scholarly journals Averaging method and two-sided bounded solutions on the axis of systems with impulsive effects at non-fixed times

2021 ◽  
Vol 104 (4) ◽  
pp. 142-150
Author(s):  
O.N. Stanzhytskyi ◽  
◽  
A.T. Assanova ◽  
M.A. Mukash ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
Averaging Method ◽  
Nonlinear Dynamical Systems ◽  
Asymptotic Properties ◽  
Stable Solution ◽  
Impulsive Systems ◽  
Nonlinear Dynamical ◽  
Systems Of Differential Equations ◽  
Averaged Systems ◽  
Fixed Times

The averaging method, originally offered by Krylov and Bogolyubov for ordinary differential equations, is one of the most widespread and effective methods for the analysis of nonlinear dynamical systems. Further, the averaging method was developed and applied for investigating of various problems. Impulsive systems of differential equations supply as mathematical models of objects that, during their evolution, they are subjected to the action of short-term forces. Many researches have been devoted to non-fixed impulse problems. For these problems, the existence, stability, and other asymptotic properties of solutions were studied and boundary value problems for impulsive systems were considered. Questions of the existence of periodic and almost periodic solutions to impulsive systems also were examined. In this paper, the averaging method is used to study the existence of two-sided solutions bounding on the axis of impulse systems of differential equations with non-fixed times. It is shown that a one-sided, bounding, asymptotically stable solution to the averaged system generates a two-sided solution to the exact system. The closeness of the corresponding solutions of the exact and averaged systems both on finite and infinite time intervals is substantiated by the first and second theorems of N.N. Bogolyubov.

scholarly journals On time transformations for differential equations with state-dependent delay

2014 ◽  
Vol 12 (2) ◽  
Author(s):  
Alexander Rezounenko
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Asymptotic Properties ◽  
The State ◽  
Delay Systems ◽  
Constant Delay ◽  
Systems Of Differential Equations ◽  
State Dependent ◽  
State Dependent Delay ◽  
Time Transformations

AbstractSystems of differential equations with state-dependent delay are considered. The delay dynamically depends on the state, i.e. is governed by an additional differential equation. By applying the time transformations we arrive to constant delay systems and compare the asymptotic properties of the original and transformed systems.

scholarly journals MEASURING AND CONTROLLING THE CHAOTIC MOTION OF PROFITS

2009 ◽  
Vol 19 (11) ◽  
pp. 3593-3604 ◽  
Author(s):  
CRISTINA JANUÁRIO ◽  
CLARA GRÁCIO ◽  
DIANA A. MENDES ◽  
JORGE DUARTE
Keyword(s):  
Dynamical Systems ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Nonlinear Dynamical Systems ◽  
Control Technique ◽  
Stable Steady State ◽  
Chaotic Motions ◽  
Nonlinear Dynamical ◽  
Firm Model ◽  
One Dimensional Maps

The study of economic systems has generated deep interest in exploring the complexity of chaotic motions in economy. Due to important developments in nonlinear dynamics, the last two decades have witnessed strong revival of interest in nonlinear endogenous business chaotic models. The inability to predict the behavior of dynamical systems in the presence of chaos suggests the application of chaos control methods, when we are more interested in obtaining regular behavior. In the present article, we study a specific economic model from the literature. More precisely, a system of three ordinary differential equations gather the variables of profits, reinvestments and financial flow of borrowings in the structure of a firm. Firstly, using results of symbolic dynamics, we characterize the topological entropy and the parameter space ordering of kneading sequences, associated with one-dimensional maps that reproduce significant aspects of the model dynamics. The analysis of the variation of this numerical invariant, in some realistic system parameter region, allows us to quantify and to distinguish different chaotic regimes. Finally, we show that complicated behavior arising from the chaotic firm model can be controlled without changing its original properties and the dynamics can be turned into the desired attracting time periodic motion (a stable steady state or into a regular cycle). The orbit stabilization is illustrated by the application of a feedback control technique initially developed by Romeiras et al. [1992]. This work provides another illustration of how our understanding of economic models can be enhanced by the theoretical and numerical investigation of nonlinear dynamical systems modeled by ordinary differential equations.

scholarly journals On eventual stability of impulsive systems of differential equations

2001 ◽  
Vol 27 (8) ◽  
pp. 485-494
Author(s):  
A. A. Soliman
Keyword(s):  
Differential Equations ◽  
Lipschitz Stability ◽  
Impulsive Systems ◽  
Systems Of Differential Equations ◽  
Eventual Stability

The notions of Lipschitz stability of impulsive systems of differential equations are extended and the notions of eventual stability are introduced. New notions called eventual and eventual Lipschitz stability. We give some criteria and results.

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