scholarly journals Application of M-Matrices for the Study of Mathematical Models of Living Systems

2018 ◽  
Vol 13 (1) ◽  
pp. 208-237 ◽  
Author(s):  
N.V. Pertsev ◽  
B.Yu. Pichugin ◽  
A.N. Pichugina
Keyword(s):  
Differential Equations ◽  
Stability Problem ◽  
Distributed Delay ◽  
Linearization Method ◽  
Living Systems ◽  
High Dimensional ◽  
Linear Differential ◽  
The Stability ◽  
Several Delays ◽  
The Right

Some results are presented of application of M-matrices to the study the stability problem of the equilibriums of differential equations used in models of living systems. The models studied are described by differential equations with several delays, including distributed delay, and by high-dimensional systems of differential equations. To study the stability of the equilibriums the linearization method is used. Emerging systems of linear differential equations have a specific structure of the right-hand parts, which allows to effectively use the properties of M-matrices. As examples, the results of studies of models arising in immunology, epidemiology and ecology are presented.

scholarly journals On the Lyapunov functionals method in the stability problem of Volterra integro-differential equations

2018 ◽  
Vol 20 (3) ◽  
pp. 260-272
Author(s):  
A. S. Andreev ◽  
O. A. Peregudova
Keyword(s):  
Differential Equations ◽  
Stability Problem ◽  
Infinite Delay ◽  
Zero Solution ◽  
Model Systems ◽  
Lyapunov Functionals ◽  
Structure Equations ◽  
Positive Limit Set ◽  
The Stability ◽  
The Right

In this paper, we consider the problem of applying the method of Lyapunov functionals to investigate the stability of non-linear integro-differential equations, the right-hand side of which is the sum of the components of the instantaneous action and also ones with a finite and infinite delay. The relevance of the problem is the widespread use of such complicated in structure equations in modeling the controllers using integral regulators for mechanical systems, as well as biological, physical and other processes. We develop the Lyapunov functionals method in the direction of revealing the limiting properties of solutions by means of Lyapunov functionals with a semi-definite derivative. We proved the theorems on the quasi-invariance of a positive limit set of bounded solution as well as ones on the asymptotic stability of the zero solution including a uniform one. The results are achieved by constructing a new structure of the topological dynamics of the equations under study. The theorems proved are applied in solving the stability problem of two model systems which are generalizations of a number of known models of natural science and technology.

scholarly journals On the Lyapunov functionals method in the stability problem of Volterra integro-differential equations

2018 ◽  
Vol 20 (3) ◽  
pp. 260-272
Author(s):  
Aleksandr S. Andreev ◽  
Olga A. Peregudova
Keyword(s):  
Differential Equations ◽  
Stability Problem ◽  
Infinite Delay ◽  
Zero Solution ◽  
Model Systems ◽  
Lyapunov Functionals ◽  
Structure Equations ◽  
Positive Limit Set ◽  
The Stability ◽  
The Right

In this paper, we consider the problem of applying the method of Lyapunov functionals to investigate the stability of non-linear integro-differential equations, the right-hand side of which is the sum of the components of the instantaneous action and also ones with a finite and infinite delay. The relevance of the problem is the widespread use of such complicated in structure equations in modeling the controllers using integral regulators for mechanical systems, as well as biological, physical and other processes. We develop the Lyapunov functionals method in the direction of revealing the limiting properties of solutions by means of Lyapunov functionals with a semi-definite derivative. We proved the theorems on the quasi-invariance of a positive limit set of bounded solution as well as ones on the asymptotic stability of the zero solution including a uniform one. The results are achieved by constructing a new structure of the topological dynamics of the equations under study. The theorems proved are applied in solving the stability problem of two model systems which are generalizations of a number of known models of natural science and technology.

scholarly journals Stability of delay differential equations in the sense of Ulam on unbounded intervals

2019 ◽  
Vol 9 (2) ◽  
pp. 125-131 ◽  
Author(s):  
Süleyman Öğrekçi
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Stability Problem ◽  
The Stability ◽  
Delay Differential ◽  
Unbounded Intervals

In this paper, we consider the stability problem of delay differential equations in the sense of Hyers-Ulam-Rassias. Recently this problem has been solved for bounded intervals, our result extends and improve the literature by obtaining stability in unbounded intervals. An illustrative example is also given to compare these results and visualize the improvement.

On the Stability Problem of Differential Equations in the Sense of Ulam

2019 ◽  
Vol 75 (1) ◽  
Author(s):  
Yasemin Başcı ◽  
Adil Mısır ◽  
Süleyman Öğrekçi
Keyword(s):  
Differential Equations ◽  
Stability Problem ◽  
The Stability
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