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 x_n: 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 \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.

Behaviour

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

Magnifying the orbit diagram

Asymmetric tent map

The asymmetric tent map is essentially a distorted, but still piecewise linear, version of the \mu = 2 case of the tent map. It is defined by for parameter a \in [0,1]. The \mu = 2 case of the tent map is the present case of. A sequence {v_n} will have the same autocorrelation function as will data from the first-order autoregressive process with {u_n} 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.

Applications

The tent map has found applications in social cognitive optimization, chaos in economics, image encryption, on risk and market sentiments for pricing, etc.