Method Of Integral Manifolds Research Articles

DECOMPOSITION OF ENZYME KINETICS SYSTEM WITH FAST AND SLOW VARIABLES IN SUICIDE SUBSTRATE PROBLEM

In this paper an example of cooperative phenomenon, in which the substrate is known as suicide substrate, because it binds to the active enzyme as a substrate, inactivates because enzyme turns it into an inhibitor and provides an irreversible reaction is considered. In this case the substrate commits suicide. The aim of the work is to apply the method of integral manifolds to the reduction of the system of kinetics of suicide substrate. Work describes in detail the rationale for the decomposition algorithm of the enzyme kinetics problem for dynamical systems with fast and slow variables and the construction of integral manifolds for such systems, this article presents the results of applying the above methods to systems of the suicide substrate kinetics and compares solutions for four equations graphically. Comparisons of solutions for four equations are given graphically, the graphs are created using Microsoft Excel.

A model of droplet evaporation: New mathematical developments

A previously developed model for mono-component droplet evaporation is revisited using new mathematical tools for its analysis. The analysis is based on steady-state mass, momentum, and energy balance equations for the vapor and air mixture surrounding a droplet. The previously obtained solution to these equations was based on the assumption that the parameter ε (proportional to the squared ratio of the diffusion coefficient and droplet radius) is equal to zero. The analysis presented in the paper is based on the method of integral manifolds, and it allowed us to present the droplet evaporation rate as the sum of the evaporation rate predicted by the model based on the assumption that ε=0 and the correction proportional to ε. The correction is shown to be particularly important in the case of small water and methanol droplets (diameters less than 5 μm) evaporating in air at low pressure (0.1 atm.). In this case, this correction could reach 35% of the original evaporation rate. In the case of evaporation of relatively large droplets (with radii more than 10 μm) in air at atmospheric and higher pressures, these corrections are shown to be small (less than 10−3 of the evaporation rate predicted by the model based on the assumption that ε=0).

Inertia Optimization Control and Transient Stability Analysis of Wind Power Grid-Connected System

The virtual inertia control effectively makes up for the insufficient inertia caused by the high penetration wind power grid connection. However, it has an impact on the mechanical part of the wind turbine and greatly increases the difficulty of the dynamic stability analysis of the system, resulting in limited engineering practicability. Therefore, the state equation of the wind power grid-connected system is established in this paper, and the influence of virtual inertia control on wind turbine shafting oscillation is analyzed based on the small-signal theory. Secondly, the nonlinear extended disturbance observer is designed as the compensation signal of inertia control to improve its dynamic stability supportability. Based on the integral manifold method, the shafting model of the wind turbine is reduced, and the transient energy function of shafting is established, which provided the basis for the design of the shafting stability controller. Finally, a grid-connected wind power system with high permeability is installed, and the results demonstrate that under the proposed control strategy, the swing stability of power angle is significantly improved, and the wind turbine shafting oscillation is suppressed.

Dynamics of Coupled Van der Pol Oscillators

In this paper, we consider the problem on the synchronization of two or three Van der Pol oscillators in the case where the oscillators are identical and constraints between them are weak. The existence and stability of two types of periodic solutions are studied under the assumption that constraints between the oscillators are dissipative or active. The analysis of the problem is based on methods of the qualitative theory of differential equations, namely, the method of integral manifolds and the method of normal Poincaré–Dulac forms. The problem is reduced to the study of normal forms. We used a version of the Krylov–Bogolyubov algorithm, which allows one to obtain asymptotic formulas for periodic solutions.

Bifurcations of Solutions to Equations with Deviating Spatial Arguments

A periodic boundary-value problem for an equation with deviating spatial argument is considered. This equation describes the phase of a light wave in light resonators with distributed feedback. Optical systems of this type are used in computer technologies and in the study of laser beams. The boundary-value problem was considered for two values of spatial deviations. In the work, bifurcation problems of codimensions 1 and 2 were analyzed by various methods of studying dynamical systems, for example, the method of normal Poincaré–Dulac forms, the method of integral manifolds, and asymptotic formulas. The problem on the stability of certain homogeneous equilibrium states is examined. Asymptotic formulas for spatially inhomogeneous solutions and conditions for their stability are obtained.

Bifurcations of Invariant Tori in Second-Order Quasilinear Evolution Equations in Hilbert Spaces and Scenarios of Transition to Turbulence

In this paper, we consider second-order quasilinear differential equations in a separable Hilbert space for which the well-known Landau–Hopf scenario of transition to turbulence can be realized. We prove increasing of the control parameter leads to the consecutive appearance of invariant tori of increasing dimensions. In this case, the invariant torus of the largest possible dimension appears to be attractive. The results are obtained by using methods of the qualitative theory of dynamical systems with an infinite-dimensional space of initial conditions: the method of integral manifolds, the theory of normal forms, and also asymptotic methods of analysis of dynamical systems.

ДЕКОМПОЗИЦИЯ МОДЕЛЕЙ ЛИНЕЙНЫХ УПРАВЛЯЕМЫХ И НАБЛЮДАЕМЫХДВУХТЕМПОВЫХ СИСТЕМ

A method of integral manifold is applied to study of linear twotempo systems. The use of this method permits us to solve a problem of decomposition of tworate controllable and observable systems. Controllability and observability of these systems are investigated. The application of the method is illustrated on example.

ДЕКОМПОЗИЦИЯ МОДЕЛЕЙ УПРАВЛЯЕМЫХ И НАБЛЮДАЕМЫХДВУХТЕМПОВЫХ СИСТЕМ

A method of integral manifold is applied to study of twotempo linear on fast variables systems. The use of this method permits us to solve a problem of decomposition of tworate controllable and observable systems. Controllability and observability of these systems are investigated. The application of the method is illustrated on example.

Квазиперманентное движение сложной механической системы

The motion of a mechanical system with a variable mass composition and variable geometric configuration is studied, continuously changing in time according to a given deterministic program. A free mechanical system moves relative to the center of mass under the influence of reactive, variational, coriolis and linear dissipative forces so that its center of mass does not move relative to the unchanging basis of the system (the carrier body). The motion of the particles of the changeable part of the system (attached masses – the working body) relative to the carrier is continuous in time and has a shock-free character determined by a given control program. We consider the problem of finding the necessary conditions for the existence of the system under which one of the components of the absolute angular velocity of the unchanging part of the sys-tem is constant (quasipermanent motion). These conditions are interpreted as control relations imposed on the mechanical system, realizing quasipermanent motion. This problem is solved using the integral manifold method of the system of equations of motion of the object under study.

Solving the problem of analytical design of the controller for a stationary discrete system with a small step

In this paper, we study the problem of designing a linear controller for a stationary discrete system with a small step. By the method of integral manifolds, the discrete optimal control problem with a small step is subdivided into two subproblems, the solutions of which are found independently of each other. Algorithms for analytical solutions of subproblems are based on the second Lyapunov method. As a result, a linear controller was designed for a controlled stationary discrete system with a small step, consisting of two subcontrollers, which separately regulate the movements of a system of lower dimension. The results obtained can be used to design an optimal linear digital controller.

Decomposition of a System of Equations of Motion of a Spacecraft with a Considerable Asymmetry in a Rarefied Atmosphere

We consider a system of equations simulating an uncontrolled descent of a spacecraft in a low-density atmosphere. The spacecraft is a solid body with a shape close to a solid of revolution. An important structural feature of the spacecraft is the presence of considerable geometric and aerodynamic asymmetry in its design. It should be noted that the initial system of equations of motion for the spacecraft is essentially nonlinear. In order to enable an effective asymptotic analysis of the evolution of motion of the spacecraft, it is necessary to decompose the initial system of equations of motion of the spacecraft. The aim of this study is to decompose the system of equations of motion of the spacecraft into two subsystems: slow-motion subsystem and fast-motion subsystem. The method of integral manifolds makes it possible to lower the order of the initial system of equations. The resulting slow subsystem can be represented in the form of a single-frequency system of equations with several slow variables. An important applied result of the study is the fact that the subsequent analysis of the evolution of slow variables in a slow subsystem can be performed using known asymptotic methods.

ДЕКОМПОЗИЦИЯ ЗАДАЧ УПРАВЛЯЕМОСТИ И НАБЛЮДАЕМОСТИ ДВУХТЕМПОВЫХ СИСТЕМ

A method of integral manifolds is applied to study of twotempo nonlinear systems. The use of this method permits us to solve a problem of decomposition of two-rate controllable and observable systems. Controllability, observability and stabilisability of these systems is investigated. The application of the method is illustrated on example.

УПРАВЛЯЕМОСТЬ, НАБЛЮДАЕМОСТЬ, СТАБИЛИЗИРУЕМОСТЬ НЕЛИНЕЙНЫХ СИСТЕМ

A method of integral manifolds is applied to study of threetempo nonlinear systems. The use of this method permits us to solve a problem of decomposition of three-rate controllable and observable systems. Local controllability, local observability and stabilisability of these systems is investigated. The application of the method is illustrated on example.

Direct and inverse secondary resonance effects in the spherical motion of an asymmetric rigid body with moving masses

Secondary resonance effects in the spherical motion of a heavy asymmetrical rigid body with moving masses are reviewed in the case close to the Lagrange top. It is known that perturbations acting on the body, particularly a small displacement of the center of mass with respect to axis of symmetry and a small perturbing moment in a connected system of coordinates, lead to non-resonant evolutions of angular velocity. These evolutionary phenomena are called direct secondary resonance effects. They are visible in a dynamic system consisting of an asymmetric rigid body and several masses fixed relative to the body. However, the presence of moving masses connected to a rigid body by springs may lead to the stabilization of the angular velocity of the body. This dynamic phenomenon should be attributed to the inverse secondary resonance effect. The aim of this paper is to study the characteristic direct and inverse secondary resonance effects in a perturbed spherical motion of a rigid body with fixed and moving masses. The method of integral manifolds and the averaging method are used for the asymptotic analysis of the secondary resonance effects. The paper presents the numerical results of modeling the direct and inverse resonance effects.

A possibility of realizing the Landau—Hopf scenario in the problem of tube oscillations under the action of a fluid flow

We consider a boundary-value problem for a quasilinear partial differential equation describing tube oscillations under the action of a fluid flow. We show that the well-known Landau—Hopf scenario of transition to turbulence is realized in the considered evolution boundary-value problem with a suitable choice of the governing parameter. To study the problem, we use the theory of infinite-dimensional dynamical systems. In particular, we use the method of integral manifolds, normal forms, and also asymptotic methods of analysis.

Reduction of the optimal control problem for a magnetoelectric power drive

A linear-quadratic optimal control problem is considered for a singularly perturbed differential system, which describes the dynamics of a magnetoelectric. The optimal control law is constructed as feedback. Using the method of integral manifolds allows us to reduce the order of the matrix differential Riccati equation.

Fractional analysis of induced resonance effects in the spherical motion of a rigid body with flywheels

The evolution of the motion of an asymmetric rigid body with flywheels can be explored by the methods of asymptotic analysis. It is shown, that the effect of small moments resulting from the displacement of the center of mass and of small control moments from the flywheels lead to the emergence of non-linear resonance evolutionary effects. The aim of this work is to investigate the resonance effects and analyze their influence on the change of the fast phase in the spherical motion of the asymmetrical rigid body with flywheels. To explore these resonance effects we applied the method of integral manifolds and the averaging method. The averaged equations showed that resonance effects can lead to the regular precession, to the formation of stationary points, or to the prolonged resonance. In addition, the characteristic features of the influence of secondary resonance effects on the change in the fast phase are revealed.

Resonant Motions of the Statically Stable Lagrange Spinning Top

In this paper, we analyze lower-order resonances during the motion of the Lagrange spinning top with a small mass asymmetry. The conditions for the implementation of long resonance modes obtained by both the averaging method in the linear case and the method of integral manifolds for considerable nutation angles are compared. The motion of a heavy rigid body with an elongated inertia ellipsoid is considered in the vicinity of the lower statically stable equilibrium position. We present a theorem that justifies the application of the method of integral manifolds for considerable nutation angles when a rigid body moves around a fixed point. The resonance capture conditions in the nonlinear case are estimated for nutation angles not exceeding ρ/2. Numerical examples illustrating the effect of nonlinearities on the resonant motion of the Lagrange spinning top are considered.

Efficient decomposition of singularly perturbed systems

The problem of the decomposition of singularly perturbed differential systems by the method of integral manifolds is studied and the application of the method to the problems of enzyme kinetics is considered.

PARAMETRIZATION OF INVARIANT MANIFOLDS OF SLOW MOTIONS

The method of integral manifolds is used to study the multidimensional systems of differential equations. This approach allows to solve an important problem of order reduction of differential systems. If a slow invariant manifold cannot be described explicitly then its parametrization is used for the system order reduction. In this case, either a part of the fast variables, or all fast variables, supplemented by a certain number of slow variables, can play a role of the parameters.