In this paper, we study the phenomenon of appearance of new resonances in a timedependent harmonic oscillator under an oscillatory decreasing force. The studied equation belongs to the class of adiabatic oscillators and arises in connection with the spectral problem for the one-dimensional Schr¨odinger equation with Wigner–von Neumann type potential. We use a specially developed method for asymptotic integration of linear systems of differential equations with oscillatory decreasing coefficients. This method uses the ideas of the averaging method to simplify the initial system. Then we apply Levinson’s fundamental theorem to get the asymptotics for its solutions. Finally, we analyze the features of a parametric resonance phenomenon. The resonant frequencies of perturbation are found and the pointwise type of the parametric resonance phenomenon is established. In conclusion, we construct an example of a time-dependent harmonic oscillator (adiabatic oscillator) in which the parametric resonances, mentioned in the paper, may occur.

We study the local dynamics of a solutions spatially distributed logistic equation in the case of a two-dimensional spatial variable. Two distribution functions important for applications are considered. It is shown, that the critical cases in the problem of equilibrium stability have an infinite dimention. For each critical case a special replacement is built, which reduces the original problem to a system of parabolic equations — a quasinormal form, the solutions behavior of which defines the local dynamics. Some of the parameters in the quasi-normal form depend on a small parameter via a discontinuous function Θ(ε), which takes an infinite number of times all the values in the interval [0, 1) for ε → 0. This gives infinite alternation of forward and backward bifurcations in the initial boundary value problem. The obtained results are compared with those for the case of a one-dimensional spatial variable. New bifurcation phenomena which occur only in the case of a two-dimensional spatial variable are revealed.

We consider a boundary problem of reaction-diffusion type in the domain consisting of two rectangular areas connected by a bridge. The bridge width is a bifurcation parameter of the problem and is changed in such way that the measure of the domain is preserved. The conditions on chaotic oscillations emergence were studied and the dependence of invariant characteristics of the attractor on the bridge width was constructed. The diffusion parameter was chosen such that in the case of widest possible bridge (corresponding to a rectangular domain) the spatially homogeneous cycle of the problem is orbitally asymptotically stable. By decreasing the bridge width the homogeneous cycle looses stability and then the spatially inhomogeneous chaotic attractor emerges. For the obtained attractor we compute Lyapunov exponents and Lyapunov dimension and notice that the dimension grows as the parameter decreases but is bounded. We show that the dimension growth is connected with the growing complexity of stable solutions distribution with respect to the space variable.

This paper presents an analytical review of a conference on the great scientist, a brilliant professor, an outstanding educator Sergei Kapitsa, held in November 2012. In the focus of this forum were problems of self-organization and a paradigm of network structures. The use of networks in the context of national defense, economics, management of mass consciousness was discussed. The analysis of neural networks in technical systems, the structure of the brain, as well as in the space of knowledge, information, and behavioral strategies plays an important role. One of the conference purposes was to an online organize community in Russia and to identify the most promising directions in this field. Some of them are presented in this paper.

Let S be a nondegenerate simplex in Rⁿ. Denote by α(S) the minimal σ > 0 such that the unit cube Q_n:= [0, 1]ⁿ is contained in a translate of σS. In the case α(S) ≠ 1 the translate of α(S)S containing Qn is a homothetic copy of S with the homothety center at some point x ∈ Rⁿ . We obtain the following computational formula for x. Denote by x (j) (j = 1, . . . , n+ 1) the vertices of S. Let A be the matrix of order n+ 1 with the rows consisting of the coordinates of x (^j) ; the last column of A consists of 1’s. Suppose that A−1 = (l_Ij ). Then the coordinates of x are the numbers

xk = Pn+1 j=1 ( Pn i=1 |lij |) x (j) k − 1 Pn i=1 Pn+1 j=1 |lij | − 2 (k = 1, . . . , n).

Since α(S) ≠ 1, the denominator from the right-hand part of this equality is not equal to zero. Also we give the estimates for norms of projections dealing with the linear interpolation of continuous functions defined on Q_n.

A logistic equation with a delay feedback circuit and with periodic perturbation of parameters is considered. The problem parameters (a coefficient of the linear growth and a delay) are chosen close to the critical values at which a cycle is bifurcated from the equilibrium point. We assume that these values have a double-frequency relation to the time, the frequency of action being close to the doubled frequency of the natural vibration. Asymptotic analysis is performed under these assumptions and leads to a two-dimensional system of ordinary differential equations. The linear part of this system is periodic. If the parameter which defines the frequency detuning of the external action is large or small, we can apply standard asymptotic methods to the resulting system. Otherwise, numerical analysis is performed. Using the results of the numerical analysis, we clarify the main scenarios of phase transformations and find the area of chaotic oscillations. The main conclusion is that in case of parametric resonance the dynamics of the problem with double-frequency perturbation is more complicated than the dynamics of the problem with single-frequency perturbation.

In this paper we consider some families of smooth rational curves of degree 2, 3 and 4 on a smooth Fano threefold X which is a linear section of the Grassmanian G(1, 4) under the Pl¨ucker embedding. We prove that these families are irreducible. The proof of the irreducibility of the families of curves of degree d is based on the study of degeneration of a rational curve of degree d into a curve which decomposes into an irreducible rational curve of degree d−1 and a projective line intersecting transversally at a point. We prove that the Hilbert scheme of curves of degree d on X is smooth at the point corresponding to such a reducible curve. Then calculations in the framework of deformation theory show that such a curve varies into a smooth rational curve of degree d. Thus, the set of reducible curves of degree d of the above type lies in the closure of a unique component of the Hilbert scheme of smooth rational curves of degree d on X. From this fact and the irreducibility of the Hilbert scheme of smooth rational curves of degree d on the Grassmannian G(1, 4) one deduces the irreducibility of the Hilbert scheme of smooth rational curves of degree d on a general Fano threefold X.

The paper discusses the problem of forecasting for samples with real-valued attributes. The goal is to estimate the effect of generated binary attributes on forecasting accuracy for the linear regression and the hybrid methods based on clustering. The initial set of attributes is expanded by binary attributes which are derived from the initial set by fuzzy classification. A comparative testing of the discussed forecasting methods on the initial samples and the resulting ones is performed. The test results on three different databases showed that the use of generated attributes for the classical linear regression resulted in the significant increase of the forecasting accuracy. In case of the linear regression with the clustering based on k-means the increase of forecasting accuracy was also observed. In case of the linear regression with the clustering based on the knn–method we registered a slight decrease, and an unstable result was obtained for the double linear regression.

In this paper is considered the extreme problem of searching for the optimal quadrature formulas in S.M. Nikolskiy sense for approximate calculation of curvilinear integrals of first kind on the class of differentiable functions, the second gradient norm of which in L_p (1 ≤ p < ∞) is bounded along the curve by which the curvilinear integral is calculated. The exact errors of optimal quadrature formula for the studied class of functions were calculated, and the explicit formulas for optimal nodes and coefficients was shown.

The paper considers such transformations of variational and optimal control problems (by a change of phase variables) that help to get the solution. Among these techniques are methods of the dimensionality reduction by a transition from an original argument to a new one, by a detection of a variable staying unchanged through equations of the system (invariant), by extension of the original problem and a transition from it to a simpler problem that gives an estimate solution of the original one. In some cases we can get a solution that is simplified by a transition from a set of conditions in the form of differential equations to the only integral condition. This relates to equations with a retarded argument. The paper gives examples of application of these techniques to real engineering problems such as an optimal control of biosynthesis, cooling of crystal systems by a laser radiation, etc.