Tent map

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

the name being due to the tent-like shape of the graph of f_μ. For the values of the parameter μ within 0 and 2, f_μmaps 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 ${\displaystyle x_{n}}$ :

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 [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 x_n in [0,1].

The ${\displaystyle \mu =2}$ 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.

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