Tent Map
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In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the tent map with parameter μ is the real-valued
function Function or functionality may refer to: Computing * Function key, a type of key on computer keyboards * Function model, a structured representation of processes in a system * Function object or functor or functionoid, a concept of object-orie ...
''f''μ defined by :f_\mu(x) := \mu\min\, the name being due to the
tent A tent is a shelter consisting of sheets of fabric or other material draped over or attached to a frame of poles or a supporting rope. While smaller tents may be free-standing or attached to the ground, large tents are usually anchored using g ...
-like shape of the
graph Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discret ...
of ''f''μ. For the values of the parameter μ within 0 and 2, ''f''μ
maps A map is a symbolic depiction of interrelationships, commonly spatial, between things within a space. A map may be annotated with text and graphics. Like any graphic, a map may be fixed to paper or other durable media, or may be displayed on ...
the
unit interval In mathematics, the unit interval is the closed interval , that is, the set of all real numbers that are greater than or equal to 0 and less than or equal to 1. It is often denoted ' (capital letter ). In addition to its role in real analysi ...
, 1into itself, thus defining a
discrete-time In mathematical dynamics, discrete time and continuous time are two alternative frameworks within which variables that evolve over time are modeled. Discrete time Discrete time views values of variables as occurring at distinct, separate "poi ...
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
on it (equivalently, a
recurrence relation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...
). In particular, iterating a point ''x''0 in , 1gives rise to a sequence x_n: :x_ = f_\mu(x_n) = \begin \mu x_n & \mathrm~~ x_n < \frac \\ \mu (1-x_n) & \mathrm~~ \frac \le x_n \end where μ is a positive real constant. Choosing for instance the parameter μ = 2, the effect of the function ''f''μ may be viewed as the result of the operation of folding the unit interval in two, then stretching the resulting interval , 1/2to get again the interval , 1 Iterating the procedure, any point ''x''0 of the interval assumes new subsequent positions as described above, generating a sequence ''x''''n'' in , 1 The \mu=2 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 The logistic map is a discrete dynamical system defined by the quadratic difference equation: Equivalently it is a recurrence relation and a polynomial mapping of degree 2. It is often referred to as an archetypal example of how complex, ...
.


Behaviour

The tent map with parameter μ = 2 and the
logistic map The logistic map is a discrete dynamical system defined by the quadratic difference equation: Equivalently it is a recurrence relation and a polynomial mapping of degree 2. It is often referred to as an archetypal example of how complex, ...
with parameter ''r'' = 4 are topologically conjugate, 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. * If μ is less than 1 the point ''x'' = 0 is an attractive 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''. * If μ is 1 all values of ''x'' less than or equal to 1/2 are fixed points of the system. * 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 (since 1.5(1 − 0.6) = 0.6) but starting at ''x'' = 0.61 we get ::0.61 \to 0.585 \to 0.6225 \to 0.56625 \to 0.650625 \ldots * If μ is between 1 and the
square root of 2 The square root of 2 (approximately 1.4142) is the positive real number that, when multiplied by itself or squared, equals the number 2. It may be written as \sqrt or 2^. It is an algebraic number, and therefore not a transcendental number. Te ...
the system maps a set of intervals between μ − μ2/2 and μ/2 to themselves. This set of intervals is the
Julia set In complex dynamics, the Julia set and the Classification of Fatou components, Fatou set are two complement set, complementary sets (Julia "laces" and Fatou "dusts") defined from a function (mathematics), function. Informally, the Fatou set of ...
of the map – that is, it is the smallest invariant subset 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 − μ2/2 to μ/2 (see bifurcation diagram). * If μ is between 1 and 2 the interval  − μ2/2, μ/2contains both periodic and non-periodic points, although all of the
orbit In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an artificial satellite around an ...
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: ::\frac \to \frac \to \frac \mbox \mu=1 ::\frac \to \frac \to \frac \to \frac \mbox \mu=\frac ::\frac \to \frac \to \frac \to \frac \to \frac \mbox \mu \approx 1.8393 * If μ equals 2 the system maps the interval , 1onto itself. There are now periodic points with every orbit length within this interval, as well as non-periodic points. The periodic points are
dense Density (volumetric mass density or specific mass) is the ratio of a substance's mass to its volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' (or ''d'') can also be use ...
in , 1 so the map has become
chaotic Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program aired on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kids, Cartoon Netwo ...
. In fact, the dynamics will be non-periodic
if and only if In logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either bo ...
x_0 is
irrational Irrationality is cognition, thinking, talking, or acting without rationality. Irrationality often has a negative connotation, as thinking and actions that are less useful or more illogical than other more rational alternatives. The concept of ...
. This can be seen by noting what the map does when x_n is expressed in
binary Binary may refer to: Science and technology Mathematics * Binary number, a representation of numbers using only two values (0 and 1) for each digit * Binary function, a function that takes two arguments * Binary operation, a mathematical op ...
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. The
autocorrelation function Autocorrelation, sometimes known as serial correlation in the discrete time case, measures the correlation of a signal with a delayed copy of itself. Essentially, it quantifies the similarity between observations of a random variable at differe ...
for a sufficiently long sequence will show zero autocorrelation at all non-zero lags.Brock, W. A., "Distinguishing random and deterministic systems: Abridged version," ''Journal of Economic Theory'' 40, October 1986, 168-195. Thus x_n cannot be distinguished from
white noise In signal processing, white noise is a random signal having equal intensity at different frequencies, giving it a constant power spectral density. The term is used with this or similar meanings in many scientific and technical disciplines, i ...
using the autocorrelation function. Note that the ''r'' = 4 case of the
logistic map The logistic map is a discrete dynamical system defined by the quadratic difference equation: Equivalently it is a recurrence relation and a polynomial mapping of degree 2. It is often referred to as an archetypal example of how complex, ...
and the \mu = 2 case of the tent map are
homeomorphic In mathematics and more specifically in topology, a homeomorphism ( from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function betw ...
to each other: Denoting the logistically evolving variable as y_n, the homeomorphism is ::x_n = \tfrac\sin^(y_^). * If μ is greater than 2 the map's Julia set becomes disconnected, and breaks up into a
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883. Throu ...
within the interval , 1 The Julia set still contains an infinite number of both non-periodic and periodic points (including orbits for any orbit length) but
almost every In measure theory (a branch of mathematical analysis), a property holds almost everywhere if, in a technical sense, the set for which the property holds takes up nearly all possibilities. The notion of "almost everywhere" is a companion notion to ...
point within , 1will now eventually diverge towards infinity. The canonical
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and mentioned by German mathematician Georg Cantor in 1883. Throu ...
(obtained by successively deleting middle thirds from subsets of the unit line) is the Julia set of the tent map for μ = 3.


Numerical errors


Magnifying the orbit diagram

* 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. * 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)


Asymmetric tent map

The asymmetric tent map is essentially a distorted, but still piecewise linear, version of the \mu = 2 case of the tent map. It is defined by :v_=\begin v_n/a &\mathrm~~ v_n \in ,a\\ (1-v_n)/(1-a) &\mathrm~~ v_n \in ,1 \end for parameter a \in ,1/math>. The \mu = 2 case of the tent map is the present case of a= \tfrac. A sequence will have the same autocorrelation function as will data from the first-order
autoregressive process In statistics, econometrics, and signal processing, an autoregressive (AR) model is a representation of a type of random process; as such, it can be used to describe certain time-varying processes in nature, economics, behavior, etc. The autoregre ...
w_ = (2a-1)w_n + u_ with 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.


Applications

The tent map has found applications in social cognitive optimization, chaos in economics, image encryption, on risk and market sentiments for pricing, etc.


See also

*
Shift space Shift may refer to: Art, entertainment, and media Gaming * ''Shift'' (series), a 2008 online video game series by Armor Games * '' Need for Speed: Shift'', a 2009 racing video game ** '' Shift 2: Unleashed'', its 2011 sequel Literature * ''S ...
*
Gray code The reflected binary code (RBC), also known as reflected binary (RB) or Gray code after Frank Gray (researcher), Frank Gray, is an ordering of the binary numeral system such that two successive values differ in only one bit (binary digit). For ...


References


External links


ChaosBook.org
{{DEFAULTSORT:Tent Map Chaotic maps