Universal Mandelbrot Set as a Model of Phase Transition Theory

The study of Mandelbrot Sets (MS) is a promising new approach to the phase transition theory. We suggest two improvements which drastically simplify the construction of MS. They could be used to modify the existing computer programs so that they start building MS properly not only for the simplest families. This allows us to add one more parameter to the base function of MS and demonstrate that this is not enough to make the phase diagram connected. The problem of stability of time evolution is one of the most important in physics. Usually one can make the motion stable or unstable by changing some parameters which characterize Hamiltonian of the system. Stability regions can be represented on the phase diagram and transitions between them are described by catastrophe theory [1], [2]. It can seem that a physical system or a mechanism can be taken from one domain of stability to any other by continuous and quasi-static variation of these parameters, i.e. that the phase diagram is connected. However sometimes this expectation is wrong, because domains of stability can be separated by points where our system is getting totally destroyed. Unfortunately today it is too difficult to explore the full phase diagram for generic physical system with many parameters. Therefore, following [3], it was proposed in [4] to consider as a simpler model the discrete dynamics of one complex variable [5]-[8]. The phase diagram in this case is known as Universal Mandelbrot Set (UMS). MS is a well-known object in mathematics [9]-[11], but its theory is too formal and not well adjusted to the use in the phase transition theory. The goal of this paper is to make MS more practical for physical applications. 1 Structure of MS First of all we remind the definition of MS and UMS from [4], which different from conventional definition in mathematical literature, see s. 3 below. Mandelbrot Set (MS) is a set of points in the complex c plane. MS includes a point c if the map x → f(x, c) has stable periodic orbits. As shown in Fig.1 MS consists of many clusters connected by trails, which in turn consist of smaller clusters and so on. Each cluster is linear connected and can be divided into elementary domains where only one periodic orbit is stable. Different elementary domains can merge and even overlap. Boundary of elementary domain of n-th order, i.e. of a domain where an n-th order orbit is stable, is a real curve c(α) given by the system: { Gn(x, c) = 0 F ′ n(x, c) + 1 = e iα (1) with Fn(x, c) = f (x, c)− x Gn(x, c) = Fn(x, c) ∏ m Gm(x, c) , n ..m Indeed, when Gn(x, c) vanishes then x belongs to the orbit of exactly the n-th order. This orbit is stable if | ∂ ∂x f(x, c)| < 1, what implies the second equation of (1). The solution of this system may give us more than a single n-th order domain. Domains of different orders merge at the points c where Resultantx ( Gn(x, c), Gk(x, c) ) = 0 (2) [email protected]