Tent map

In mathematics, the tent map is an iterated function, in the shape of a tent, forming a discrete-time dynamical system. It takes a point $x\_n$ on the real line and maps it to another point:

:

where μ is a positive real constant.

Behaviour

The tent map and the logistic map are topologically conjugate, and thus the behaviour 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 (because 1.5(1 − 0.6) = 0.6) but starting at "x" = 0.61 we get

::$0.61\; o\; 0.585\; o\; 0.6225\; o\; 0.56625\; o\; 0.650625\; ldots$

* If μ is between 1 and the square root of 2 the system maps a set of intervals between μ − μ²/2 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 − μ²/2 to μ/2 (see bifurcation diagram).

* If μ is between 1 and 2 the interval [μ − μ²/2, μ/2] contains both periodic and non-periodic points, although all of the orbits 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:

::$frac\{mu\}\{mu^2+1\}\; o\; frac\{mu^2\}\{mu^2+1\}\; o\; frac\{mu\}\{mu^2+1\}\; mbox\{\; appears\; at\; \}\; mu=1$

::$frac\{mu\}\{mu^3+1\}\; o\; frac\{mu^2\}\{mu^3+1\}\; o\; frac\{mu^3\}\{mu^3+1\}\; o\; frac\{mu\}\{mu^3+1\}\; mbox\{\; appears\; at\; \}\; mu=frac\{1+sqrt\{5\{2\}$

::$frac\{mu\}\{mu^4+1\}\; o\; frac\{mu^2\}\{mu^4+1\}\; o\; frac\{mu^3\}\{mu^4+1\}\; o\; frac\{mu^4\}\{mu^4+1\}\; o\; frac\{mu\}\{mu^4+1\}\; mbox\{\; appears\; at\; \}\; mu\; approx\; 1.8393$

* If μ equals 2 the system maps the interval [0,1] to itself. There are now periodic points with every orbit length within this interval, as well as non-periodic points. The periodic points are dense in [0,1] , so the map has become chaotic.

* 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 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.

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)