Browsing by Author "Tunc, O."

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Bifurcation Analysis and Chaos of a Discrete-Time Kolmogorov Model
(Taylor & Francis Ltd, 2021) Khan, A. Q.; Khaliq, S.; Tunc, O.; Khaliq, A.; Javaid, M. B.; Ahmed, I
In this paper, we explore local dynamical characteristics with different topological classifications at fixed points, bifurcations and chaos in the discrete Kolmogorov model. More precisely, we investigate the existence of trivial, boundary and interior fixed points of the discrete Kolmogorov model by algebraic techniques. We prove that for all involved parameters, the discrete Kolmogorov model has trivial and two boundary fixed points, and the interior fixed point under specific parametric condition. Further we explore the local dynamics with topological classifications at fixed points and existence of periodic points of the discrete Kolmogorov model simultaneously. We also explore the occurrence of bifurcation at fixed points and prove that at boundary points there exists no flip bifurcation but it occurs at the interior fixed point. Moreover, we utilize feedback control method to stabilize chaos appears in the Kolmogorov model. Finally, we present numerical simulations to verify corresponding theoretical results and also reveal some new dynamics.
Construction of a Lyapunov Function for a Linear Large-Scale Periodic System With Possibly Unstable Subsystems
(Pergamon-elsevier Science Ltd, 2022) Slynko, V; Tunc, O.; Atamas, I
This article proposes an approach to construct a Lyapunov function for a linear large-scale periodic system. In this case, in contrast to various variants of small-gain stability conditions for large-scale systems, the presence of the asymptotic stability property of independent subsystems is not assumed. To analyze the asymptotic stability of a large-scale system, the direct Lyapunov method is used in combination with the discretization method and identities of the commutator calculus. The main results are illustrated by means of examples. (C) 2022 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.
New Results on the Qualitative Analysis of Solutions of Vides by the Lyapunov-Razumikhin Technique
(Springer, 2023) Tunc, O.; Korkmaz, E.
A new mathematical model described by a Volterra integrodifferential equation (VIDE) with constant delay is examined. New agreeable conditions on the uniformly asymptotic stability, boundedness, and square integrability of solutions of the VIDE are obtained by using the Lyapunov-Razumikhin technique. The established conditions improve some former results. They can be also regarded as nonlinear generalizations of these results. Moreover, they are weaker than some available results cited in the bibliography. Two examples are presented to demonstrate possible applications of these results and the introduced concepts. The application of the Lyapunov-Razumikhin technique leads to a significant difference and gives certain advantages over the related methods used in the books and papers cited in the bibliography.
On the New Qualitative Results in Integro-Differential Equations With Caputo Fractional Derivative and Multiple Kernels and Delays
(Yokohama Publ, 2022) Tunc, C.; Tunc, O.; Yao, J. C.
In this paper, qualitative properties such as uniform stability (US), asymptotic stability (AS) and Mittag-Leffler stability (MLS) of trivial solution and boundedness of nonzero solutions of a system of non-linear fractional integrodelay differential equations (FFIDDEs) with Caputo fractional derivative, multiple kernels and multiple delays are investigated. Four new theorems including sufficient conditions are proved on these qualitative concepts of solutions. The established conditions depend upon the verification of the basic qualitative results of fractional calculus and our main results, which are proved via the Lyapunov-Razumikhin method (LRM). In the end, as numerical applications of the proved theorems, an example is given to demonstrate the effectiveness of the applied method and obtained results. Our results improve and generalize the known ones in this direction.