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A dynamic process modeled by ordinary differential equations is considered. If a nonautonomous system of ordinary differential equations has a general solution in a certain area, than the system can be simplified by nonautonomous substitution of variables: right parts turn to zeroes. Right parts of an autonomous system of ordinary differential equations in the neighborhood of nonsingular points can be linearized. A separable system where the right part contains linear combination of autonomous vector fields and factors are functions of independent variable is considered. If the fields commute than they can be linearized by general substitution of variables.

The article discusses the emergence of chaotic attractors in the system of three ordinary differential equations arising in the theory of «reaction-diffusion» systems. The dynamics of the corresponding one- and two-dimensional maps and Lyapunov exponents of such attractors are studied. It is shown that the transition to chaos is in accordance with a non-traditional scenario of repeated birth and disappearance of chaotic regimes, which had been previously studied for one-dimensional maps with a sharp apex and a quadratic minimum. Some characteristic features of the system — zones of bistability and hyperbolicity, the crisis of chaotic attractors — are studied by means of numerical analysis.

We construct necessary and sufficient conditions for the existence of solution of seminonlinear matrix boundary value problem for a parametric excitation system of ordinary differential equations. The convergent iteration algorithms for the construction of the solutions of the semi-nonlinear matrix boundary value problem for a parametric excitation system differential equations in the critical case have been found. Using the convergent iteration algorithms we expand solution of seminonlinear periodical boundary value problem for a parametric excitation Riccati type equation in the neighborhood of the generating solution. Estimates for the value of residual of the solutions of the seminonlinear periodical boundary value problem for a parametric excitation Riccati type equation are found.

A hierarchical method of mathematical and computer modeling of interval-stochastic thermal processes in complex electronic systems for various purposes is developed. The developed concept of hierarchical structuring reflects both the constructive hierarchy of a complex electronic system and the hierarchy of mathematical models of heat exchange processes. Thermal processes that take into account various physical phenomena in complex electronic systems are described by systems of stochastic, unsteady, and nonlinear partial differential equations and, therefore, their computer simulation encounters considerable computational difficulties even with the use of supercomputers. The hierarchical method avoids these difficulties. The hierarchical structure of the electronic system design, in general, is characterized by five levels: Level 1 — the active elements of the ES (microcircuits, electro-radio-elements); Level 2 — electronic module; Level 3 — a panel that combines a variety of electronic modules; Level 4 — a block of panels; Level 5 — stand installed in a stationary or mobile room. The hierarchy of models and modeling of stochastic thermal processes is constructed in the reverse order of the hierarchical structure of the electronic system design, while the modeling of interval-stochastic thermal processes is carried out by obtaining equations for statistical measures. The hierarchical method developed in the article allows to take into account the principal features of thermal processes, such as the stochastic nature of thermal, electrical and design factors in the production, assembly and installation of electronic systems, stochastic scatter of operating conditions and the environment, non-linear temperature dependencies of heat exchange factors, unsteady nature of thermal processes. The equations obtained in the article for statistical measures of stochastic thermal processes are a system of 14 non-stationary nonlinear differential equations of the first order in ordinary derivatives, whose solution is easily implemented on modern computers by existing numerical methods. The results of applying the method for computer simulation of stochastic thermal processes in electron systems are considered. The hierarchical method is applied in practice for the thermal design of real electronic systems and the creation of modern competitive devices.

A nonlinear oscillatory system described by ordinary differential equations with variable coefficients is considered, in which terms that are linearly dependent on coordinates, velocities and accelerations are explicitly distinguished; nonlinear terms are written as implicit functions of these variables. For the numerical solution of the initial problem described by such a system of differential equations, the one-step Galerkin method is used. At the integration step, unknown functions are represented as a sum of linear functions satisfying the initial conditions and several given correction functions in the form of polynomials of the second and higher degrees with unknown coefficients. The differential equations at the step are satisfied approximately by the Galerkin method on a system of corrective functions. Algebraic equations with nonlinear terms are obtained, which are solved by iteration at each step. From the solution at the end of each step, the initial conditions for the next step are determined.

The corrective functions are taken the same for all steps. In general, 4 or 5 correction functions are used for calculations over long time intervals: in the first set — basic power functions from the 2nd to the 4th or 5th degrees; in the second set — orthogonal power polynomials formed from basic functions; in the third set — special linear-independent polynomials with finite conditions that simplify the “docking” of solutions in the following steps.

Using two examples of calculating nonlinear oscillations of systems with one and two degrees of freedom, numerical studies of the accuracy of the numerical solution of initial problems at various time intervals using the Galerkin method using the specified sets of power-law correction functions are performed. The results obtained by the Galerkin method and the Adams and Runge –Kutta methods of the fourth order are compared. It is shown that the Galerkin method can obtain reliable results at significantly longer time intervals than the Adams and Runge – Kutta methods.

In the last decades, the development of methods for increasing the efficiency of hydrocarbon extraction in fields with unconventional reserves containing large amounts of gas condensate is of great importance. This makes important the development of methods of mathematical modeling that realistically describe physical processes in a gas-condensate mixture in a porous medium.

In the paper, a mathematical model which describes the dynamics of the pressure, velocity and concentration of the components of a two-component two-phase mixture entering a laboratory model of plast filled with a porous substance with known physicochemical properties is considered. The mathematical model is based on a system of nonlinear spatially one-dimensional partial differential equations with the corresponding initial and boundary conditions. Laboratory experiments show that during a finite time the system stabilizes, what gives a basis to proceed to the stationary formulation of the problem.

The numerical solution of the formulated system of ordinary differential equations is realized in the Maple environment on the basis of the Runge–Kutta procedure. It is shown that the physical parameters of the gascondensate mixture, which characterize the modeled system in the stabilization regime, obtained on this basis, are in good agreement with the available experimental data. This confirms the correctness of the chosen approach and the validity of its further application and development for computer modeling of physical processes in gas-condensate mixtures in a porous medium. The paper presents a mathematical formulation of the system of partial differential equations and of respective system stationary equations, describes the numerical approach, and discusses the numerical results obtained in comparison with experimental data.

In this paper we consider the problem of modeling of flow in the far regions of an axisymmetric momentumless turbulent wake that occurs when a body is flowed. The main features of the studied flow are its almost shear-free nature and its proximity to the self-similar regime, which is confirmed by the available experimental data. A semi-empirical $e-\varepsilon$ model of turbulence based on the Rodi algebraic model of Reynolds stresses is invoked to describe a flow in the far wake. Using group-theoretic analysis, similarity reduction of the invoked model of turbulence to a system of ordinary differential equations is obtained. An approximate self-similar solution of the second kind of the corresponding boundary value problem for the reduced system of equations was constructed using a method based on matching the truncations of asymptotic expansions of its solution found in neighborhoods of the boundary points. The involved approach also allowed for the extraction of an approximate value of the self-similarity exponent and other unknown parameters of the problem during the procedure of matching of the asymptotic expansion. Comparison of the approximate solution of the given boundary value problem constructed in the form of Puiseux series, which includes 16 approximating terms, with the numerical solution found using the shooting method, showed that the maximum relative error between the solutions does not exceed 2%. The approximate value of the self-similarity exponent of the problem established in this work is in a good agreement with the exact value obtained earlier by other authors as a result of the analytical solution of the corresponding nonlinear eigenvalue problem. The approximate solution was compared with the available experimental data on the axisymmetric momentumless turbulent wake obtained at the Lavrentyev Institute of Hydrodynamics of the Siberian Branch of the Russian Academy of Sciences, and a satisfactory agreement was found. The approximate formulas developed in this work have practical significance and can be used for quick estimation of the values of hydrodynamic quantities in the far regions of the axisymmetric momentumless turbulent wake. The developed method for constructing approximate solutions to the boundary value problems of free turbulence can also be used in other applied fields that are not related to turbulence theory.

Rugose spiraling whitefly (RSW) is one of the major pests which affects the coconut trees. It feeds on the tree by sucking up the water content as well as the essential nutrients from leaves. It also forms sooty mold in leaves due to which the process of photosynthesis is inhibited. Biocontrol of pest is harmless for trees and crops. The experimental results in literature reveal that Pseudomallada astur is a potential predator for this pest. We investigate the dynamics of predator, Pseudomallada astur’s interaction with rugose spiralling whitefly, Aleurodicus rugioperculatus in coconut trees using a mathematical model. In this system of ordinary differential equation, the pest-predator interaction is modeled using Holling type III functional response. The parametric values are calculated from the experimental results and are tabulated. An approximate analytical solution for the system has been derived. The homotopy analysis method proves to be a suitable method for creating solutions that are valid even for moderate to large parameter values, hence we employ the same to solve this nonlinear model. The $\hbar$-curves, which give the admissible region of $\hbar$, are provided to validate the region of convergence. We have derived the approximate solution at fifth order and stopped at this order since we obtain a more approximate solution in this iteration. Numerical simulation is obtained through MATLAB. The analytical results are compared with numerical simulation and are found to be in good agreement. The biological interpretation of figures implies that the use of a predator reduces the whitefly’s growth to a greater extent.

The viscoelastic fluid flow model across a porous medium has captivated the interest of many contemporary researchers due to its industrial and technical uses, such as food processing, paper and textile coating, packed bed reactors, the cooling effect of transpiration and the dispersion of pollutants through aquifers. This article focuses on the influence of variable viscosity and viscoelasticity on the magnetohydrodynamic oscillatory flow of second-order fluid through thermally radiating wavy walls. A mathematical model for this fluid flow, including governing equations and boundary conditions, is developed using the usual Boussinesq approximation. The governing equations are transformed into a system of nonlinear ordinary differential equations using non-similarity transformations. The numerical results obtained by applying finite-difference code based on the Lobatto IIIa formula generated by bvp4c solver are compared to the semi-analytical solutions for the velocity, temperature and concentration profiles obtained using the homotopy perturbation method (HPM). The effect of flow parameters on velocity, temperature, concentration profiles, skin friction coefficient, heat and mass transfer rate, and skin friction coefficient is examined and illustrated graphically. The physical parameters governing the fluid flow profoundly affected the resultant flow profiles except in a few cases. By using the slope linear regression method, the importance of considering the viscosity variation parameter and its interaction with the Lorentz force in determining the velocity behavior of the viscoelastic fluid model is highlighted. The percentage increase in the velocity profile of the viscoelastic model has been calculated for different ranges of viscosity variation parameters. Finally, the results are validated numerically for the skin friction coefficient and Nusselt number profiles.