Stable Arcs Connecting Polar Cascades on a Torus

In the article, the components of the stable isotopic connection of polar gradient-like diffeomorphisms on a two-dimensional torus are found under the assumption that all non-wandering points are fixed and have a positive orientation type.

Nonlinear Processes in Safety Systems for Substances with Parameters Close to a Critical State

The paper presents a modification of the digital method by S. K. Godunov for calculating real gas flows under conditions close to a critical state. The method is generalized to the case of the Van der Waals equation of state using the local approximation algorithm. Test calculations of flows in a shock tube have shown the validity of this approach for the mathematical description of gas-dynamic processes in real gases with shock waves and contact discontinuity both in areas with classical and nonclassical behavior patterns. The modified digital scheme by Godunov with local approximation of the Van der Waals equation by a two-term equation of state was used for simulating a spatial flow of real gas based on Navier – Stokes equations in the area of a complex shape, which is characteristic of the internal space of a safety valve. We have demonstrated that, under near-critical conditions, areas of nonclassical gas behavior may appear, which affects the nature of flows. We have studied nonlinear processes in a safety valve arising from the movement of the shut-off element, which are also determined by the device design features and the gas flow conditions.

Vibroimpact Mobile Robot

A simple model of a capsule robot is studied. The device moves upon a rough horizontal plane and consists of a capsule with an embedded motor and an internal moving mass. The motor generates a harmonic force acting on the bodies. Capsule propulsion is achieved by collisions of the inner body with the right wall of the shell. There is Coulomb friction between the capsule and the support, it prevents a possibility of reversal motion. A periodic motion is constructed such that the robot gains the maximal average velocity.

Symmetry and Relative Equilibria of a Bicycle System

In this paper, we study the symmetry of a bicycle moving on a flat, level ground. Applying the Gibbs – Appell equations to the bicycle dynamics, we previously observed that the coefficients of these equations appeared to depend on the lean and steer angles only, and in one such equation, a term quadratic in the rear wheel’s angular velocity and a pseudoforce term would always vanish. These properties indeed arise from the symmetry of the bicycle system. From the point of view of the geometric mechanics, the bicycle’s configuration space is a trivial principal fiber bundle whose structure group plays the role of a symmetry group to keep the Lagrangian and constraint distribution invariant. We analyze the dimension relationship between the space of admissible velocities and the tangent space to the group orbit, and then employ the reduced nonholonomic Lagrange – d’Alembert equations to directly prove the previously observed properties of the bicycle dynamics. We then point out that the Gibbs – Appell equations give the local representative of the reduced dynamic system on the reduced constraint space, whose relative equilibria are related to the bicycle’s uniform upright straight or circular motion. Under the full rank condition of a Jacobian matrix, these relative equilibria are not isolated, but form several families of one-parameter solutions. Finally, we prove that these relative equilibria are Lyapunov (but not asymptotically) stable under certain conditions. However, an isolated asymptotically stable equilibrium may be achieved by restricting the system to an invariant manifold, which is the level set of the reduced constrained energy.

On the Organization of Homoclinic Bifurcation Curves in Systems with Shilnikov Spiral Attractors

We study spiral chaos in the classical Rössler and Arneodo – Coullet – Tresser systems. Special attention is paid to the analysis of bifurcation curves that correspond to the appearance of Shilnikov homoclinic loop of saddle-focus equilibrium states and, as a result, spiral chaos. To visualize the results, we use numerical methods for constructing charts of the maximal Lyapunov exponent and bifurcation diagrams obtained using the MatCont package.

Artificial Neural Network as a Universal Model of Nonlinear Dynamical Systems

We suggest a universal map capable of recovering the behavior of a wide range of dynamical systems given by ODEs. The map is built as an artificial neural network whose weights encode a modeled system. We assume that ODEs are known and prepare training datasets using the equations directly without computing numerical time series. Parameter variations are taken into account in the course of training so that the network model captures bifurcation scenarios of the modeled system. The theoretical benefit from this approach is that the universal model admits applying common mathematical methods without needing to develop a unique theory for each particular dynamical equations. From the practical point of view the developed method can be considered as an alternative numerical method for solving dynamical ODEs suitable for running on contemporary neural network specific hardware. We consider the Lorenz system, the Rössler system and also the Hindmarch – Rose model. For these three examples the network model is created and its dynamics is compared with ordinary numerical solutions. A high similarity is observed for visual images of attractors, power spectra, bifurcation diagrams and Lyapunov exponents.

Incorporation of Fluid Compressibility into the Calculation of the Stationary Mode of Operation of a Hydraulic Device at High Fluid Pressures

This paper is concerned with assessing the correctness of applying various mathematical models for the calculation of the hydroshock phenomena in technical devices for modes close to critical parameters of the fluid. We study the applicability limits of the equation of state for an incompressible fluid (the assumption of constancy of the medium density) to the simulation of processes of the safety valve operation for high values of pressures in the valve. We present a scheme for adapting the numerical method of S. K. Godunov for calculation of flows of incompressible fluids. A generalization of the method for the Mie – Grüneisen equation of state is made using an algorithm of local approximation. A detailed validation and verification of the developed numerical method is provided, and relevant schemes and algorithms are given. Modeling of the hydroshock phenomenon under the valve actuation within the incompressible fluid model is carried out by the openFoam software. The comparison of the results for the weakly compressible

Calculation and Experimental Study of Heat Exchange in a System of Plane-Parallel Channels with Surface Intensifiers

This paper presents the results of numerical investigation, calculation analysis and experimental study of heat exchange in a system of plane-parallel channels formed by rectangular fins, which are applied in a heat removal device using heat tubes for power semiconductor energy converters. Passive cooling (heat removal by radiation and natural convection) and active cooling (heat removal by radiation and forced convection) are investigated for various velocities of air cooling of fins by spherical vortex generators applied to its surface. A comparative analysis of the results is carried out for the average effective heat removal resistance and for the average temperature at the ends of the fins. The application of numerical modeling to solve such problems confirms the effectiveness of computational technologies. The difference between the results of the study ranges from 10 to 16% depending on the airflow rate.

A Two-Contour System with Two Clusters of Different Lengths

A system belonging to the class of dynamical systems such as Buslaev contour networks is investigated. On each of the two closed contours of the system there is a segment, called a cluster, which moves with constant velocity if there are no delays. The contours have two common points called nodes. Delays in the motion of the clusters are due to the fact that two clusters cannot pass through a node simultaneously. The main characteristic we focus on is the average velocity of the clusters with delays taken into account. The contours have the same length, taken to be unity. The nodes divide each contour into parts one of which has length d, and the other, length 1-d. Previously, this system was investigated under the assumption that the clusters have the same length. It turned out that the behavior of the system depends qualitatively on how the directions of motion of the clusters correlate with each other. In this paper we explore the behavior of the system in the case where the clusters differ in length.

Modelling of the Thermodynamic Properties of the Plasma Mixture

Numerical modelling of the thermodynamic properties of plasma mixture is performed using the Thomas – Fermi model with two different approaches. For this purpose, a numerical algorithm, as well as program realization, is developed to solve the Thomas – Fermi equations with quantum-exchange corrections. For the first time a comparison between different methods for taking account of the heterogeneous composition of plasma is made and an algorithm for estimating the corrections for mixtures is developed.