We consider the meaning, the role and the prospects of interdisciplinary investigations in the world scientific knowledge. We discuss the theory of self-organization or the synergetics development and its reflection in the book series “Synergetics: From the Past to the Future”. We show that fundamental questions of the modern science and a number of key high technologies bounded with the future of the economy directly depend on the level of interdisciplinary works performed in the country. Synergetics talks now in terms of mathematical models. So their construction, investigation and use are the main way in the development of interdisciplinary approaches.

We discuss the dynamics of the Hutchinson’s equation and its generalizations. An estimate of the global stability region of a positive steady state is obtained. The main results refer to existence, stability and asymptotics of a slow oscillating solution. New asymptotic methods are applied to a problem of dynamical properties of ODE system describing Belousov — Zhabotinsky reaction.

The operator inclusion 0 ∈ A(x)+N(x) is studied. The main results refer to the case, when A – a bounded operator of monotone type from a reflexive space into conjugate to it, N – a conevalued operator. No solution criterion of the viewed inclusion is set up. Integer characteristics of multivalued mappings with homotopy invariance and additivity are introduced. Application to the theory of variational inequalities with multivalued operators is identified.

The dynamics of one-dimensional piecewise linear mapping with one rupture is investigated. In the space of parameters we defined regions with different dynamical effects.

The statistical analysis of random quantities derived from the dynamics of interaction of a pair of neuron-type oscillators are presented. It is shown that computation of some statistical characteristics of the process allows, with enough precision, to distinguish two types of orbits, while their phase portraits, Lyapunov dimension and time plots are slightly different.

We discuss properties of functions which are covered by spreading the differentiation operator from the space W₁¹.

The dynamics of interaction of three neurons is studied. One of possible options of the link is a feedback loop arising at modeling neuronets is considered.

We give a survey of some results in the geometry of convex bodies and their applications.

In 1978 R. Mercle and M. Hellman offered to use the subset sum problem for constructing cryptographic systems. The proposed cryptosystems were based on a class of the knapsacks with super-increasing vectors. This class is a subset of the set of knapsacks with injective (cryptographic) vectors that allow the single-valued decoding (decryption) result. In this paper we consider the problems related to the order in the growth of the injective vectors knapsacks quantity and to the order in the growth of knapsacks quantity with the super-increasing vectors through the knapsack maximal element increasing.

We study the local dynamics of the Stuart–Landau equation with large delay in the neibourhood of periodic solutions. We find sufficient conditions of instability of periodic solutions and sufficient conditions of their stability.

This paper is dedicated to the study of ideals in semi-lattice of the enumeration degrees.