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| [[Image:Tent map.png|thumb|right|Graph of tent map function]]
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| In [[mathematics]], the '''tent map''' with parameter μ is the real-valued function f<sub>μ</sub> defined by
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| :<math>f_\mu:=\mu\min\{x,\,1-x\},</math>
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| the name being due to the [[tent]]-like shape of the [[graph (mathematics)|graph]] of f<sub>μ</sub>. For the values of the parameter μ within 0 and 2, f<sub>μ</sub> [[image (mathematics)|maps]] the [[unit interval]] [0, 1] into itself, thus
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| defining a [[discrete-time]] [[dynamical system]] on it (equivalently, a [[recurrence relation]]). In particular, [[iterated function|iterating]] a point x<sub>0</sub> in [0, 1] gives rise to a sequence <math>x_n</math> :
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| :<math>
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| x_{n+1}=f_\mu(x_n)=\begin{cases}
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| \mu x_n & \mathrm{for}~~ x_n < \frac{1}{2} \\ \\
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| \mu (1-x_n) & \mathrm{for}~~ \frac{1}{2} \le x_n
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| \end{cases}
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| </math>
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| where μ is a positive real constant. Choosing for instance the parameter μ=2, the effect of the function f<sub>μ</sub> may be viewed as the result of the operation of folding the unit interval in two, then stretching the resulting interval [0,1/2] to get again the interval [0,1]. Iterating the procedure, any point x<sub>0</sub> of the interval assumes new subsequent positions as described above, generating a sequence x<sub>n</sub> in [0,1].
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| The <math>\mu=2</math> case of the tent map is a non-linear transformation of both the [[bit shift map]] and the ''r''=4 case of the [[logistic map]].
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| ==Behaviour==
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| [[Image:Tent-map.png|thumb|right|Orbits of unit-height tent map]]
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| [[Image:TentMap BifurcationDiagram.png|thumb|right|Bifurcation diagram for the tent map. Higher density indicates increased probability of the x variable acquiring that value for the given value of the μ parameter.]]
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| The tent map and the [[logistic map]] are [[topologically conjugate]],<ref>[http://www.math.lsa.umich.edu/~rauch/558/logisticconjugation.pdf Conjugating the Tent and Logistic Maps], Jeffrey Rauch, University of Michigan</ref> and thus the behaviours of the two maps are in this sense identical under iteration.
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| Depending on the value of μ, the tent map demonstrates a range of dynamical behaviour ranging from predictable to chaotic.
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| * If μ is less than 1 the point ''x'' = 0 is an [[attractor|attractive]] [[fixed point (mathematics)|fixed point]] of the system for all initial values of ''x'' i.e. the system will converge towards ''x'' = 0 from any initial value of ''x''.
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| * If μ is 1 all values of ''x'' less than or equal to 1/2 are fixed points of the system.
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| * If μ is greater than 1 the system has two fixed points, one at 0, and the other at μ/(μ + 1). Both fixed points are unstable i.e. a value of ''x'' close to either fixed point will move away from it, rather than towards it. For example, when μ is 1.5 there is a fixed point at ''x'' = 0.6 (because 1.5(1 − 0.6) = 0.6) but starting at ''x'' = 0.61 we get
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| ::<math>0.61 \to 0.585 \to 0.6225 \to 0.56625 \to 0.650625 \ldots</math>
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| * If μ is between 1 and the square root of 2 the system maps a set of intervals between μ − μ<sup>2</sup>/2 and μ/2 to themselves. This set of intervals is the [[Julia set]] of the map i.e. it is the smallest invariant sub-set of the real line under this map. If μ is greater than the square root of 2, these intervals merge, and the Julia set is the whole interval from μ − μ<sup>2</sup>/2 to μ/2 (see bifurcation diagram).
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| * If μ is between 1 and 2 the interval [μ − μ<sup>2</sup>/2, μ/2]contains both periodic and non-periodic points, although all of the [[orbit (dynamics)|orbit]]s are unstable (i.e. nearby points move away from the orbits rather than towards them). Orbits with longer lengths appear as μ increases. For example:
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| ::<math>\frac{\mu}{\mu^2+1} \to \frac{\mu^2}{\mu^2+1} \to \frac{\mu}{\mu^2+1} \mbox{ appears at } \mu=1</math>
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| ::<math>\frac{\mu}{\mu^3+1} \to \frac{\mu^2}{\mu^3+1} \to \frac{\mu^3}{\mu^3+1} \to \frac{\mu}{\mu^3+1} \mbox{ appears at } \mu=\frac{1+\sqrt{5}}{2}</math>
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| ::<math>\frac{\mu}{\mu^4+1} \to \frac{\mu^2}{\mu^4+1} \to \frac{\mu^3}{\mu^4+1} \to \frac{\mu^4}{\mu^4+1} \to \frac{\mu}{\mu^4+1} \mbox{ appears at } \mu \approx 1.8393</math>
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| * If μ equals 2 the system maps the interval [0,1] onto itself. There are now periodic points with every orbit length within this interval, as well as non-periodic points. The periodic points are [[dense set|dense]] in [0,1], so the map has become [[chaos theory|chaotic]]. In fact, the dynamics will be non-periodic if and only if <math>x_0</math> is irrational. This can be seen by noting what the map does when <math>x_n</math> is expressed in binary notation: It shifts the binary point one place to the right; then, if what appears to the left of the binary point is a "one" it changes all ones to zeroes and vice versa (with the exception of the final bit "one" in the case of a finite binary expansion); starting from an irrational number, this process goes on forever without repeating itself. The invariant measure for ''x'' is the uniform density over the unit interval.<ref>Collett, Pierre, and [[Jean-Pierre Eckmann|Eckmann, Jean-Pierre]], ''Iterated Maps on the Interval as Dynamical Systems'', Boston: Birkhauser, 1980.</ref> The [[autocorrelation function]] for a sufficiently long sequence {<math>x_n</math>} will show zero autocorrelation at all non-zero lags.<ref name="Brock">Brock, W. A., "Distinguishing random and deterministic systems: Abridged version," ''Journal of Economic Theory'' 40, October 1986, 168-195.</ref> Thus <math>{x_n}</math> cannot be distinguished from [[white noise]] using the autocorrelation function. Note that the r=4 case of the [[logistic map]] and the <math>\mu =2</math> case of the tent map are transformations of each other: Denoting the logistically evolving variable as <math>y_n</math>, we have <math>x_n = \tfrac{2}{\pi}sin^{-1}(y_{n}^{1/2})</math>.
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| * If μ is greater than 2 the map's Julia set becomes disconnected, and breaks up into a [[Cantor set]] within the interval [0,1]. The Julia set still contains an infinite number of both non-periodic and periodic points (including orbits for any orbit length) but [[almost everywhere|almost every]] point within [0,1] will now eventually diverge towards infinity. The canonical [[Cantor set]] (obtained by successively deleting middle thirds from subsets of the unit line) is the Julia set of the tent map for μ = 3.
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| ==Magnifying the orbit diagram==
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| [[Image:TentMagnification.JPG|thumb|right|Magnification near the tip shows more details.]]
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| * A closer look at the orbit diagram shows that there are 4 separated regions at μ ≈ 1. For further magnification, 2 reference lines (red) are drawn from the tip to suitable x at certain μ (e.g., 1.10) as shown.
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| [[Image:TentTipDetail.JPG|thumb|right|Further magnification shows 8 separated regions.]]
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| * With distance measured from the corresponding reference lines, further detail appears in the upper and lower part of the map. (total 8 separated regions at some μ)
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| ==Asymmetric tent map==
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| The asymmetric tent map is essentially a distorted, but still piecewise linear, version of the <math>\mu=2</math> case of the tent map. It is defined by
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| <math>
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| v_{n+1}=\begin{cases}
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| v_n/a &\mathrm{for}~~ v_n \in [0,a) \\ \\
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| (1-v_n)/(1-a) &\mathrm{for}~~ v_n \in [a,1]
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| \end{cases}</math>
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| for parameter <math>a \in [0,1]</math>. The <math>\mu=2</math> case of the tent map is the present case of <math>a= \tfrac{1}{2}</math>. A sequence {<math>v_n</math>} will have the same autocorrelation function <ref name="Brock" /> as will data from the first-order [[autoregressive process]] <math>w_{n+1} = (2a-1)w_n + u_{n+1}</math> with {<math>u_n</math>} [[independent and identically-distributed random variables|independently and identically distributed]]. Thus data from an asymmetric tent map cannot be distinguished, using the autocorrelation function, from data generated by a first-order autoregressive process.
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| ==References==
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| {{reflist}}
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| ==External links==
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| * [http://chaosbook.org/ ChaosBook.org]
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| {{Chaos theory}}
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| {{DEFAULTSORT:Tent Map}}
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| [[Category:Chaotic maps]]
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I'm Chi and was born on 27 October 1981. My hobbies are Radio-Controlled Car Racing and Association football.
Look into my web blog Fifa coin Generator