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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク logistic map ： ウィキペディア英語版 logistic map The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized in a seminal 1976 paper by the biologist Robert May,〔May, Robert M. 1976. "Simple mathematical models with very complicated dynamics." Nature 261(5560):459-467.〕 in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.〔"〕 Mathematically, the logistic map is written :$(1)\backslash qquad\; x\_\; =\; r\; x\_n\; (1-x\_n)$ where $x\_n$ is a number between zero and one that represents the ratio of existing population to the maximum possible population. The values of interest for the parameter ''r'' are those in the interval (0, 4]. This nonlinear difference equation is intended to capture two effects: * reproduction where the population will increase at a rate proportional to the current population when the population size is small. * starvation (density-dependent mortality) where the growth rate will decrease at a rate proportional to the value obtained by taking the theoretical "carrying capacity" of the environment less the current population. However, as a demographic model the logistic map has the pathological problem that some initial conditions and parameter values lead to negative population sizes. This problem does not appear in the older Ricker model, which also exhibits chaotic dynamics. The $r=4$ case of the logistic map is a nonlinear transformation of both the bit-shift map and the $\backslash mu\; =2$ case of the tent map. ==Behavior dependent on ''r''== The image below shows the amplitude and frequency content of some logistic map iterates for parameter values ranging from 2 to 4. By varying the parameter ''r'', the following behavior is observed: * With ''r'' between 0 and 1, the population will eventually die, independent of the initial population. * With ''r'' between 1 and 2, the population will quickly approach the value $\backslash frac$, independent of the initial population. * With ''r'' between 2 and 3, the population will also eventually approach the same value $\backslash frac$, but first will fluctuate around that value for some time. The rate of convergence is linear, except for ''r''=3, when it is dramatically slow, less than linear (see Bifurcation memory). * With ''r'' between 3 and $1+\backslash sqrt$ (approximately 3.44949), from almost all initial conditions the population will approach permanent oscillations between two values. These two values are dependent on ''r''. * With ''r'' between 3.44949 and 3.54409 (approximately), from almost all initial conditions the population will approach permanent oscillations among four values. The latter number is a root of a 12th degree polynomial . * With ''r'' increasing beyond 3.54409, from almost all initial conditions the population will approach oscillations among 8 values, then 16, 32, etc. The lengths of the parameter intervals that yield oscillations of a given length decrease rapidly; the ratio between the lengths of two successive such bifurcation intervals approaches the Feigenbaum constant δ = 4.66920$\backslash dots$. This behavior is an example of a period-doubling cascade. * At ''r'' approximately 3.56995 is the onset of chaos, at the end of the period-doubling cascade. From almost all initial conditions, we no longer see oscillations of finite period. Slight variations in the initial population yield dramatically different results over time, a prime characteristic of chaos. * Most values beyond 3.56995 exhibit chaotic behaviour, but there are still certain isolated ranges of ''r'' that show non-chaotic behavior; these are sometimes called ''islands of stability''. For instance, beginning at $1+\backslash sqrt$〔Zhang, Cheng, "Period three begins", ''Mathematics Magazine'' 83, October 2010, 295-297.〕 (approximately 3.82843) there is a range of parameters ''r'' that show oscillation among three values, and for slightly higher values of ''r'' oscillation among 6 values, then 12 etc. * The development of the chaotic behavior of the logistic sequence as the parameter ''r'' varies from approximately 3.56995 to approximately 3.82843 is sometimes called the Pomeau–Manneville scenario, characterized by a periodic (laminar) phase interrupted by bursts of aperiodic behavior. Such a scenario has an application in semiconductor devices.〔 〕 There are other ranges that yield oscillation among 5 values etc.; all oscillation periods occur for some values of ''r''. A period-doubling window with parameter ''c'' is a range of ''r''-values consisting of a succession of sub-ranges. The ''k''th sub-range contains the values of ''r'' for which there is a stable cycle (a cycle that attracts a set of initial points of unit measure) of period $c2^.$ This sequence of sub-ranges is called a cascade of harmonics. In a sub-range with a stable cycle of period $c2^$ for all * Beyond ''r'' = 4, almost all initial values eventually leave the interval () and diverge. For any value of ''r'' there is at most one stable cycle. If a stable cycle exists, it is globally stable, attracting almost all points.〔Collet, Pierre, and Jean-Pierre Eckmann, ''Iterated Maps on the Interval as Dynamical Systems'', Birkhauser, 1980.〕 Some values of ''r'' with a stable cycle of some period have infinitely many unstable cycles of various periods. The bifurcation diagram at right summarizes this. The horizontal axis shows the possible values of the parameter ''r'' while the vertical axis shows the set of values of ''x'' visited asymptotically from almost all initial conditions by the iterates of the logistic equation with that ''r'' value. The bifurcation diagram is a self-similar: if you zoom in on the above-mentioned value ''r'' = 3.82843 and focus on one arm of the three, the situation nearby looks like a shrunk and slightly distorted version of the whole diagram. The same is true for all other non-chaotic points. This is an example of the deep and ubiquitous connection between chaos and fractals. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「logistic map」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.