Tawfiqullah Ayoubi - Academia.edu (original) (raw)
Papers by Tawfiqullah Ayoubi
Applied Mathematics, 2019
Current research is concerned with the stability of stochastic logistic equation with Ornstein-Uh... more Current research is concerned with the stability of stochastic logistic equation with Ornstein-Uhlenbeck process. First, this research proves that the stochastic logistic model with Ornstein-Uhlenbeck process has a positive solution. After that, it also introduces the sufficient conditions for stochastically stability of stochastic logistic model for cell growth of microorganism in fermentation process for positive equilibrium point by using Lyapunov function. In addition, this research establishes the sufficient conditions for zero solution as mentioned in Appendix A due to the cell growth of microorganism max µ , which cannot be negative in fermentation process. Furthermore, for numerical simulation, current research uses the 4-stage stochastic Runge-Kutta (SRK4) method to show the reality of the results.
Nonlinear Analysis: Modelling and Control, 2016
In this paper, deterministic and stochastic SEIRI epidemic models featuring a distributed latent ... more In this paper, deterministic and stochastic SEIRI epidemic models featuring a distributed latent period and general, unspecified nonlinear incidence and growth rates for the susceptible class are proposed and investigated from a stability viewpoint. By applying Lyapunov-LaSalle invariance principle, we first obtain sufficient conditions for the global stability of equilibria of the deterministic model. On the basis of this result, we subsequently derive sufficient conditions for asymptotic stability of the stochastic model. Finally, numerical simulations are given to illustrate the previously obtained theoretical framework.
Mathematical Problems in Engineering, 2017
This paper overviews the research investigations pertaining to stability and stabilization of con... more This paper overviews the research investigations pertaining to stability and stabilization of control systems with time-delays. The prime focus is the fundamental results and recent progress in theory and applications. The overview sheds light on the contemporary development on the linear matrix inequality (LMI) techniques in deriving both delay-independent and delay-dependent stability results for time-delay systems. Particular emphases will be placed on issues concerned with the conservatism and the computational complexity of the results. Key technical bounding lemmas and slack variable introduction approaches will be presented. The results will be compared and connections of certain delay-dependent stability results are also discussed.
SOP Transactions on Statistics and Analysis, 2014
In this paper, the stochastic theta-logistic population growth model is introduced. The existence... more In this paper, the stochastic theta-logistic population growth model is introduced. The existence, uniqueness and asymptotic stability of the solution are discussed. In order to represent the effect of theta on logistic curve and the procedure of population growth, the nonlinear stochastic differential equation of this model is solved numerically and for more clarification Irans population data in the period between 1921 to 2011 is applied. The theta-logistic is a simple and flexible model for describing how the growth rate of population increases as theta increases.
AIP Conference Proceedings, 2014
In this paper, the stochastic growth logistic model with aftereffect for the cell growth of C. ac... more In this paper, the stochastic growth logistic model with aftereffect for the cell growth of C. acetobutylicum P262 and Luedeking-Piret equations for solvent production in batch fermentation system is introduced. The parameters values of the mathematical models are estimated via Levenberg-Marquardt optimization method of non-linear least squares. We apply Milstein scheme for solving the stochastic models numerically. The effciency of mathematical models is measured by comparing the simulated result and the experimental data of the microbial growth and solvent production in batch system. Low values of Root Mean-Square Error (RMSE) of stochastic models with aftereffect indicate good fits.
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2011
Statistical complexity measures are used to quantify the degree of complexity of the delayed logi... more Statistical complexity measures are used to quantify the degree of complexity of the delayed logistic map, with linear and nonlinear feedback. We employ two methods for calculating the complexity measures, one with the ‘histogram-based’ probability distribution function and the other one with ordinal patterns. We show that these methods provide complementary information about the complexity of the delay-induced dynamics: there are parameter regions where the histogram-based complexity is zero while the ordinal pattern complexity is not, and vice versa. We also show that the time series generated from the nonlinear delayed logistic map can present zero missing or forbidden patterns, i.e. all possible ordinal patterns are realized into orbits.
Journal of Inequalities and Applications, 2010
In this paper, the exponential stability analysis problem is considered for a class of recurrent ... more In this paper, the exponential stability analysis problem is considered for a class of recurrent neural networks RNNs with random delay and Markovian switching. The evolution of the delay is modeled by a continuous-time homogeneous Markov process with a finite number of states. The main purpose of this paper is to establish easily verifiable conditions under which the random delayed recurrent neural network with Markovian switching is exponentially stable. The analysis is based on the Lyapunov-Krasovskii functional and stochastic analysis approach, and the conditions are expressed in terms of linear matrix inequalities, which can be readily checked by using some standard numerical packages such as the Matlab LMI Toolbox. A numerical example is exploited to show the usefulness of the derived LMI-based stability conditions.
Advances in Difference Equations, 2015
In this paper, we consider the logistic equation with piecewise constant argument of generalized ... more In this paper, we consider the logistic equation with piecewise constant argument of generalized type. We analyze the stability of the trivial fixed point and the positive fixed point after reducing the equation into a nonautonomous difference equation. We also discuss the existence of bounded solutions for the reduced nonautonomous difference equation. Then we investigate the stability of the positive fixed point by means of Lyapunov's second method developed for nonautonomous difference equations. We find conditions formulated through the parameters of the model and the argument function. We also present numerical simulations to validate our findings.
Journal of Computational and Applied Mathematics, 2010
In this paper, by using the concept of differential equations with piecewise constant arguments o... more In this paper, by using the concept of differential equations with piecewise constant arguments of generalized type [1-4], a model of cellular neural networks (CNNs) [5,6] is developed. The Lyapunov-Razumikhin technique is applied to find sufficient conditions for the uniform asymptotic stability of equilibria. Global exponential stability is investigated by means of Lyapunov functions. An example with numerical simulations is worked out to illustrate the results.
Journal of Mathematics Research, 2020
In this research, we first prove that the stochastic logistic model (10) has a positive global so... more In this research, we first prove that the stochastic logistic model (10) has a positive global solution. Subsequently, we introduce the sufficient conditions for the stochastically stability of the general form of stochastic differential equations (SDEs) in terms of equation (1), for zero solution by using the Lyapunov function. This result is verified via several examples in Appendix A. Besides; we prove that the stochastic logistic model, by incorporating the Ornstein-Uhlenbeck process is stable in zero solution. Furthermore, the simulated results are displayed via the 4-stage stochastic Runge-Kutta (SRK4) numerical method.
This paper is concerned about the stability of stochastic delay logistic model with Ornsttein-Uhl... more This paper is concerned about the stability of stochastic delay logistic model with Ornsttein-Uhlenbeck process. First, current paper introduces a general theory to determine the stability of SDDEs for zero solution in term of equation (1) via Lyapunov function. Aftermath, this result verified by stochastic logistic model with delayed feedback and several equations. Furthermore, this research uses the 4-stage stochastic Runge-Kutta (SRK4) method to evaluate the numerical solution and reflects the reality of the results.
Applied Mathematics and Computation
Current research is concerned with the stability of stochastic logistic equation with Ornstein-Uh... more Current research is concerned with the stability of stochastic logistic equation with Ornstein-Uhlenbeck process. First, this research proves that the stochas-tic logistic model with Ornstein-Uhlenbeck process has a positive solution. After that, it also introduces the sufficient conditions for stochastically stability of stochastic logistic model for cell growth of microorganism in fermentation process for positive equilibrium point by using Lyapunov function. In addition, this research establishes the sufficient conditions for zero solution as mentioned in Appendix A due to the cell growth of microorganism max µ , which cannot be negative in fermentation process. Furthermore, for numerical simulation, current research uses the 4-stage stochastic Runge-Kutta (SRK4) method to show the reality of the results.
In this paper, the stochastic theta-logistic population growth model is introduced. The existence... more In this paper, the stochastic theta-logistic population growth model is introduced. The existence, uniqueness and asymptotic stability of the solution are discussed. In order to represent the effect of theta on logistic curve and the procedure of population growth, the nonlinear stochastic differential equation of this model is solved numerically and for more clarification Irans population data in the period between 1921 to 2011 is applied. The theta-logistic is a simple and flexible model for describing how the growth rate of population increases as theta increases.
In this paper, by using the concept of differential equations with piecewise constant argument, t... more In this paper, by using the concept of differential equations with piecewise constant argument, the model of Hopfield neural networks with constant delay is developed. Sufficient conditions for the existence of an equilibrium as well as its global exponential stability by means of Lyapunov functionals and a linear matrix inequality (LMI) are obtained. An example is given to illustrate our results.
Differential equations with a piecewise constant argument of generalized type Lyapunov-Razumikhin... more Differential equations with a piecewise constant argument of generalized type Lyapunov-Razumikhin technique Method of Lyapunov functions Linear matrix inequality a b s t r a c t
In this paper, we apply the method of Lyapunov functions for differential equations with piecewis... more In this paper, we apply the method of Lyapunov functions for differential equations with piecewise constant argument of generalized type to a model of recurrent neural networks (RNNs). The model involves both advanced and delayed arguments. Sufficient conditions are obtained for global exponential stability of the equilibrium point. Examples with numerical simulations are presented to illustrate the results.
In this paper, we consider the logistic equation with piecewise constant argument of generalized ... more In this paper, we consider the logistic equation with piecewise constant argument of generalized type. We analyze the stability of the trivial fixed point and the positive fixed point after reducing the equation into a nonautonomous difference equation. We also discuss the existence of bounded solutions for the reduced nonautonomous difference equation. Then we investigate the stability of the positive fixed point by means of Lyapunov's second method developed for nonautonomous difference equations. We find conditions formulated through the parameters of the model and the argument function. We also present numerical simulations to validate our findings.
Global exponential stability of a class of neural networks with periodic coefficients and piecewi... more Global exponential stability of a class of neural networks with periodic coefficients and piecewise constant arguments is investigated in this paper. A new definition of exponential stability and a novel differential inequality with piecewise constant arguments are introduced to obtain sufficient conditions for the globally exponential stability of the periodic solution of neural networks. The stability criteria are independent on the upper bound of the adjacent element difference of the discontinuous switching moments. According to the new definition of exponential stability and the novel differential inequality, not only it is not necessary to establish any relationship between the norms of the states with/without piecewise constant arguments, but also the stability criteria for the neural networks can be obtained just in terms of the original differential equation, rather than the equivalent integral equation which is widely used in the early works. Typical numerical examples are utilized to illustrate the validity and improvement in less conservatism of the theoretical results. (D. Cao).
I hereby declare that all information in this document has been obtained and presented in accorda... more I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. Name, Last Name: MELTEM KARACAÖREN Signature : iv ABSTRACT STABILITY ANALYSIS OF NEURAL NETWORKS WITH PIECEWISE CONSTANT ARGUMENT Karacaören, Meltem
Applied Mathematics, 2019
Current research is concerned with the stability of stochastic logistic equation with Ornstein-Uh... more Current research is concerned with the stability of stochastic logistic equation with Ornstein-Uhlenbeck process. First, this research proves that the stochastic logistic model with Ornstein-Uhlenbeck process has a positive solution. After that, it also introduces the sufficient conditions for stochastically stability of stochastic logistic model for cell growth of microorganism in fermentation process for positive equilibrium point by using Lyapunov function. In addition, this research establishes the sufficient conditions for zero solution as mentioned in Appendix A due to the cell growth of microorganism max µ , which cannot be negative in fermentation process. Furthermore, for numerical simulation, current research uses the 4-stage stochastic Runge-Kutta (SRK4) method to show the reality of the results.
Nonlinear Analysis: Modelling and Control, 2016
In this paper, deterministic and stochastic SEIRI epidemic models featuring a distributed latent ... more In this paper, deterministic and stochastic SEIRI epidemic models featuring a distributed latent period and general, unspecified nonlinear incidence and growth rates for the susceptible class are proposed and investigated from a stability viewpoint. By applying Lyapunov-LaSalle invariance principle, we first obtain sufficient conditions for the global stability of equilibria of the deterministic model. On the basis of this result, we subsequently derive sufficient conditions for asymptotic stability of the stochastic model. Finally, numerical simulations are given to illustrate the previously obtained theoretical framework.
Mathematical Problems in Engineering, 2017
This paper overviews the research investigations pertaining to stability and stabilization of con... more This paper overviews the research investigations pertaining to stability and stabilization of control systems with time-delays. The prime focus is the fundamental results and recent progress in theory and applications. The overview sheds light on the contemporary development on the linear matrix inequality (LMI) techniques in deriving both delay-independent and delay-dependent stability results for time-delay systems. Particular emphases will be placed on issues concerned with the conservatism and the computational complexity of the results. Key technical bounding lemmas and slack variable introduction approaches will be presented. The results will be compared and connections of certain delay-dependent stability results are also discussed.
SOP Transactions on Statistics and Analysis, 2014
In this paper, the stochastic theta-logistic population growth model is introduced. The existence... more In this paper, the stochastic theta-logistic population growth model is introduced. The existence, uniqueness and asymptotic stability of the solution are discussed. In order to represent the effect of theta on logistic curve and the procedure of population growth, the nonlinear stochastic differential equation of this model is solved numerically and for more clarification Irans population data in the period between 1921 to 2011 is applied. The theta-logistic is a simple and flexible model for describing how the growth rate of population increases as theta increases.
AIP Conference Proceedings, 2014
In this paper, the stochastic growth logistic model with aftereffect for the cell growth of C. ac... more In this paper, the stochastic growth logistic model with aftereffect for the cell growth of C. acetobutylicum P262 and Luedeking-Piret equations for solvent production in batch fermentation system is introduced. The parameters values of the mathematical models are estimated via Levenberg-Marquardt optimization method of non-linear least squares. We apply Milstein scheme for solving the stochastic models numerically. The effciency of mathematical models is measured by comparing the simulated result and the experimental data of the microbial growth and solvent production in batch system. Low values of Root Mean-Square Error (RMSE) of stochastic models with aftereffect indicate good fits.
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2011
Statistical complexity measures are used to quantify the degree of complexity of the delayed logi... more Statistical complexity measures are used to quantify the degree of complexity of the delayed logistic map, with linear and nonlinear feedback. We employ two methods for calculating the complexity measures, one with the ‘histogram-based’ probability distribution function and the other one with ordinal patterns. We show that these methods provide complementary information about the complexity of the delay-induced dynamics: there are parameter regions where the histogram-based complexity is zero while the ordinal pattern complexity is not, and vice versa. We also show that the time series generated from the nonlinear delayed logistic map can present zero missing or forbidden patterns, i.e. all possible ordinal patterns are realized into orbits.
Journal of Inequalities and Applications, 2010
In this paper, the exponential stability analysis problem is considered for a class of recurrent ... more In this paper, the exponential stability analysis problem is considered for a class of recurrent neural networks RNNs with random delay and Markovian switching. The evolution of the delay is modeled by a continuous-time homogeneous Markov process with a finite number of states. The main purpose of this paper is to establish easily verifiable conditions under which the random delayed recurrent neural network with Markovian switching is exponentially stable. The analysis is based on the Lyapunov-Krasovskii functional and stochastic analysis approach, and the conditions are expressed in terms of linear matrix inequalities, which can be readily checked by using some standard numerical packages such as the Matlab LMI Toolbox. A numerical example is exploited to show the usefulness of the derived LMI-based stability conditions.
Advances in Difference Equations, 2015
In this paper, we consider the logistic equation with piecewise constant argument of generalized ... more In this paper, we consider the logistic equation with piecewise constant argument of generalized type. We analyze the stability of the trivial fixed point and the positive fixed point after reducing the equation into a nonautonomous difference equation. We also discuss the existence of bounded solutions for the reduced nonautonomous difference equation. Then we investigate the stability of the positive fixed point by means of Lyapunov's second method developed for nonautonomous difference equations. We find conditions formulated through the parameters of the model and the argument function. We also present numerical simulations to validate our findings.
Journal of Computational and Applied Mathematics, 2010
In this paper, by using the concept of differential equations with piecewise constant arguments o... more In this paper, by using the concept of differential equations with piecewise constant arguments of generalized type [1-4], a model of cellular neural networks (CNNs) [5,6] is developed. The Lyapunov-Razumikhin technique is applied to find sufficient conditions for the uniform asymptotic stability of equilibria. Global exponential stability is investigated by means of Lyapunov functions. An example with numerical simulations is worked out to illustrate the results.
Journal of Mathematics Research, 2020
In this research, we first prove that the stochastic logistic model (10) has a positive global so... more In this research, we first prove that the stochastic logistic model (10) has a positive global solution. Subsequently, we introduce the sufficient conditions for the stochastically stability of the general form of stochastic differential equations (SDEs) in terms of equation (1), for zero solution by using the Lyapunov function. This result is verified via several examples in Appendix A. Besides; we prove that the stochastic logistic model, by incorporating the Ornstein-Uhlenbeck process is stable in zero solution. Furthermore, the simulated results are displayed via the 4-stage stochastic Runge-Kutta (SRK4) numerical method.
This paper is concerned about the stability of stochastic delay logistic model with Ornsttein-Uhl... more This paper is concerned about the stability of stochastic delay logistic model with Ornsttein-Uhlenbeck process. First, current paper introduces a general theory to determine the stability of SDDEs for zero solution in term of equation (1) via Lyapunov function. Aftermath, this result verified by stochastic logistic model with delayed feedback and several equations. Furthermore, this research uses the 4-stage stochastic Runge-Kutta (SRK4) method to evaluate the numerical solution and reflects the reality of the results.
Applied Mathematics and Computation
Current research is concerned with the stability of stochastic logistic equation with Ornstein-Uh... more Current research is concerned with the stability of stochastic logistic equation with Ornstein-Uhlenbeck process. First, this research proves that the stochas-tic logistic model with Ornstein-Uhlenbeck process has a positive solution. After that, it also introduces the sufficient conditions for stochastically stability of stochastic logistic model for cell growth of microorganism in fermentation process for positive equilibrium point by using Lyapunov function. In addition, this research establishes the sufficient conditions for zero solution as mentioned in Appendix A due to the cell growth of microorganism max µ , which cannot be negative in fermentation process. Furthermore, for numerical simulation, current research uses the 4-stage stochastic Runge-Kutta (SRK4) method to show the reality of the results.
In this paper, the stochastic theta-logistic population growth model is introduced. The existence... more In this paper, the stochastic theta-logistic population growth model is introduced. The existence, uniqueness and asymptotic stability of the solution are discussed. In order to represent the effect of theta on logistic curve and the procedure of population growth, the nonlinear stochastic differential equation of this model is solved numerically and for more clarification Irans population data in the period between 1921 to 2011 is applied. The theta-logistic is a simple and flexible model for describing how the growth rate of population increases as theta increases.
In this paper, by using the concept of differential equations with piecewise constant argument, t... more In this paper, by using the concept of differential equations with piecewise constant argument, the model of Hopfield neural networks with constant delay is developed. Sufficient conditions for the existence of an equilibrium as well as its global exponential stability by means of Lyapunov functionals and a linear matrix inequality (LMI) are obtained. An example is given to illustrate our results.
Differential equations with a piecewise constant argument of generalized type Lyapunov-Razumikhin... more Differential equations with a piecewise constant argument of generalized type Lyapunov-Razumikhin technique Method of Lyapunov functions Linear matrix inequality a b s t r a c t
In this paper, we apply the method of Lyapunov functions for differential equations with piecewis... more In this paper, we apply the method of Lyapunov functions for differential equations with piecewise constant argument of generalized type to a model of recurrent neural networks (RNNs). The model involves both advanced and delayed arguments. Sufficient conditions are obtained for global exponential stability of the equilibrium point. Examples with numerical simulations are presented to illustrate the results.
In this paper, we consider the logistic equation with piecewise constant argument of generalized ... more In this paper, we consider the logistic equation with piecewise constant argument of generalized type. We analyze the stability of the trivial fixed point and the positive fixed point after reducing the equation into a nonautonomous difference equation. We also discuss the existence of bounded solutions for the reduced nonautonomous difference equation. Then we investigate the stability of the positive fixed point by means of Lyapunov's second method developed for nonautonomous difference equations. We find conditions formulated through the parameters of the model and the argument function. We also present numerical simulations to validate our findings.
Global exponential stability of a class of neural networks with periodic coefficients and piecewi... more Global exponential stability of a class of neural networks with periodic coefficients and piecewise constant arguments is investigated in this paper. A new definition of exponential stability and a novel differential inequality with piecewise constant arguments are introduced to obtain sufficient conditions for the globally exponential stability of the periodic solution of neural networks. The stability criteria are independent on the upper bound of the adjacent element difference of the discontinuous switching moments. According to the new definition of exponential stability and the novel differential inequality, not only it is not necessary to establish any relationship between the norms of the states with/without piecewise constant arguments, but also the stability criteria for the neural networks can be obtained just in terms of the original differential equation, rather than the equivalent integral equation which is widely used in the early works. Typical numerical examples are utilized to illustrate the validity and improvement in less conservatism of the theoretical results. (D. Cao).
I hereby declare that all information in this document has been obtained and presented in accorda... more I hereby declare that all information in this document has been obtained and presented in accordance with academic rules and ethical conduct. I also declare that, as required by these rules and conduct, I have fully cited and referenced all material and results that are not original to this work. Name, Last Name: MELTEM KARACAÖREN Signature : iv ABSTRACT STABILITY ANALYSIS OF NEURAL NETWORKS WITH PIECEWISE CONSTANT ARGUMENT Karacaören, Meltem