Introduction

The theory of complex systems is a modern field of natural sciences, which considers systems that consist of a large number of interacting parts. There is not a large number of quality properties of a system that are observed in the study of its individual parts; they arise only as a result of their interactions.

The theory of complex systems is based on methods of systems analysis as an applied scientific methodology, consisting, in particular, of mathematical methods, algorithmic software and computing tools, which provide the formation of holistic knowledge about the object, as a set of interconnected processes of different natures for further decision-making on its development and behavior, taking into account conflicting criteria, the presence of risk factors and inaccurate information. Due to the application of systems analysis methods, the theory of complex systems has been actively developing in recent decades and consists, in particular, of the development of methods for structuring, modeling, analysis and synthesis of deterministic and stochastic systems in problems of mathematical (including statistical) physics, applied questions of probability theory and fractal analysis and mathematical problems of biology, sociology, ecology, economics and medicine. The main motivation that unites all these diverse mathematical and natural sciences in a single discipline is closeness, as well as the kinship laws and methods of studying the collective behavior of such systems.

Mathematical problems associated with the theory of complex systems relate to the development of mathematical methods of system simplification, which can be very complex, even for computer analysis. In this case, the simplified system should be such that, first, its local characteristics are determined by simple enough functionals of the local characteristics of the original system, and, second, that the simplified model could be qualitatively analyzed by mathematical methods and its global characteristics are an effective approximation of the corresponding characteristics of the original system.

Typical examples of complex systems are evolutionary systems, which, in particular, are modeled using random evolutionary systems and random processes. For example, impulsive processes describe various models in the queuing theory, namely, network structures and production systems; processes with independent increments are used in modeling conflict systems, such as birth-death processes in ecological and biological systems, as well as in models of statistical physics; and stochastic evolutionary systems model problems that arise in reliability theory, control theory, financial mathematics and so on.

We provide an analysis of the asymptotic properties of such models, which are considered and studied from different points of view, using several approximation schemes. That is, the question of the convergence of evolutionary systems in Poisson approximation schemes is analyzed, the problem of large deviations for evolutionary systems in different approximation schemes is analyzed and some applications are discussed: the probability of leaving an interval, the conditions of equilibrium, stationarity, the classification of equilibria and so on.

The development of the theory of random evolutions began in the late 1960s, probably with the work of R. Griego and R. Hersh. They introduced the concept of random evolution in Griego and Hersh (1971).

Applications of this model follow from the work of R.Z. Khasminskii (1966a, 1966b), which were stimulated by the problems of stability of stochastic systems, in particular, by the works of R.L. Stratonovich (1967) on the problems of the nonlinear theory of oscillations in the presence of noise. In the 1960s and 1970s, problems related to the theory of random evolutions were actively studied by American mathematicians R. Hersh, M. Pinsky, G. Papanikolaou, T. Kurtz, R. Griego, L. Gorostiza (Pinsky 1968; Griego and Hersh 1971; Papanicolaou and Kellek 1971; Gorostiza 1972, 1973a, 1973b; Hersh and Papanicolaou 1972; Hersh and Pinsky 1972; Kurtz 1973; Hersh 1974; Papanicolaou 1975) and others. In particular, G. Papanicolaou, D. Strook and S. Varadhan proposed a martingale approach for proving limit theorems (Papanicolaou et al. 1977) using methods similar to solving the singular perturbation problem.

An effective method for proving the limit theorems in the theory of random evolutions is the theory of phase merging of complex systems, developed by V.S. Korolyuk and A.F. Turbin (1975, 1993). A.V. Skorokhod (2008) investigated several important problems of the theory of dynamic systems, denoted by stochastic differential equations. The problem of the stability of dynamic systems in the case of random perturbation of their parameters was investigated by R.Z. Khasminskii (2012) and V.S. Korolyuk (1991, 1998). Problems of stability and problems of stochastic approximation of evolutionary systems with switching were investigated by V.S. Korolyuk and Y.M. Chabanyuk (2007, 2007). V.S. Korolyuk and A.V. Swishchuk also developed a theory of semi-Markov random evolutions, based on martingale theory (Korolyuk and Swishchuk 1995a, 1995b). Many applications of processes with switching to the analysis of network systems can be found in the works of V.V. Anisimov (2008).

Asymptotic methods are collected and described in detail in the book by V.S. Korolyuk and N. Limnios (2005), where these methods are mainly applied to models in the schemes of averaging and diffusion approximation. At increasing time intervals, the averaging scheme demonstrates the deterministic averaging behavior of the system, and the diffusion approximation scheme shows stochastic fluctuations around the deterministic averaged trajectory. These two schemes differ in the normalization of the switching process, namely, in the case of the averaging scheme, the acceleration of time by the parameter e-1 is considered, while, in the case of diffusion, approximation by the parameter e-2 or e-3 is considered. An important element of the algorithm is the assumption of ergodicity of the switching (Markov or semi-Markov) process. In their book, V.S. Korolyuk and N. Limnios (2005) also considered several models for processes with independent increments and impulsive processes with switching in the schemes of Poisson and Lévy approximation. Here we generalize the models in the schemes of Poisson and Lévy approximation to the case of processes with locally independent increments, impulsive recurrent processes and so on.

We also present another important area of research, namely, the solution of the large deviations problem, using the methods of asymptotic analysis of nonlinear exponential generators, associated with the singular perturbation problem. The large deviations problem arose as a method of solving statistical problems, related to estimating the probabilities of rare events. The first work in this direction was, obviously, the article by H. Cramér (1938), but this idea was finally fully formed in the work of G. Chernoff (1952).

The main method of studying such problems is the technique of substituting the measure and using the Chebyshev inequality to obtain estimates of the probabilities of rare events, and to calculate the rate functional. In this case, a new measure is introduced, in relation to which the studied events have a high probability, while the probabilities of these events relative to the initial measure are determined in terms of the Radon-Nicodemus derivative, which connects the two mentioned measures. An important role in determining the rate functional is also played by the Legendre transform, which links the rate functional and the cumulant of the process. The most famous works in this technique, which relate to Markov processes, are the works of M. Donsker and S. Varadan (1975a, 1975b, 1976) and M.Y. Freidlin and O.D. Wentzell (1976, 1978, 1979, 1990) (see also Deuschel and Stroock 1989; Dupuis and Ellis 1997; Dembo and Zeitouni 2010). These works contain comprehensive bibliographic references on this topic.

Another approach to solving the problem is related to the convergence and compactness of probabilistic measures. The works in which such methods are used include articles by A.A. Puhalsky (1991), G. O'Brien and W. Vervaat (1995) and A. de Acosta (1997).

Here, we use a method that has emerged and developed recently and is closely related to the control problem. The idea of such an approach was presented in Hopf (1950) and Çinlar et al. (1980) and was finally developed by Fleming and Suganidis (1986). The most developed and generalized version of this approach can be found in the book by J. Feng and T. Kurtz (2006), which also contains a relevant bibliography and historical review. It should be noted that the classical normalization scheme in which the large deviations problem is studied is the small diffusion scheme (see Feng and Kurtz (2006) and Freidlin and Ventzell (2012) and the corresponding bibliographic sections of these works). One of the rare models where another possible normalization for the process with independent increments was investigated is the work of A.A. Mogulsky (1993) (compare this to other works by A.A. Borovkov and A.A. Mogulsky (1992, 2012)).

We investigate the large deviations problem for processes with independent increments and impulsive processes with switching in the schemes of the Poisson and Lévy approximations. Models with phase merging are studied for the first time. This formulation of the problem is completely new and allows us to obtain some significant generalizations. For example, the presence of a diffusion...