Parametric-homogeneity and self-similar phenomena

Abstract

A mathematical tool for describing the symmetry of natural phenomena is the theory of groups and the group of coordinate dilations is one of the most important groups when studying natural phenomena because it is often known in advance that the group is applicable to the model under consideration. Dimensional analysis [l] and the concept of self-similarity in the usual sense [2], which are used in all branches of physics and mechanics, are based on the use of this group. The concept of groups of coordinate dilations is closely connected to the concept of homogeneous functions because the eigenvectors of quasi-homogeneous (homogeneous) coordinate dilation operators are the quasi-homogeneous (in particular homogeneous) functions [2]. The quasi-homogeneous functions are often called self-similar because they make it possible to reduce searching for a function of n-variables to searching for a function of n l-variables. It follows from the definition that homogeneous functions are smooth functions of arguments. However, the property of self-similarity often belongs to non-smooth objects. Thus, the following question arises: Which functions can describe some kinds of non-smooth or discrete self-similarity? Parametric-homogeneity, which includes parametric-homogeneous (PH) and parametric-qusshomogeneous (PQH) functions, PH-sets, and corresponding transformations, is a kind of discrete selfsimilarity. PH-fuhctions and PQH-functions are natural generalisations of concepts of homogeneous and quasi-homogeneous functions when instead of the continuous group we consider the discrete group of coordinate dilations. The idea to introduce PH-functions arose in 1992 with applications to contact problems being discussed in [3-51. It has been shown that the similarity approach (see, e.g., [6]) is applicable to nonsmooth surfaces, in particular to some fractal surfaces. Some properties of the PH-functions were described in [7]. Clearly, real natural phenomena do not exhibit the pure mathematical PH-properties. However, PH-features can be exhibited by some processes on their intermediate stage when the behaviour of the processes has ceased to depend on the details of the boundary conditions or initial conditions. The idea, that various processes possess an intermediate self-similar stage of their development, was successfully used in the study of continuous self-similarity [l]. We can expect that some selforganized processes possess the PH-features. Indeed, log-periodicity, which as we will see below is a particular kind of parametric-homogeneity, was considered in various papers mainly in applications to critical phenomena (see, e.g., [8-121).