Tent map

Graph of tent map function

In mathematics, the tent map with parameter μ is the real-valued function fμ defined by

the name being due to the tent-like shape of the graph of fμ. For the values of the parameter μ within 0 and 2, fμ maps the unit interval [0, 1] into itself, thus defining a discrete-time dynamical system on it (equivalently, a recurrence relation). In particular, iterating a point x0 in [0, 1] gives rise to a sequence  :

where μ is a positive real constant. Choosing for instance the parameter μ=2, the effect of the function fμ 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 x0 of the interval assumes new subsequent positions as described above, generating a sequence xn in [0,1].

The 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.


Orbits of unit-height tent map
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.

The tent map with parameter μ=2 and the logistic map with parameter r=4 are topologically conjugate,[1] 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.

Magnifying the orbit diagram

Magnification near the tip shows more details.
Further magnification shows 8 separated regions.

Asymmetric tent map

The asymmetric tent map is essentially a distorted, but still piecewise linear, version of the case of the tent map. It is defined by

for parameter . The case of the tent map is the present case of . A sequence {} will have the same autocorrelation function [3] as will data from the first-order autoregressive process 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.


  1. Conjugating the Tent and Logistic Maps, Jeffrey Rauch, University of Michigan
  2. Collett, Pierre, and Eckmann, Jean-Pierre, Iterated Maps on the Interval as Dynamical Systems, Boston: Birkhauser, 1980.
  3. 1 2 Brock, W. A., "Distinguishing random and deterministic systems: Abridged version," Journal of Economic Theory 40, October 1986, 168-195.

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