MULTIPERIODIC SOLUTIONS OF SECOND-ORDER SYSTEMS WITH DIFFERENTIATION OPERATOR ON THE LYAPUNOV'S VECTOR FIELD

2020 ◽  
Vol 69 (1) ◽  
pp. 155-163
Author(s):  
B.Zh. Omarova ◽  
Keyword(s):  
Integral Representation ◽  
Vector Fields ◽  
Homogeneous System ◽  
Differentiation Operator ◽  
Second Order ◽  
Inhomogeneous System ◽  
Representation Of Solutions ◽  
Real Analytic ◽  
Time Variables ◽  
Second Order Systems

The problem of the existence and integral representation of a unique multiperiodic solution of a second-order linear inhomogeneous system with constant coefficients and a differentiation operator on the direction of the main diagonal of the space of time variables and of the vector fields in the form of Lyapunov systems with respect to space variables were considered. The multiperiodicity of zeros of this operator and the condition for the absence of a nonzero multiperiodic and real-analytic solution of the homogeneous system corresponding to the given system are established. An integral representation of solutions of an inhomogeneous linear autonomous system that multiperiodic in time variables and realanalytic in space variables is obtained. The existence theorem of a unique multiperiodic in time variables and real-analytic in space variables solutions of the original linear system in terms of the Green's function under sufficiently general conditions is substantiated.

scholarly journals INTEGRAL REPRESENTATION OF SOLUTIONS OF AN ORDINARY DIFFERENTIAL EQUATION AND THE LOEWNER– KUFAREV EQUATION

2020 ◽  
pp. 28-39
Author(s):  
Olga V. ZADOROZHNAYA ◽  
◽  
Vladimir K. KOCHETKOV ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Equations ◽  
Integral Representation ◽  
Univalent Functions ◽  
Integral Form ◽  
Second Order ◽  
Partial Differential ◽  
Representation Of Solutions ◽  
Order Part

The article presents a method of integral representation of solutions of ordinary differential equations and partial differential equations with a polynomial right-hand side part, which is an alternative to the construction of solutions of differential equations in the form of different series. The method is based on the introduction of additional analytical functions establishing the equation of connection between the introduced functions and the constituent components of the original differential equation. The implementation of the coupling equations contributes to the representation of solutions of the differential equation in the integral form, which allows solving some problems of mathematics and mathematical physics. The first part of the article describes the coupling equation for an ordinary differential equation of the first order with a special polynomial part of a higher order. Here, the integral representation of the solution of a differential equation with a second-order polynomial part is indicated in detail. In the second part of the paper, we consider the integral representation of the solution of a partial differential equation with the polynomial second-order part of the Loewner–Kufarev equation, which is an equation for univalent functions.

scholarly journals The Lie point symmetry generators admitted by systems of linear differential equations

2014 ◽  
Vol 470 (2166) ◽  
pp. 20130779 ◽  
Author(s):  
Robert J. Gray
Keyword(s):  
Differential Equations ◽  
Homogeneous System ◽  
Second Order ◽  
Linear Differential Equations ◽  
Degree Zero ◽  
Inhomogeneous System ◽  
First Order ◽  
Dependent Variables ◽  
Linear Differential ◽  
Symmetry Generators

Computing the Lie point symmetries of systems of linear differential equations can be prohibitively difficult. For homogeneous systems in Kovalevskaya form of order two or higher, this paper proves the existence of a basis of infinitesimal generators (as determined by Lie's algorithm) whose characteristic forms are homogeneous in the dependent variables of degree zero, one or two. Suppose Lie's algorithm yields a characteristic form of degree two; in this case, the system is second order. If it contains only ordinary differential equations (ODEs), its general solution is constructed from those of a given first-order linear homogeneous system, of the same dimension, and a second-order linear scalar equation. Otherwise, coordinates are given in which its dimension (necessarily two or higher now) is lowered by one, leaving an inhomogeneous system of parametrized ODEs. Its homogeneous part is solved as in the previous case.

scholarly journals ℋ∞ Control Tunning to Guarantee the Output Performance of LTI Second-Order Systems

2020 ◽  
Vol 53 (2) ◽  
pp. 4611-4616
Author(s):  
Ramón I. Verdés ◽  
Luis T. Aguilar ◽  
Yury Orlov
Keyword(s):  
Second Order ◽  
Output Performance ◽  
Second Order Systems

scholarly journals Search Patterns Based on Trajectories Extracted from the Response of Second-Order Systems

2021 ◽  
Vol 11 (8) ◽  
pp. 3430
Author(s):  
Erik Cuevas ◽  
Héctor Becerra ◽  
Héctor Escobar ◽  
Alberto Luque-Chang ◽  
Marco Pérez ◽  
...  
Keyword(s):  
Convergence Rates ◽  
Optimization Problems ◽  
Search Space ◽  
Second Order ◽  
Order System ◽  
Search Patterns ◽  
Multiple Behaviors ◽  
Complex Optimization ◽  
Second Order Systems ◽  
Order Algorithm

Recently, several new metaheuristic schemes have been introduced in the literature. Although all these approaches consider very different phenomena as metaphors, the search patterns used to explore the search space are very similar. On the other hand, second-order systems are models that present different temporal behaviors depending on the value of their parameters. Such temporal behaviors can be conceived as search patterns with multiple behaviors and simple configurations. In this paper, a set of new search patterns are introduced to explore the search space efficiently. They emulate the response of a second-order system. The proposed set of search patterns have been integrated as a complete search strategy, called Second-Order Algorithm (SOA), to obtain the global solution of complex optimization problems. To analyze the performance of the proposed scheme, it has been compared in a set of representative optimization problems, including multimodal, unimodal, and hybrid benchmark formulations. Numerical results demonstrate that the proposed SOA method exhibits remarkable performance in terms of accuracy and high convergence rates.

An Asymptotical Stability in Variational Sense for Second Order Systems

2010 ◽  
Vol 10 (1) ◽  
Author(s):  
Dariusz Idczak ◽  
Stanisław Walczak
Keyword(s):  
Sobolev Space ◽  
Differential System ◽  
Sufficient Conditions ◽  
Zero Solution ◽  
Second Order ◽  
Functional Parameter ◽  
Asymptotical Stability ◽  
Parameter Control ◽  
Initial Condition ◽  
Second Order Systems

AbstractIn this paper, a new, variational concept of asymptotical stability of zero solution to an ordinary differential system of the second order, considered in Sobolev space, is presented. Sufficient conditions for an asymptotical stability in a variational sense with respect to initial condition and functional parameter (control) are given. Relation to the classical asymptotical stability is illustrated.

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