ISSN:
1089-7658
Source:
AIP Digital Archive
Topics:
Mathematics
,
Physics
Notes:
When xt and xt+1 represent two random variables, each belonging to a real interval [0,1] and being related by a first-order difference equation xt+1=F(xt), called a discrete-time map, the probability density distribution connected with xt can be translated into that associated with xt+1. This yields an evolution equation by means of which one can construct an infinite sequence {wt(x)||t∈N, x∈[0,1]} starting from an integrable function w0(x) normalized to unity on [0,1]. The question of the convergence of the sequence toward a so-called invariant density function w(x) as t→+∞ and the problem of finding this limit were examined by a number of authors, mostly studying isolated cases. The present paper solves the problem for a class of discrete-time maps characterized by xt+1 =f(||sn[l sn−1f−1(xt)]||), l∈{2,3,...}, whereby f is a real, continuous, monotonically increasing function mapping [0,1] onto itself and sn is the usual symbol for the sinelike Jacobian elliptic function with modulus k∈[0,1] (including the sine function). Convergence is proven under very general conditions on w0(x) and an explicit formula to calculate w(x) is established. Some properties of w(x) are discussed. A necessary and sufficient condition for the symmetry of w(x) about x= 1/2 is obtained and attention is also devoted to the inverse problem, leading to a reformulation of the discrete-time map of the type cited above which corresponds to a given invariant density. The examples of practical application considered here cover almost all special cases which were treated in the literature thus far, as well as new cases.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1063/1.527528
Permalink
