Monte-Carlo for solving large linear systems of ordinary differential equations

Monte-Carlo approach towards solving Cauchy problem for large systems of linear differential equations is being proposed in this paper. Firstly, a quick overlook of previously obtained results from applying the approach towards Fredholm-type integral equations is being made. In the main part of the paper, a similar method is being applied towards a linear system of ODE. It is transformed into an equivalent system of Volterra-type integral equations, which relaxes certain limitations being present due to necessary conditions for convergence of majorant series. The following theorems are being stated. Theorem 1 provides necessary compliance conditions that need to be imposed upon initial and transition distributions of a required Markov chain, for which an equality between estimate’s expectation and a desirable vector product would hold. Theorem 2 formulates an equation that governs estimate’s variance, while theorem 3 states a form for Markov chain parameters that minimise the variance. Proofs are given, following the statements. A system of linear ODEs that describe a closed queue made up of ten virtual machines and seven virtual service hubs is then solved using the proposed approach. Solutions are being obtained both for a system with constant coefficients and time-variable coefficients, where breakdown intensity is dependent on t. Comparison is being made between Monte-Carlo and Rungge Kutta obtained solutions. The results can be found in corresponding tables.

Retrospective analysis of the orbits of asteroids colliding with the Earth

In this paper, we consider the trajectories of real and model asteroids that lead to collisions with the Earth. They highlight the close approaches to the Earth that precede the impact. The presence of such approaches allows you to detect a dangerous object in advance, clarify its orbit, and also use the effect of a gravitational maneuver to economically prevent an asteroid from hitting the Earth. The article considers various families of collision trajectories: possible trajectories of real dangerous asteroids, as well as model ones trajectories that are not linked to a specific object. It is shown that in the first case, the approaches preceding the collision are noticeably greater.

Search of weights in the problem of finite-rank signal estimation in presence of noise

The problem of weighted finite-rank time-series approximation is considered for signal estimation in “signal plus noise” model, where the inverse covariance matrix of noise is (2p+1)-diagonal. Finding of weights, which improve the estimation accuracy, is examined. An effective method for the numerical search of the weights is constructed and proved. Numerical simulations are performed to study the improvement of the estimation accuracy for several noise models.

The effect of distributions symmetrization on their peakedness

Indirect transformations of a one-dimensional, two-dimensional, and multidimensional random variable are proposed. They are based on various symmetrizations of the density function. The focus is on changing peakedness of a distribution about the origin.

Fixed point theorems for new contractions with application in dynamic programming

In this study, we give a generalization of the well-known Reich fixed point in the setting of general topological spaces with τ -distances. As applications of the obtained result, we prove some fixed point theorems for new contraction types in metric spaces. Moreover, we establish the existence and the uniqueness of solutions for a class of functional equations arising in dynamic programming.

Generalized normal forms of the systems of ordinary differential equations with a quasi-homogeneous polynomial (ax1^2 + x2, x1x2) in the unperturbed part

In this paper a study on constructive construction of the generalized normal forms (GNF) is continued. The planar real-analytical at the origin system is considered. Its unperturbed part forms a first degree quasi-homogeneous first degree polynomial (αx21 + x2, x1x2) of type (1, 2) where parameter α ∈ 2 (-1/2, 0)[(0, 1/2]. For given value of this polynomial is a canonical form, that is an element of a class of equivalence relative to quasi-homogeneous substitutions of zero order into which any first order quasi-homogeneous polynomial of type (1, 2) is divided in accordance with the chosen structural principles due to it only making sense to reduce the systems with the various canonical forms in their unperturbed part to GNF. Based on the constructive method of resonance equations and sets, the resonance equations are derived. Perturbations of the acquired system satisfies these equations if an almost identity quasi-homogeneous substitution in the given system is applied. Their validity guarantees formal equivalence of the systems. Besides, resonance sets of coefficients are specified that allows to get all possible GNF structures and prove reducibility of the given system to a GNF with any of specified structures. In addition, some examples of characteristic GNFs are provided including that with the parameter leading to appearance of an additional resonance equation and the second nonzero coefficient of the appropriate orders in GNFs.

The inverse problem of stabilization of a spherical pendulum in a given position under oblique vibration

The inverse problem is posed of stabilizing a spherical pendulum (a mass point at the end of a weightless solid rod of length l ) in a given position using high-frequency vibration of the suspension point. The position of the pendulum is determined by the angle between the pendulum rod and the gravity acceleration vector. For any given position of the pendulum, a series of oblique vibration parameters (amplitude of the vibration velocity and the angle between the vibration velocity vector and the vertical) were found that stabilize the pendulum in this position. From the obtained series of solutions, the parameters of optimal vibration (vibration with a minimum amplitude of velocity) are selected depending on the position of the pendulum. The region of initial conditions is studied, of which the optimal vibration leads the pendulum to a predetermined stable position after a sufficiently long time. This area, following N. F.Morozov et al., called the area of attraction.

Averaging technique in the problem of satellite attitude stabilization in indirect position in the orbital reference frame with the use of Lorentz torque

A dynamically symmetric satellite in a circular orbit of small inclination is considered. The problem of the Lorentzian stabilization of the satellite in the orbital coordinate system in the indirect equilibrium position is solved under the conditions of the perturbing effect of the gravitational torque. To solve this problem, which is characterized by incomplete control, a method of averaging differential equations is developed. Using the original construction of the unsteady Lyapunov function, we obtain sufficient conditions for the asymptotic stability of the programmed satellite motion mode in the form of constructive inequalities with respect to the control parameters.

On convergence and compactness in variation with shift of discrete probability laws

We consider a class of discrete distribution functions, whose characteristic functions are separated from zero, i. e. their absolute values are greater than positive constant on the real line. The class is rather wide, because it contains discrete infinitely divisible distribution functions, functions of lattice distributions, whose characteristic functions have no zeroes on the real line, and also distribution functions with a jump greater than 1/2. Recently the authors showed that characteristic functions of elements of this class admit the Lévy-Khinchine type representations with non-monotonic spectral function. Thus our class is included in the set of so called quasi-infinitely divisible distribution functions. Using these representation the authors also obtained limit and compactness theorems with convergence in variation for the sequences from this class. This note is devoted to similar results concerning convergence and compactness but with weakened convergence in variation. Replacing of type of convergence notably expands applicability of the results.

Multi-pass stable periodic points of diffeomorphism of a plane with a homoclinic orbit

A diffeomorphism of a plane into itself with a fixed hyperbolic point and a nontransversal point homoclinic to it is studied. There are various ways of touching a stable and unstable manifold at a homoclinic point. Periodic points whose trajectories do not leave the vicinity of the trajectory of a homoclinic point are divided into a countable set of types. Periodic points of the same type are called n-pass periodic points if their trajectories have n turns that lie outside a sufficiently small neighborhood of the hyperbolic point. Earlier in the articles of Sh. Newhouse, L. P. Shil’nikov, B. F. Ivanov and other authors, diffeomorphisms of the plane with a nontransversal homoclinic point were studied, it was assumed that this point is a tangency point of finite order. In these papers, it was shown that in a neighborhood of a homoclinic point there can be infinite sets of stable two-pass and three-pass periodic points. The presence of such sets depends on the properties of the hyperbolic point. In this paper, it is assumed that a homoclinic point is not a point with a finite order of tangency of a stable and unstable manifold. It is shown in the paper that for any fixed natural number n, a neighborhood of a nontransversal homolinic point can contain an infinite set of stable n-pass periodic points with characteristic exponents separated from zero.