Tent map

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

${\displaystyle f_{\mu }(x):=\mu \min\{x,\,1-x\},}$

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_μ 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 ${\displaystyle x_{n}}$:

${\displaystyle x_{n+1}=f_{\mu }(x_{n})={\begin{cases}\mu x_{n}&\mathrm {for} ~~x_{n}<{\frac {1}{2}}\\\mu (1-x_{n})&\mathrm {for} ~~{\frac {1}{2}}\leq x_{n}\end{cases}}}$

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.