76 results on '"Felix Sadyrbaev"'
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2. On the Period-Amplitude Relation by Reduction to Liénard Quadratic Equation
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Svetlana Atslega and Felix Sadyrbaev
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General Mathematics - Abstract
We apply Sabatini’s transformation for the study of a class of nonlinear oscillators, dependent on quadratic terms. As a result, an initial equation is reduced to Newtonian form, for which in a standard way the period-amplitude relation can be established.
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- 2023
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3. Comparative Analysis of Models of Gene and Neural Networks
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Inna Samuilik, Felix Sadyrbaev, and Diana Ogorelova
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General Medicine ,General Chemistry - Abstract
In the language of mathematics, the method of cognition of the surrounding world in which the description of the object is carried out the name is mathematical modeling. The study of the model is carried out using certain mathematical methods. The systems of the ordinary differential equations modeling artificial neuronal networks and the systems modeling the gene regulatory networks are considered. The one system consists of a sigmoidal function which depends on linear combinations of the arguments minus the linear part. The other system consists of a sigmoidal function which depends on the hyperbolic tangent function. The linear combinations and hyperbolic tangent functions of the arguments are described by one regulatory matrix. For the three-dimensional cases, two types of matrices are considered and the behavior of the solutions of the system is analyzed. The attracting sets are constructed for several cases. Illustrative examples are provided. The list of references consists of 19 items.
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- 2023
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4. On a three-dimensional neural network model
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Diana Ogorelova and Felix Sadyrbaev
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Materials Science (miscellaneous) ,Business and International Management ,Industrial and Manufacturing Engineering - Abstract
The dynamics of a model of neural networks is studied. It is shown that the dynamical model of a three-dimensional neural network can have several attractors. These attractors can be in the form of stable equilibria and stable limit cycles. In particular, the model in question can have two three-dimensional limit cycles.
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- 2022
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5. Genetic engineering – construction of a network of arbitrary dimension with periodic attractor
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Inna Samuilik and Felix Sadyrbaev
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Materials Science (miscellaneous) ,Business and International Management ,Industrial and Manufacturing Engineering - Abstract
It is shown, how to construct a system of ordinary differential equations of arbitrary order, which has the periodic attractor and models some genetic network of arbitrary size. The construction is carried out by combining of multiple systems of lower dimensions with known periodic attractors. In our example the six-dimensional system is constructed, using two identical three-dimensional systems, which have stable periodic solutions.
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- 2022
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6. On a Dynamical Model of Genetic Networks
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Inna Samuilik and Felix Sadyrbaev
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Economics and Econometrics ,Business and International Management ,Finance - Abstract
e consider the model of a four-dimensional gene regulatory network (GRN in short). This model consists of ordinary differential equations of a special kind, where the nonlinearity is represented by a sigmoidal function and the linear part is present also. The evolution of GRN is described by the solution vector X(t), depending on time. We describe the changes that the system undergoes if the entries of the regulatory matrix are perturbed in some way. The sensitive dependence of solutions on the initial data is revealed by the analysis using the Lyapunov exponents.
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- 2022
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7. On Targeted Control over Trajectories of Dynamical Systems Arising in Models of Complex Networks
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Diana Ogorelova, Felix Sadyrbaev, and Inna Samuilik
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General Mathematics ,network control ,attracting sets ,dynamical system ,phase portrait ,gene regulatory networks ,artificial neural systems ,Computer Science (miscellaneous) ,Engineering (miscellaneous) - Abstract
The question of targeted control over trajectories of systems of differential equations encountered in the theory of genetic and neural networks is considered. Examples are given of transferring trajectories corresponding to network states from the basin of attraction of one attractor to the basin of attraction of the target attractor. This article considers a system of ordinary differential equations that arises in the theory of gene networks. Each trajectory describes the current and future states of the network. The question of the possibility of reorienting a given trajectory from the initial state to the assigned attractor is considered. This implies an only partial control of the network. The difficulty lies in the selection of parameters, the change of which leads to the goal. Similar problems arise when modeling the response of the body’s gene networks to serious diseases (e.g., leukemia). Solving such problems is the first step in the process of applying mathematical methods in medicine and pharmacology.
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- 2023
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8. On a System Without Critical Points Arising in Heat Conductivity Theory
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Inna Samuilik and Felix Sadyrbaev
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General Physics and Astronomy - Abstract
A two-point boundary value problem for the second order nonlinear ordinary differential equation, arising in the heat conductivity theory, is considered. Multiplicity and existence results are established for this problem, where the equation contains two parameters.
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- 2022
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9. Examples of Periodic Biological Oscillators: Transition to a Six-dimensional System
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Inna Samuilik, Felix Sadyrbaev, and Valentin Sengileyev
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General Computer Science ,General Engineering - Abstract
We study a genetic model (including gene regulatory networks) consisting of a system of several ordinary differential equations. This system contains a number of parameters and depends on the regulatory matrix that describes the interactions in this multicomponent network. The question of the attracting sets of this system, which depending on the parameters and elements of the regulatory matrix, isconsidered. The consideration is mainly geometric, which makes it possible to identify and classify possible network interactions. The system of differential equations contains a sigmoidal function, which allows taking into account the peculiarities of the network response to external influences. As a sigmoidal function, a logistic function is chosen, which is convenient for computer analysis. The question of constructing attractors in a system of arbitrary dimension is considered by constructing a block regulatory matrix, the blocks of which correspond to systems of lower dimension and have been studied earlier. The method is demonstrated with an example of a three-dimensional system, which is used to construct a system of dimensions twice as large. The presentation is provided with illustrations obtained as a result of computer calculations, and allowing, without going into details, to understand the formulation of the issue and ways to solve the problems that arise in this case.
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- 2022
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10. On nehari solutions
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null Felix Sadyrbaev
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Mathematics::Complex Variables - Abstract
The Nehari solutions are solutions of a superlinear second order ordinary differential equation that possess remarkable properties. This property is the minimization of a certain functional associated with the equation. In this paper we review these properties, recall some problems about the Nehari solutions and solve some unsolved ones.
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- 2022
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11. Remarks on Inhibition
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Felix Sadyrbaev and Valentin Sengileyev
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In networks, which arise in multiple applications, the inhibitory connection between elements occur. These networks appear in genetic regulation, neuronal interactions, telecommunication designs, electronic devices. Mathematical modelling of such networks is an efficient tool for their studying. We consider the specific mathematical model, which uses systems of ordinary differential equations of a special form. The solution vector X(t) describes the current state of a network. Future states are dependent on the structure of the phase space and emerging attractive sets. Attractors, their properties and locations depend on the parameters in a system. Some of these parameters are adjustable. The important problem of managing and control over the system, is considered also.
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- 2022
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12. Mathematical Modeling of Three - Dimensional Genetic Regulatory Networks Using Logistic and Gompertz Functions
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Inna Samuilik, Felix Sadyrbaev, and Diana Ogorelova
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Artificial Intelligence ,Control and Systems Engineering ,General Mathematics - Abstract
Mathematical modeling is a method of cognition of the surrounding world in which the description of the object is carried out in the language of mathematics, and the study of the model is performed using certain mathematical methods. Mathematical models based on ordinary differential equations (ODE) are used in the study of networks of different kinds, including the study of genetic regulatory networks (GRN). The use of ODE makes it possible to predict the evolution of GRN in time. Nonlinearity in these models is included in the form of a sigmoidal function. There are many of them, and in the literature, there are models that use different sigmoidal functions. The article discusses the models that use the logistic function and Gompertz function. The comparison of the results, related to three-dimensional networks, has been made. The text is accompanied by examples and illustrations.
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- 2022
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13. Linear Stability of a Combined Convective Flow in an Annulus
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Armands Gritsans, Valentina Koliskina, Andrei Kolyshkin, and Felix Sadyrbaev
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Fluid Flow and Transfer Processes ,Mechanical Engineering ,Condensed Matter Physics ,linear stability ,convective flow ,bifurcation analysis ,collocation method - Abstract
Linear stability analysis of a combined convective flow in an annulus is performed in the paper. The base flow is generated by two factors: (a) different constant wall temperatures and (b) heat release as a result of a chemical reaction that takes place in the fluid. The nonlinear boundary value problem for the distribution of the base flow temperature is analyzed using bifurcation analysis. The linear stability problem is solved numerically using a collocation method. Two separate cases are considered: Case 1 (non-zero different constant wall temperatures) and Case 2 (zero wall temperatures). Numerical calculations show that the development of instability is different for Cases 1 and 2. Multiple minima on the marginal stability curves are found for Case 1 as the Prandtl number increases. Concurrence between local minima leads to the selection of the global minimum in such a way that a finite jump in the value of the wave number is observed for some values of the Prandtl number. All marginal stability curves for Case 2 have one minimum in the range of the Prandtl numbers considered. The corresponding critical values of the Grashof number decrease monotonically as the Prandtl number grows.
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- 2023
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14. Modeling the evolution of complex networks arising in applications
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Felix Sadyrbaev
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- 2023
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15. On trajectories of a system modeling evolution of genetic networks
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Inna Samuilik and Felix Sadyrbaev
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Computational Mathematics ,Applied Mathematics ,Modeling and Simulation ,General Medicine ,General Agricultural and Biological Sciences - Abstract
A system of ordinary differential equations is considered, which arises in the modeling of genetic networks and artificial neural networks. Any point in phase space corresponds to a state of a network. Trajectories, which start at some initial point, represent future states. Any trajectory tends to an attractor, which can be a stable equilibrium, limit cycle or something else. It is of practical importance to answer the question of whether a trajectory exists which connects two points, or two regions of phase space. Some classical results in the theory of boundary value problems can provide an answer. Some problems cannot be answered and require the elaboration of new approaches. We consider both the classical approach and specific tasks which are related to the features of the system and the modeling object.
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- 2022
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16. Modelling Three Dimensional Gene Regulatory Networks
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Inna Samuilik and Felix Sadyrbaev
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Artificial Intelligence ,Control and Systems Engineering ,Quantitative Biology::Molecular Networks ,General Mathematics ,Quantitative Biology::Genomics - Abstract
We consider the three-dimensional gene regulatory network (GRN in short). This model consists of ordinary differential equations of a special kind, where the nonlinearity is represented by a sigmoidal function and the linear part is present also. The evolution of GRN is described by the solution vector X(t), depending on time. We describe the changes that system undergoes if the entries of the regulatory matrix are perturbed in some way.
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- 2021
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17. Mathematical Modelling of Leukemia Treatment
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Inna Samuilika and Felix Sadyrbaev
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Oncology ,medicine.medical_specialty ,Leukemia ,General Computer Science ,Computer science ,Internal medicine ,medicine ,medicine.disease ,Research data - Abstract
Leukemia is a cancer that can be treated in a variety of ways: chemotherapy, radiation therapy and stem cell transplant. Recovery rates for this disease are relatively high, the treatment itself has a painful effect on the body and is accompanied by numerous side effects that can persist years after the patient is cured. For this reason, efforts are underway worldwide to develop more selective therapies that will only affect leukemia cells and not healthy cells. Knowledge of developmental GRN is yet scarce, and it is early for a systematic comparative effort. We consider mathematical model of genetic regulatory networks. This model consists of a nonlinear system of ordinary differential equations. We describe the changes that system undergoes if the entries of the regulatory matrix are perturbed in some way. We discuss, how attractors for high-dimensional systems can be constructed, using known attractors of low-dimensional systems. Examples and visualizations are provided.
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- 2021
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18. On Modelling of Genetic Regulatory Net Works
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Inna Samuilik, Valentin Sengileyev, and Felix Sadyrbaev
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0303 health sciences ,Current (mathematics) ,Differential equation ,Computer science ,Ode ,Net (mathematics) ,01 natural sciences ,Value of time ,010305 fluids & plasmas ,03 medical and health sciences ,Nonlinear system ,Ordinary differential equation ,0103 physical sciences ,Attractor ,Applied mathematics ,Electrical and Electronic Engineering ,030304 developmental biology - Abstract
We consider mathematical model of genetic regulatory networks (GRN). This model consists of a nonlinear system of ordinary differential equations. The vector of solutions X(t) is interpreted as a current state of a network for a given value of time t: Evolution of a network and future states depend heavily on attractors of system of ODE. We discuss this issue for low dimensional networks and show how the results can be applied for the study of large size networks. Examples and visualizations are provided
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- 2021
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19. Mathematical modelling of nonlinear dynamic systems
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Inna Samuilik, Felix Sadyrbaev, and Svetlana Atslega
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- 2022
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20. A two-point boundary value problem for third order asymptotically linear systems
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Felix Sadyrbaev and Armands Gritsans
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Asymptotically linear ,index of isolated singular point ,Applied Mathematics ,Physics::Classical Physics ,spectrum ,Point boundary ,Third order ,asymptotically linear vector field ,boundary value problem ,Physics::Space Physics ,QA1-939 ,Applied mathematics ,Value (mathematics) ,Mathematics - Abstract
We consider a third order system ${\boldsymbol x}'''={\boldsymbol f}({\boldsymbol x})$ with the two-point boundary conditions ${\boldsymbol x}(0)={\boldsymbol 0}$, ${\boldsymbol x}'(0)={\boldsymbol 0}$, ${\boldsymbol x}(1)={\boldsymbol 0}$, where ${\boldsymbol f}({\boldsymbol 0})={\boldsymbol 0}$ and the vector field ${\boldsymbol f}\in C^1(\mathbb{R}^n,\mathbb{R}^n)$ is asymptotically linear with the derivative at infinity ${\boldsymbol f}'(\infty)$. We introduce an asymptotically linear vector field ${\boldsymbol \phi}$ such that its singular points (zeros) are in a one-to-one correspondence with the solutions of the boundary value problem. Using the vector field rotation theory, we prove that under the non-resonance conditions for the linearized problems at zero and infinity the indices of ${\boldsymbol \phi}$ at zero and infinity can be expressed in the terms of the eigenvalues of the matrices ${\boldsymbol f}'({\boldsymbol 0})$ and ${\boldsymbol f}'(\infty)$, respectively. This proof constitutes an essential part of our article. If these indices are different, then standard arguments of the vector field rotation theory ensure the existence of at least one nontrivial solution to the boundary value problem. At the end of the article we consider the consequences for the scalar case.
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- 2019
21. On modelling of complex networks
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Svetlana Atslega, Felix Sadyrbaev, and Inna Samuilik
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- 2021
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22. Solutions of nonlinear boundary value problem with applications to biomass thermal conversion
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Armands Gritsans, Andrei Kolyshkin, Diana Ogorelova, Felix Sadyrbaev, Inna Samuilik, and Inara Yermachenko
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- 2021
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23. On the Stability of a Steady Convective Flow in a Vertical Layer of a Chemically Reacting Fluid
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Armands Gritsans, Valentina Koliskina, Felix Sadyrbaev, and Andrei Kolyshkin
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Physics::Fluid Dynamics ,Convective flow ,Materials science ,Mechanics ,Layer (electronics) ,Stability (probability) - Abstract
Linear stability of a steady flow of a chemically reacting fluid located in a vertical fluid layer bounded by two infinite parallel planes is investigated. Steady convective flow in the vertical direction is initiated due to the combined effect of internal heat generation and the temperature difference between the planes. Imposing small perturbations on the base flow, linearizing equations of thermal convection under the Boussinesq approximation in the neighbourhood of the base flow and using the method of normal modes we obtain an eigenvalue problem for a system of ordinary differential equations. Collocation method is used to discretize the problem. Numerical calculations are performed in Matlab. Fluid velocity, pressure, and temperature are the solutions of a nonlinear boundary value problem. Properties of the nonlinear boundary value problem for the base flow are investigated numerically using bifurcation analysis. It is shown that both the temperature difference between the planes and intensity of internal heat generation have a destabilizing influence on the base flow. The intensity of heat transfer in the direction perpendicular to the main flow can promote instability and leads to more intensive mixing. This fact can be used in design of bioreactors for biomass thermal conversion.
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- 2021
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24. Control in Inhibitory Genetic Regulatory Network Models
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Felix Sadyrbaev, D. Ogorelova, and V. Sengileyev
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General Medicine ,General Chemistry ,Computational biology ,Control (linguistics) ,Inhibitory postsynaptic potential ,Network model ,Mathematics - Abstract
The system of two the first order ordinary differential equations arising in the gene regulatory networks theory is studied. The structure of attractors for this system is described for three important behavioral cases: activation, inhibition, mixed activation-inhibition. The geometrical approach combined with the vector field analysis allows treating the problem in full generality. A number of propositions are stated and the proof is geometrical, avoiding complex analytic. Although not all the possible cases are considered, the instructions are given what to do in any particular situation.
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- 2020
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25. On modelling of artificial networks arising in applications
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Felix Sadyrbaev and Svetlana Atslega
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Computer science ,Distributed computing ,Artificial networks - Published
- 2020
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26. Networks Describing Dynamical Systems
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Eduard Brokan and Felix Sadyrbaev
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0303 health sciences ,03 medical and health sciences ,Dynamical systems theory ,Quantitative Biology::Molecular Networks ,General Mathematics ,010102 general mathematics ,Statistical physics ,0101 mathematics ,01 natural sciences ,030304 developmental biology ,Mathematics - Abstract
We consider systems of ordinary differential equations that arise in the theory of gene regulatory networks. These systems can be of arbitrary size but of definite structure that depends on the choice of regulatory matrices. Attractors play the decisive role in behaviour of elements of such systems. We study the structure of simple attractors that consist of a number of critical points for several choices of regulatory matrices.
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- 2018
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27. Attraction in n ‐dimensional differential systems from network regulation theory
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Felix Sadyrbaev and Eduard Brokan
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0301 basic medicine ,Dynamical systems theory ,N dimensional ,General Mathematics ,General Engineering ,02 engineering and technology ,Differential systems ,Attraction ,03 medical and health sciences ,020210 optoelectronics & photonics ,030104 developmental biology ,Attractor ,0202 electrical engineering, electronic engineering, information engineering ,Statistical physics ,Mathematics - Published
- 2018
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28. On boundary value problem for equations with cubic nonlinearity and step-wise coefficient
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A. Kirichuka and Felix Sadyrbaev
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Cubic nonlinearity ,Organic Chemistry ,Mathematical analysis ,Boundary value problem ,Biochemistry ,Mathematics - Published
- 2018
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29. A remark on attracting sets in genetic regulatory networks
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Svetlana Atslega and Felix Sadyrbaev
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Computer science ,Quantitative Biology::Molecular Networks ,Attractor ,Topology ,Quantitative Biology::Genomics - Abstract
We consider a three-dimensional model of Genetic regulatory network (GRN). We show that if the system has not sinks (attracting equilibria) then a stable structures (attractors) can appear. The example is given.
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- 2020
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30. On a differential system arising in the network control theory
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Eduard Brokan and Felix Sadyrbaev
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Physics ,Network control ,Pure mathematics ,network control ,Phase portrait ,attracting sets ,Applied Mathematics ,010102 general mathematics ,lcsh:QA299.6-433 ,Boundary (topology) ,phase portrait ,lcsh:Analysis ,02 engineering and technology ,01 natural sciences ,dynamical system ,Set (abstract data type) ,Matrix (mathematics) ,Systems theory ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,0101 mathematics ,Dynamical system (definition) ,Analysis ,Complement (set theory) - Abstract
We investigate the three-dimensional dynamical system occurring in the network regulatory systems theory for specific choices of regulatory matrix { { 0, 1, 1 } { 1, 0, 1 } { 1, 1, 0 } } and sigmoidal regulatory function f(z) = 1 / (1 + e-μz), where z = ∑ Wij xj - θ. The description of attracting sets is provided. The attracting sets consist of respectively one, two or three critical points. This depends on whether the parameters (μ,θ) belong to a set Ω or to the complement of Ω or to the boundary of Ω, where Ω is fully defined set.
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- 2016
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31. Solutions of two-point boundary value problems via phase-plane analysis
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Felix Sadyrbaev and Svetlana Atslega
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Point boundary ,period annulus ,Mathematical analysis ,QA1-939 ,Phase plane analysis ,phase portrait ,Value (mathematics) ,neumann boundary conditions ,Mathematics ,multiplicity of solutions - Abstract
We consider period annuli (continua of periodic solutions) in equations of the type $x''+g(x)=0$ and $x''+f(x) x'^2 + g(x)= 0,$ where $g$ and $f$ are polynomials. The conditions are provided for existence of multiple nontrivial (encircling more than one critical point) period annuli. The conditions are obtained (by phase-plane analysis of period annuli) for existence of families of solutions to the Neumann boundary value problems.
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- 2016
32. On controllability of nonlinear dynamical network
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Felix Sadyrbaev and Eduard Brokan
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Scheme (programming language) ,Controllability ,Nonlinear system ,Computer science ,Control theory ,Attractor ,Structure (category theory) ,Process (computing) ,State (computer science) ,Dynamical system ,computer ,computer.programming_language - Abstract
We provide the scheme for driving a network modelled by dynamical system from one state (“undesired”) to another one. This can be done by changing in time adjustable parameters and require knowledge of the structure of attractors of a system. The process is explained and illustrated by analyzing the two-element network.We provide the scheme for driving a network modelled by dynamical system from one state (“undesired”) to another one. This can be done by changing in time adjustable parameters and require knowledge of the structure of attractors of a system. The process is explained and illustrated by analyzing the two-element network.
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- 2019
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33. On a problem for a system of two second-order differential equations via the theory of vector fields
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Inara Yermachenko and Felix Sadyrbaev
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Cauchy problem ,multiple solutions ,critical points ,Differential equation ,Applied Mathematics ,Mathematical analysis ,rotation of a vector field ,lcsh:QA299.6-433 ,lcsh:Analysis ,winding number ,Elliptic boundary value problem ,Dirichlet boundary value problem ,Nonlinear system ,Bounded function ,Cauchy boundary condition ,planar vector field ,Boundary value problem ,Analysis ,Numerical partial differential equations ,Mathematics - Abstract
We consider Dirichlet boundary value problem for systems of two second-order differential equations with nonlinear continuous and bounded functions in right-hand sides. We prove the existence of a nontrivial solution to the problem comparing behaviors of solutions of auxiliary Cauchy problems at zero solution and at infinity.
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- 2015
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34. Extension of the example by Moore-Nehari
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Armands Gritsans and Felix Sadyrbaev
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Discrete mathematics ,Continuous function (set theory) ,Simple (abstract algebra) ,General Mathematics ,Mathematical analysis ,Interval (graph theory) ,Boundary value problem ,Extension (predicate logic) ,Mathematics - Abstract
R. Moore and Z. Nehari developed the variational theory for superlinear boundary value problems of the form x'' = p(t) |x|2εx, x(a) = 0 = x(b), where ε > 0 and p(t) is a positive continuous function. They constructed simple example of the equation considered in the interval [0, b] so that the problem had three positive solutions. We show that this example can be extended so that the respective BVP has infinitely many groups of solutions with a presribed number of zeros.
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- 2015
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35. Remarks on GRN-type systems
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Eduard Brokan and Felix Sadyrbaev
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0301 basic medicine ,Computer science ,lcsh:R ,General Engineering ,Gene regulatory network ,attractors ,lcsh:Medicine ,Type (model theory) ,Nullcline ,03 medical and health sciences ,030104 developmental biology ,0302 clinical medicine ,ordinary differential equations ,030220 oncology & carcinogenesis ,Ordinary differential equation ,Attractor ,genetic regulatory networks ,Applied mathematics ,lcsh:Q ,lcsh:Science ,Focus (optics) - Abstract
Systems of ordinary differential equations that appear in gene regulatory networks theory are considered. We are focused on asymptotical behavior of solutions. There are stable critical points as well as attractive periodic solutions in two-dimensional and three-dimensional systems. Instead of considering multiple parameters (10 in a two-dimensional system) we focus on typical behaviors of nullclines. Conclusions about possible attractors are made.
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- 2020
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36. On Types of Solutions of the Second Order Nonlinear Boundary Value Problems
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Inara Yermachenko, Felix Sadyrbaev, Maria Dobkevich, and Nadezhda Sveikate
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Nonlinear system ,Article Subject ,Differential form ,lcsh:Mathematics ,Applied Mathematics ,Multiplicity results ,Mathematical analysis ,Order (group theory) ,Nonlinear boundary value problem ,Boundary value problem ,lcsh:QA1-939 ,Analysis ,Mathematics - Abstract
We review the results concerning types of solutions of boundary value problems for the second order nonlinear equation(l2x)(t)=f(t,x,x′),where(l2x)(t)is the second order linear differential form. The existence results and the multiplicity results are stated in terms of types of solutions.
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- 2014
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37. Attracting sets in network regulatory theory
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Felix Sadyrbaev, Dmitrijs Finaskins, and Eduard Brokan
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Intelligent computer network ,Dynamic network analysis ,business.industry ,Computer science ,Distributed computing ,Network segment ,business ,Network topology ,Traffic generation model ,Network traffic control ,Network simulation ,Computer network ,Network formation - Abstract
Modern telecommunication networks are very complex and they should be able to deal with rapid and unpredictable changes in traffic flows. Virtual Network Topology is used to carry IP traffic over Wavelength-division multiplexing optical network. To use network resources in the most optimal way, there is a need for an algorithm, which will dynamically re-share resources among all devices in the particular network segment, based on links utilization between routers. Attractor selection mechanism could be used to dynamically control such Virtual Network Topology. The advantage of this algorithm is that it can adopt to very rapid, unknown and unpredictable changes in traffic flows. This mechanism was adopted from biology - gene regulatory networks in cells use it to adopt to unknown changes in the environment. The dynamical system modeling the gene regulatory network is considered. The relations between genes are described by the regulatory matrix W with the entries from the set {−1, 0, 1} meaning inhibition, no relation and activation respectively. The structure of the attracting set in some specific case is studied. It is remarkable that the attracting set contains a focus-type critical point.
- Published
- 2016
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38. On Different Type Solutions of Boundary Value Problems
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Felix Sadyrbaev and Maria Dobkevich
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Discrete mathematics ,multiple solutions ,existence ,010103 numerical & computational mathematics ,Type (model theory) ,01 natural sciences ,010101 applied mathematics ,Set (abstract data type) ,Modeling and Simulation ,boundary value problem ,QA1-939 ,Interval (graph theory) ,Boundary value problem ,0101 mathematics ,Analysis ,Mathematics - Abstract
We consider boundary value problems of the type x'' = f(t, x, x'), (∗) x(a) = A, x(b) = B. A solution ξ(t) of the above BVP is said to be of type i if a solution y(t) of the respective equation of variations y'' = fx(t, ξ(t), ξ' (t))y + fx' (t, ξ(t), ξ' (t))y' , y(a) = 0, y' (a) = 1, has exactly i zeros in the interval (a, b) and y(b) 6= 0. Suppose there exist two solutions x1(t) and x2(t) of the BVP. We study properties of the set S of all solutions x(t) of the equation (∗) such that x(a) = A, x'1(a) ≤ x' (a) ≤ x'2(a) provided that solutions extend to the interval [a, b].
- Published
- 2016
39. On a Planar Dynamical System Arising in the Network Control Theory
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Felix Sadyrbaev, Svetlana Atslega, and Dmitrijs Finaskins
- Subjects
0301 basic medicine ,Dynamical systems theory ,Phase portrait ,attractor selection ,020206 networking & telecommunications ,phase portraits ,02 engineering and technology ,Dynamical system ,networks control ,dynamical system ,Linear dynamical system ,03 medical and health sciences ,030104 developmental biology ,Projected dynamical system ,Control theory ,Modeling and Simulation ,Attractor ,0202 electrical engineering, electronic engineering, information engineering ,QA1-939 ,Statistical physics ,Limit set ,Random dynamical system ,Analysis ,Mathematics - Abstract
We study the structure of attractors in the two-dimensional dynamical system that appears in the network control theory. We provide description of the attracting set and follow changes this set suffers under the changes of positive parametersµ andΘ. 
- Published
- 2016
40. Qualitative Analysis of Differential, Difference Equations, and Dynamic Equations on Time Scales
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Josef Diblík, Qi-Ru Wang, Yuriy V. Rogovchenko, Felix Sadyrbaev, Tongxing Li, and Alexander Domoshnitsky
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Article Subject ,Differential equation ,lcsh:Mathematics ,Applied Mathematics ,Finite difference method ,lcsh:QA1-939 ,Stochastic partial differential equation ,Nonlinear system ,Multigrid method ,Kolmogorov equations (Markov jump process) ,Simultaneous equations ,Applied mathematics ,Analysis ,Numerical partial differential equations ,Mathematics - Abstract
and Applied Analysis 3 thank Guest Editors Josef Dibĺik, Alexander Domoshnitsky, Yuriy V. Rogovchenko, Felix Sadyrbaev, and Qi-Ru Wang for their unfailing support with editorial work that ensured timely preparation of this special edition. Tongxing Li Josef Dibĺik Alexander Domoshnitsky Yuriy V. Rogovchenko Felix Sadyrbaev Qi-Ru Wang
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- 2015
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41. TWO-PARAMETER NONLINEAR OSCILLATIONS: THE NEUMANN PROBLEM
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Felix Sadyrbaev and Armands Gritsans
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Dirichlet problem ,solution surfaces ,Neumann–Dirichlet method ,Mathematical analysis ,Solution set ,α-branch ,Mixed boundary condition ,Function (mathematics) ,solution curves ,Elliptic boundary value problem ,centro-affine equivalence ,Nonlinear oscillations ,Dirichlet boundary value problem ,Modeling and Simulation ,QA1-939 ,α-spectrum ,Neumann boundary condition ,Neumann boundary value problem ,Boundary value problem ,Mathematics ,Analysis - Abstract
Boundary value problems of the formare considered, whereIn our considerations functionsfandgare generally nonlinear. We give a description of a solution set of the problem (i), (ii). It consist of all triples () such that (λ,μ,x(t)) nontrivially ′solves the problem(i),(ii) and |x(z)| = α at zero pointszof the functionx(t) (iii). We show that this solution set is a union of solution surfaces which are centro-affine equivalent. Each solution surface is associated with nontrivial solutions with definite nodal type. Properties of solution surfaces are studied. It is shown, in particular, that solution surface associated with solutions with exactlyizeroes in the interval (a,b) is centro-affne equivalent to a solution surface of the Dirichlet problem (i),x(a) = 0 =x(b), (iii) corresponding to solutions with odd number of zeros 2j− 1 (i≠ 2j)in the interval (a,b).
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- 2011
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42. On a Maximal Number of Period Annuli
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Felix Sadyrbaev and Yelena Kozmina
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Combinatorics ,Polynomial ,Article Subject ,Degree (graph theory) ,lcsh:Mathematics ,Applied Mathematics ,lcsh:QA1-939 ,Mathematics::Geometric Topology ,Analysis ,Period (music) ,Mathematics - Abstract
We consider equationx′′+g(x)=0, whereg(x)is a polynomial, allowing the equation to have multiple period annuli. We detect the maximal number of possible period annuli for polynomials of odd degree and show how the respective optimal polynomials can be constructed.
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- 2011
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43. Multiple period annuli in Liénard type equations
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Svetlana Atslega and Felix Sadyrbaev
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Liénard equation ,Phase portrait ,Applied Mathematics ,Mathematical analysis ,Critical point (mathematics) ,Mathematics - Abstract
We consider the equation x ″ x 1 − x 2 x ′ 2 + g ( x ) = 0 , where g ( x ) is a polynomial. We provide the conditions for existence of multiple period annuli enclosing several critical points.
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- 2010
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44. Multiple solutions of two-point nonlinear boundary value problems
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Inara Yermachenko and Felix Sadyrbaev
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Shooting method ,Applied Mathematics ,Mathematical analysis ,Free boundary problem ,Boundary value problem ,Mixed boundary condition ,Singular boundary method ,Laplace operator ,Analysis ,Robin boundary condition ,Elliptic boundary value problem ,Mathematics - Abstract
A quasi-linear boundary value problem has a solution with properties induced by oscillatory properties of the linear part of an equation. This result is proved for two dimensional systems. Consequences for Φ -Laplacian equations and problems with resonant linear parts are discussed.
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- 2009
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45. Two-parametric nonlinear eigenvalue problems
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Armands Gritsans and Felix Sadyrbaev
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Combinatorics ,Dirichlet problem ,Nonlinear system ,QA1-939 ,Cauchy distribution ,Monotonic function ,Positive and negative parts ,Linear equation ,Spectral line ,Eigenvalues and eigenvectors ,Mathematics - Abstract
Eigenvalue problems of the form $x'' = -\lambda f(x^+) + \mu g(x^-),$ $\quad (i),$ $x(0) = 0, \; x(1) = 0,$ $\quad (ii)$ are considered, where $x^+$ and $x^-$ are the positive and negative parts of $x$ respectively. We are looking for $(\lambda, \mu)$ such that the problem $(i), (ii)$ has a nontrivial solution. This problem generalizes the famous Fu\v{c}\'{i}k problem for piece-wise linear equations. In our considerations functions $f$ and $g$ may be nonlinear functions of super-, sub- and quasi-linear growth in various combinations. The spectra obtained under the normalization condition $|x'(0)|=1$ are sometimes similar to usual Fu\v{c}\'{i}k spectrum for the Dirichlet problem and sometimes they are quite different. This depends on monotonicity properties of the functions $\xi t_1 (\xi)$ and $\eta \tau_1 (\eta),$ where $t_1 (\xi)$ and $\tau_1 (\eta)$ are the first zero functions of the Cauchy problems $x''= -f(x),$ $\: x(0)=0,$ $\: x'(0)=\xi> 0,$ $y''= g(y),$ $\: y(0)=0,$ $\: y'(0)=-\eta,$ $(\eta > 0)$ respectively.
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- 2008
46. Oscillatory Solutions of Boundary Value Problems
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Felix Sadyrbaev
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Physics ,Pure mathematics ,Boundary value problem ,Type (model theory) ,Dynamical system (definition) - Abstract
We consider boundary value problems of the form $$\displaystyle\begin{array}{rcl} & x'' = f(t,x,x'), & {}\\ & x(a) = A,\quad x(b) = B,& {}\\ \end{array}$$ assuming that f is continuous together with f x and fx′. We study also equations in a quasi-linear form $$\displaystyle{x'' + p(t)x' + q(t)x = F(t,x,x').}$$ Introducing types of solutions of boundary value problems as an oscillatory type of the respective equation of variations, we show that for a solution of definite type, the problem can be reformulated in a quasi-linear form. Resonant problems are considered separately. Any resonant problem that has no solutions of indefinite type is in fact nonresonant. The ways of how to detect solutions of definite types are discussed.
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- 2016
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47. Dirichlet Boundary Value Problem for the Second Order Asymptotically Linear System
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Inara Yermachenko, Felix Sadyrbaev, and Armands Gritsans
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Article Subject ,Dirichlet conditions ,lcsh:Mathematics ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Mixed boundary condition ,Dirichlet's energy ,lcsh:QA1-939 ,01 natural sciences ,010101 applied mathematics ,symbols.namesake ,Dirichlet eigenvalue ,Generalized Dirichlet distribution ,Dirichlet's principle ,Dirichlet boundary condition ,symbols ,Boundary value problem ,0101 mathematics ,Analysis ,Mathematics - Abstract
We consider the second order system x′′=f(x) with the Dirichlet boundary conditions x(0)=0=x(1), where the vector field f∈C1(Rn,Rn) is asymptotically linear and f(0)=0. We provide the existence and multiplicity results using the vector field rotation theory.
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- 2016
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48. Multiplicity of Solutions for Second Order Two-Point Boundary Value Problems with Asymptotically Asymmetric Nonlinearities at Resonance
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Felix Sadyrbaev
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Asymptotically linear ,Nonlinear system ,Point boundary ,Sublinear function ,General Mathematics ,Mathematical analysis ,Multiplicity (mathematics) ,Angular function ,Boundary value problem ,Mixed boundary condition ,Mathematics - Abstract
Estimations of the number of solutions are given for various resonant cases of the boundary value problem 𝑥″ + 𝑔(𝑡, 𝑥) = 𝑓(𝑡, 𝑥, 𝑥′), 𝑥(𝑎) cos α – 𝑥′(𝑎) sin α = 0, 𝑥(𝑏) cos β – 𝑥′(𝑏) sin β = 0, where 𝑔(𝑡, 𝑥) is an asymptotically linear nonlinearity, and 𝑓 is a sublinear one. We assume that there exists at least one solution to the BVP.
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- 2007
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49. Nonlinear Spectra for Parameter Dependent Ordinary Differential Equations
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Armands Gritsans and Felix Sadyrbaev
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Dirichlet problem ,Sublinear function ,Applied Mathematics ,Fuchik spectrum ,Mathematical analysis ,Continuous spectrum ,lcsh:QA299.6-433 ,lcsh:Analysis ,jumping nonlinearity ,Spectral line ,Nonlinear system ,asymptotically asymmetric nonlinearities ,Ordinary differential equation ,Analysis ,Linear equation ,Eigenvalues and eigenvectors ,nonlinear spectra ,Mathematics - Abstract
Eigenvalue problems of the form x'' = −λf(x) + µg(x) (i), x(0) = 0, x(1) = 0 (ii) are considered. We are looking for (λ, µ) such that the problem (i), (ii) has a nontrivial solution. This problem generalizes the famous Fuchik problem for piece-wise linear equations. In our considerations functions f and g may be super-, sub- and quasi-linear in various combinations. The spectra obtained under the normalization condition (otherwise problems may have continuous spectra) structurally are similar to usual Fuchik spectrum for the Dirichlet problem. We provide explicit formulas for Fuchik spectra for super and super, super and sub, sub and super, sub and sub cases, where superlinear and sublinear parts of equations are of the form |x|2αx and |x|1/(2β+1) respectively (α > 0, β > 0.)
- Published
- 2007
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50. The Nehari solutions and asymmetric minimizers
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Felix Sadyrbaev and Armands Gritsans
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Mathematical analysis ,Boundary value problem ,Value (mathematics) ,Bifurcation ,Mathematics - Abstract
We consider the boundary value problem $x'' = -q(t,h) x^3,$ $x(-1)=x(1)=0$ which exhibits bifurcation of the Nehari solutions. The Nehari solution of the problem is a solution which minimizes certain functional. We show that for $h$ small there is exactly one Nehari solution. Then under the increase of $h$ there appear two Nehari solutions which supply the functional smaller value than the remaining symmetrical solution does. So the bifurcation of the Nehari solutions is observed and the previously studied in the literature phenomenon of asymmetrical Nehari solutions is confirmed.
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- 2015
- Full Text
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