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Vol 229, No 4 (2018)
- Year: 2018
- Articles: 10
- URL: https://journals.rcsi.science/1072-3374/issue/view/14891
Article
On Periodic Solutions of Autonomous Systems
Abstract
We propose a method for the investigation of periodic solutions of autonomous dynamical systems described by ordinary differential equations with phase and integral restrictions. We formulate the general problem of periodic solutions as a boundary-value problem with restrictions. By introducing a fictitious control, we transform the boundary-value problem into a control problem for dynamical systems with phase and integral restrictions. The control problem is solved by reducing it to an integral Fredholm equation of the first kind. We establish necessary and sufficient conditions for the existence of periodic solutions and propose an algorithm for finding periodic solution according to the limit points of the minimizing sequences.
335-353
354-366
367-389
On the Optimal Stabilization of an Integral Manifold
Abstract
By the method of Lyapunov functions, we study the problem of optimal stabilization of an analytically defined integral manifold in the class of stochastic differential equations in the case where random perturbations belong to the class of processes with independent increments.
390-402
403-411
412-424
Weakly Perturbed Fredholm Integral Equations with Degenerate Kernels in Banach Spaces
Abstract
We consider weakly perturbed Fredholm equations with degenerate kernels in Banach spaces and establish conditions for ε = 0 to be a bifurcation point for the solutions of weakly perturbed operator equations X in Banach spaces. A convergent iterative scheme for finding solutions in the form of series \( {\Sigma}_{i=-1}^{+\infty }{\varepsilon}^i{z}_i(t) \) in powers of ε is proposed.
425-438
Explicitly Solvable Models of Redistribution of the Conflict Space
Abstract
We analyze a class of explicitly solvable models for the problems of redistribution of the conflict space between two alternative opponents. The existence of equilibrium state is proved for a complex nonlinear system whose time evolution is generated by the conflict interaction between its components. Explicit formulas are obtained for the limit compromise distributions in terms of the densities of probability measures. We consider a number of specific model examples of the dynamics of redistribution of the conflict territory and formation of an equilibrium (compromise) distribution of the space. We propose an interpretation of the results for the case of social and territorial conflicts.
439-454
On the Behavior of Solutions of Some Systems of Differential Equations Partially Solved with Respect to the Derivatives in the Presence of a Pole
Abstract
We study the existence of analytic solutions for some systems of ordinary differential equations partially solvable with respect to the derivatives. We establish sufficient conditions for the existence of analytic solutions of the Cauchy problem in the presence of a pole. An estimate for these solutions is obtained in a certain domain and the problem of the number of solutions is investigated.
455-469
Vibrations of Elastic Shells of Revolution Partially Filled with Ideal Liquid
Abstract
We propose an algorithm for the numerical analyses of vibrations of elastic shells of revolution partially filled with ideal incompressible liquids. In the solution of this problem, the wave motions of liquid on its free surface are taken into account. The solution of the problem of hydroelasticity is based on the application of the method of decomposition of the domain of integration of equations of the theory of shells with the use of the variational statement of the problem and on the approximate construction of the operator inverse to the operator of the hydrodynamic part of the problem. We construct a generalized functional of displacements of the shell for which the role of conditions of matching for the solutions in different subdomains is played by the natural boundary conditions. The obtained numerical results are compared with the existing exact solutions of the analyzed problem for a shell in the form of a straight circular cylinder.
470-486