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Graph of tent map function
Example of iterating the initial condition x0 = 0.4 over the tent map with μ = 1.9.

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.

Numerical errors

Time series of the Tent map for the parameter m = 2.0 which shows numerical error: "the plot of time series (plot of x variable with respect to number of iterations) stops fluctuating and no values are observed after n = 50". Parameter m = 2.0, initial point is random.

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.


The tent map has found applications in social cognitive optimization,[4] chaos in economics,[5][6] image encryption,[7] on risk and market sentiments for pricing,[8] etc.

See also


  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. ^ a b Brock, W. A., "Distinguishing random and deterministic systems: Abridged version," Journal of Economic Theory 40, October 1986, 168-195.
  4. ^ Sun, Jiaze; Li, Yang (January 2019). "Social cognitive optimization with tent map for combined heat and power economic dispatch". International Transactions on Electrical Energy Systems. 29 (1): e2660. arXiv:1809.03616. doi:10.1002/etep.2660.
  5. ^ Brock, William A.; Dechert, W. Davis (1991-01-01), "Chapter 40 Non-linear dynamical systems: Instability and chaos in economics", Handbook of Mathematical Economics, vol. 4, Elsevier, pp. 2209–2235, retrieved 2023-09-29
  6. ^ "Nonlinearities in Economics". SpringerLink. doi:10.1007/978-3-030-70982-2#editorsandaffiliations.
  7. ^ Li, Chunhu; Luo, Guangchun; Qin, Ke; Li, Chunbao (2017-01-01). "An image encryption scheme based on chaotic tent map". Nonlinear Dynamics. 87 (1): 127–133. doi:10.1007/s11071-016-3030-8. ISSN 1573-269X.
  8. ^ Lampart, Marek; Lampartová, Alžběta; Orlando, Giuseppe (2023-09-01). "On risk and market sentiments driving financial share price dynamics". Nonlinear Dynamics. 111 (17): 16585–16604. doi:10.1007/s11071-023-08702-5. ISSN 1573-269X.