stability, i.e. linear stabil ity, was not good enough, because if you waited long enough the small errors due to linearization would pile up and make the approximation invalid. Lyap unov developed concepts (now called Lyapunov Stability) to overcome these di fficulties. Lyapunov exponents measure the rate at which nearby orbits converge or diver ge. There are as many Lyapunov exponents as there are dimensions in the stat e space of the system, but the largest is usually the most important. Roughl y speaking the (maximal) Lyapunov exponent is the time constant, lambda, in the expression for the distance between two nearby orbits, exp(lambda * t). If lambda is negative, then the orbits converge in time, and the dynamical s ystem is insensitive to initial conditions. However, if lambda is positive, then the distance between nearby orbits grows exponentially in time, and the system exhibits sensitive dependence on initial conditions. There are basically two ways to compute Lyapunov exponents. In one way one c hooses two nearby points, evolves them in time, measuring the growth rate of the distance between them. This is useful when one has a time series, but h as the disadvantage that the growth rate is really not a local effect as the points separate. A better way is to measure the growth rate of tangent vect ors to a given orbit. More precisely, consider a map f in an m dimensional phase space, and its de rivative matrix Df(x). Let v be a tangent vector at the point x. Then we def ine a function 1 n L(x,v) = lim --- ln |( Df (x)v )| n -> oo n Now the Multiplicative Ergodic Theorem of Oseledec states that this limit ex ists for almost all points x and all tangent vectors v. There are at most m distinct values of L as we let v range over the tangent space. These are the Lyapunov exponents at x. For more information on computing the exponents see Wolf, A., J. B. Swift, et al. (1985). "Determining Lyapunov Exponents from a Time Series." Physica D 16: 285-317. Eckmann, J.-P., S. O. Kamphorst, et al. (1986). "Liapunov exponents from tim e series." Phys. Rev. A 34: 4971-4979. -- 在細雨的午後 書頁裡悉哩哩地傳來 " 週期3 = ? " 然而我知道 當我正在日耳曼深處的黑森林 繼續發掘海森堡未曾做過的夢時 康德的諾言早已遠離......... 遠來的傳教士靜靜地看著山澗不斷反覆疊代自己的 過去 現在 和 未來 於是僅以 一顆量子渾沌 一本符號動力學 祝那發生在週一下午的新生 --

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