Tent map

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 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. 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. If μ is less than 1 the point x = 0 is an attractive fixed point of the system for all initial values of x i.e. the system will converge towards x = 0 from any initial value of x. If μ is 1 all values of x less than or equal to 1/2 are fixed points of the system. If μ is greater than 1 the system has two fixed points, one at 0, and the other at μ/(μ + 1). Both fixed points are unstable, i.e. a value of x close to either fixed point will move away from it, rather than towards it. For example, when μ is 1.5 there is a fixed point at x = 0.6 (since 1.5(1 − 0.6) = 0.6) but starting at x = 0.61 we get If μ is between 1 and the square root of 2 the system maps a set of intervals between μ − μ2/2 and μ/2 to themselves. This set of intervals is the Julia set of the map – that is, it is the smallest invariant subset of the real line under this map.