We consider a finite-dimensional model of phase oscillators with inertia in the case of star configuration of coupling. The system of equations is reduced to a nonlinearly coupled system of pendulum equations. We prove that the transition from synchronous to asynchronous oscillations occurs via bifurcation of saddle-node equilibrium. In this connection the asynchronous regime can be partially synchronous rotations. We find that the reverse transition from asynchronous to synchronous regime occurs via bifurcation of homoclinic orbit both of the saddle equilibrium point and of the saddle periodic orbit. In the case of homoclinic loop of the saddle point the synchrony appears only from asynchronous mode without partially synchronized rotations. In the case of the homoclinic curve of the saddle periodic orbit the system undergoes a chaotic rotation regime which results in a random return to synchrony. We establish that return transitions are hysteretic in the case of large inertia.

We consider the problem of density wave propagation of a logistic equation with deviation of the spatial variable and diffusion (Fisher-Kolmogorov equation with deviation of the spatial variable). A Ginzburg–Landau equation was constructed in order to study the qualitative behavior of the solution near the equilibrium state. We analyzed the profile of the wave equation and found conditions for the appearance of oscillatory regimes. The numerical analysis of wave propagation shows that for a suficiently small spatial deviation this equation has a solution similar to the solution of the classical Fisher–Kolmogorov equation. The spatial deviation increasing leads to the existence of the oscillatory component in the spatial distribution of solutions. A further increase of the spatial deviation leads to destruction of the traveling wave. That is expressed in the fact that undamped spatio-temporal fluctuations exist in a neighborhood of the initial perturbation. These fluctuations are close to the solution of the corresponding boundary value problem with periodic boundary conditions. Finally, when the spatial deviation is suficiently large we observe intensive spatio-temporal fluctuations in the whole area of wave propagation.

We continue the study of the compactification of the moduli scheme for Gieseker-semistable vector bundles on a nonsingular irreducible projective algebraic surface S with polarization L, by locally free sheaves. The relation of main components of the moduli functor or admissible semistable pairs and main components of the Gieseker – Maruyama moduli functor (for semistable torsion-free coherent sheaves) with the same Hilbert polynomial on the surface S is investigated. The compactification of interest arises when families of Gieseker-semistable vector bundles E on the nonsingular polarized projective surface (S, L) are completed by vector bundles E on projective polarized schemes (S, L) of special form. The form of the scheme S, of its polarization L and of the vector bundle E is described in the text. The collection ((S, L), E) is called a semistable admissible pair. Vector bundles E on the surface (S, L) and E on schemes (S, L) are supposed to have equal ranks and Hilbert polynomials which are compute with respect to polarizations L and L, respectively. Pairs of the form ((S, L), E) named as S-pairs are also included into the class under the scope. Since the purpose is to study the compactification of moduli space for vector bundles, only families which contain S-pairs are considered. We build up the natural transformation of the moduli functor for admissible semistable pairs to the Gieseker – Maruyama moduli functor for semistable torsion-free coherent sheaves on the surface (S, L), with same rank and Hilbert polynomial. It is demonstrated that this natural transformation is inverse to the natural transformation built in the preceding paper and defined by the standard resolution of a family of torsion-free coherent sheaves with a possibly nonreduced base scheme. The functorial isomorphism constructed determines the scheme isomorphism of compactifications of moduli space for semistable vector bundles on the surface (S, L).

A version of the method of collocations and least residuals is proposed for the numerical solution of the Poisson equation in polar coordinates on non-uniform grids. By introducing general curvilinear
coordinates the original Poisson equation is reduced to the Beltrami equation. A uniform grid is used in curvilinear coordinates. The grid non-uniformity in the plane of the original polar coordinates is ensured with the aid of functions which control the grid stretching and entering the formulas of the passage from polar coordinates to the curvilinear ones. The method was verified on two test problems having exact analytic solutions. The examples of numerical computations show that if the radial coordinate axis origin lies outside the computational region, the proposed method has the second order of accuracy. If the computational region contains the singularity, the application of a non-uniform grid along the radial coordinate enables an increase in the numerical solution accuracy by factors from 1.7 to 5 in comparison with the uniform grid case at the same number of grid nodes.

We consider a periodic boundary-value problem for a nonlinear equation with the deviating spatial argument in the case when the deviation is small. This equation is called a spatially nonlocal erosion equation. It describes the formation of undulating surface relief under the influence of ion bombardment and can be interpreted as a development of the well-known Bradley-Harper model. It is shown that the nonhomogeneous surface relief can occur when the stability of the homogeneous states of equilibrium changes. In this boundary value problem the loss of stability can occur at the higher modes and a number of such modes. The mode number depends on many factors. For example, it depends on the angle of incidence. It is also shown that the nonlinear boundary value problem can be included into the class of 
abstract parabolic equations. Solvability of this problem was studied in the works by P.E. Sobolevsky, and this method assumes to use the analytical theory of semigroups of bounded linear operators. In 
order to solve the occurring bifurcation problems there were used the investigation methods of dynamical systems with an infinite-dimensional phase space (a space of initial conditions) such as: the method of integral manifolds, the method of Poincare–Dulac normal forms and asymptotic methods of analysis. Both possible in the given situation problems were studied: in codimension one and in codimension two. In particular, asymptotic formulas were obtained for solutions which describe nonhomogeneous undulating surface relief. The question about the stability of these solutions was studied. And the analysis of normal form was given. Also the asymptotic formulas for the nonhomogeneous undulating solutions were obtained. In conclusion some possible interpretations of the obtained results are indicated.

We consider a linear differential equation of second order with a small factor at the highest derivative. We study the problem of the asymptotic behavior of the eigenvalues of the first boundary value problem (task Dirichlet) in situation when the turning points (points where the coefficient at the first derivative equals to zero) exist. It is shown that only the behavior of coefficients of the equation in a small 
neighborhood of the turning points is essential. The main result is a theorem on the limit values of the eigenvalues of the first boundary value problem.

In this paper, a first-order equation with state-dependent delay and with a nonlinear right-hand side is considered. Conditions of existence and uniqueness of the solution of initial value problem are
supposed to be executed. The task is to study the behavior of solutions of the considered equation in a small neighborhood of its zero equilibrium. Local dynamics depends on real parameters which are coefficients of equation right-hand side decomposition in a Taylor series. The parameter which is a coefficient at the linear part of this decomposition has two critical values which determine a stability domain of zero equilibrium. We introduce a small positive parameter and use the asymtotic method of normal forms in order to investigate local dynamics modifications of the equation near each two critical values. We show that the stability exchange bifurcation occurs in the considered equation near the first of these critical values, and the supercritical Andronov – Hopf bifurcation occurs near the second of them (if the sufficient condition is executed). Asymptotic decompositions according to correspondent small parameters are obtained for each stable solution. Next, a logistic equation with state-dependent delay is considered as an example. The bifurcation parameter of this equation has one critical value. A simple sufficient condition of Andronov – Hopf bifurcation occurence in the considered equation near a critical value is obtained as a result of applying the method of normal forms.

Recall Lebesgue’s singular function. Imagine flipping a biased coin with probability p of heads and probability q = 1 − p of tails. Let the binary expansion of ξ ∈ [0, 1]: ξ = ∑∞ k=1 ck2−k be determined by flipping the coin infinitely many times, that is, ck = 1 if the k-th toss is heads and ck = 0 if it is tails. We define Lebesgue’s singular function L(t) as the distribution function of the random variable ξ: L(t) = Prob{ξ < t}. It is well-known that L(t) is strictly increasing and its derivative is zero almost everywhere (p ̸= q). The moments of Lebesque’ singular function are defined as Mn = Eξn. The main result of this paper is the following: Mn = O(nlog2 p).