arXiv subjects

Fatma Tokmak Fen

Publications and source records attributed to Fatma Tokmak Fen.

16 recordsLinked to original sources

arXiv2026

Anti-periodic Solutions of Quasilinear Impulsive Systems with Piecewise Constant Argument of Generalized Type

This study is devoted to anti-periodic solutions of quasilinear impulsive systems with piecewise constant argument of generalized type. We rigorously prove the existence and uniqueness of an anti-periodic solution making use of the Banach fixed point theorem. The presence of a term with piecewise constant argument is the main novelty. Appropriate examples which support the theoretical findings are provided.

math.DS↗

arXiv2025

Homoclinic and Heteroclinic Trajectories of Differential Equations with Piecewise Constant Arguments of Generalized Type

Quasilinear systems with piecewise constant arguments of generalized type are under investigation from the asymptotic point of view. The systems have discontinuous right-hand sides which are identified via a discrete-time map. It is rigorously proved that homoclinic and heteroclinic solutions are generated, and they are taken into account in the functional sense. The Banach fixed point theorem is used for the verification. The hyperbolic set of solutions is also discussed, and an example supporting the theoretical findings is provided.

math.DS↗

arXiv2024

Asymptotically Unpredictable Solutions of Quasilinear Impulsive Systems with Regular Discontinuity Moments

The notion of asymptotic unpredictability was recently introduced in (Commun. Nonlinear Sci. Numer. Simul. 134, 108029, 2024) for semiflows. Likewise unpredictable trajectories, asymptotically unpredictable ones are also capable of producing sensitivity in a dynamics, which is an indispensable feature of chaos. In the present study, we newly propose piecewise continuous asymptotically unpredictable functions, and investigate the existence and uniqueness of such solutions in a quasilinear impulsive system of differential equations comprising a term which is periodic in the time argument. The class of functions and the impulsive system under discussion admit regular discontinuity moments. Some techniques for obtaining discontinuous asymptotically unpredictable functions are additionally provided. Even though piecewise continuous unpredictable functions are asymptotically unpredictable, it is demonstrated that the converse is not true. In other words, the set of discontinuous unpredictable functions is properly contained in the set of asymptotically unpredictable ones. Appropriate examples are given with regard to all theoretical results.

math.DS↗

arXiv2024

Continuous-Time and Discrete-Time Quasilinear Systems with Asymptotically Unpredictable Solutions

A novel type of trajectory on semiflows, called asymptotically unpredictable, was proposed by Fen and Tokmak Fen [15]. The presence of sensitivity, which is an indispensable feature of chaotic dynamics, is a crucial property that arises from such trajectories. In the present paper, we show the existence and uniqueness of asymptotically unpredictable solutions for quasilinear systems with delay making benefit of the contraction mapping principle. Additionally, we introduce the notion of an asymptotically unpredictable sequence. It is verified that there exist asymptotically unpredictable sequences which are not unpredictable. Discrete-time equations possessing asymptotically unpredictable orbits are also under investigation. Examples of continuous-time and discrete-time systems with asymptotically unpredictable solutions are provided.

math.DS↗

arXiv2024

Persistence and Doubling of Chaotic Attractors in Coupled 3-Cell Hopfield Neural Networks

Two novel phenomena for unidirectionally coupled 3-cell Hopfield neural networks (HNNs) are investigated. The first one is the persistence of chaos, which means the permanency of sensitivity and infinitely many unstable periodic oscillations in the response HNN even if the networks are not synchronized in the generalized sense. Doubling of chaotic attractors is the second phenomenon realized in this study. It can be achieved when the response network possesses two stable point attractors in the absence of the driving. This feature leads to the formation of two coexisting chaotic attractors with disjoint basins. Lyapunov functions are utilized to deduce the presence of an invariant region, and the sensitivity is rigorously proved. The absence of synchronization is approved via the auxiliary system approach and analysis of conditional Lyapunov exponents. Additionally, quadruple and octuple coexisting chaotic attractors are demonstrated, and the formation of hyperchaos is discussed.

nlin.AO↗

arXiv2023

Period-Doubling Route to Chaos and Intermittency in a Hybrid Rössler Model

A Rössler model perturbed with a piecewise constant function is investigated. The perturbation function used in the model is constructed by means of the logistic map. In the absence of the perturbation the system is assumed to possess two equilibrium points one of which is linearly stable. The occurrences of period-doubling cascade and intermittency are numerically investigated. Extensions of the aforementioned phenomena among coupled Rössler systems are also shown. Our results reveal that discontinuous perturbations are capable of generating continuous chaos.

nlin.CD↗

arXiv2022

Modulo periodic Poisson stable solutions of dynamic equations on a time scale

The existence, uniqueness, and asymptotic stability of modulo periodic Poisson stable solutions of dynamic equations on a periodic time scale are investigated. The model under investigation involves a term which is constructed via a Poisson stable sequence. Novel definitions for Poisson stable as well as modulo periodic Poisson stable functions on time scales are provided, and the reduction technique to systems of impulsive differential equations is utilized to achieve the main result. An example which confirms the theoretical results is provided.

math.DS↗

arXiv2022

Generation of Synchronous Unpredictable Oscillations by Coupled Hopfield Neural Networks

A new criterion based on generalized synchronization is provided for the extension of unpredictable oscillations among coupled Hopfield neural networks (HNNs). It is shown that if a drive HNN possesses an unpredictable oscillation, then a response HNN also possesses such an oscillation provided that they are synchronized in the generalized sense. Extension of unpredictability in coupled 4D HNNs are exemplified with simulations. The auxiliary system approach and conditional Lyapunov exponents are utilized to demonstrate the presence of synchronization.

nlin.CD↗

arXiv2022

A Novel Criterion for Unpredictable Motions

We demonstrate the extension of unpredictable motions in coupled autonomous systems with skew product structure in the case that generalized synchronization takes place. Sufficient conditions for the existence of unpredictable motions in the dynamics of the response system are provided. The theoretical results are exemplified for coupled autonomous systems in which the drive is a hybrid dynamical system and the response is a Lorenz system. The auxiliary system approach and conditional Lyapunov exponents are utilized to detect the presence of generalized synchronization.

nlin.CD↗

arXiv2021

Unpredictability in Perturbed Quasilinear Systems with Regular Moments of Impulses

It is rigorously proved that quasilinear impulsive systems possess unpredictable solutions when a perturbation generated by an unpredictable sequence is applied. The existence, uniqueness, as well as asymptotic stability of such solutions are demonstrated. The system under consideration is with regular moments of impulses, and for that reason a novel definition for unpredictable functions with regular discontinuity moments is provided. To show the existence of an unpredictable solution a Gronwall type inequality for piecewise continuous functions is utilized. The theoretical results are supported with an illustrative example.

math.DS↗

arXiv2021

Unpredictable Solutions of Quasilinear Systems with Discontinuous Right-Hand Sides

It is rigorously proved under certain assumptions that a quasilinear system with discontinuous right-hand side possesses a unique unpredictable solution. The discontinuous perturbation function on the right-hand side is defined by means of an unpredictable sequence. A Gronwall-Coppel type inequality is utilized to achieve the main result, and the stability of the unpredictable solution is discussed. Examples with exponentially asymptotically stable and unstable unpredictable solutions are provided.

nlin.CD↗

arXiv2021

Replication of Period-Doubling Route to Chaos in Coupled Systems with Delay

In this study, replication of a period-doubling cascade in coupled systems with delay is rigorously proved under certain assumptions, which guarantee the existence of bounded solutions and replication of sensitivity. A novel definition for replication of sensitivity is utilized, in which the proximity of solutions is considered in an interval instead of a single point. Examples with simulations supporting the theoretical results concerning sensitivity and period-doubling cascade are provided.

nlin.CD↗

arXiv2018

Period-doubling route to chaos in driven impulsive systems

In the present study, we investigate the dynamics of impulsive differential equations driven by a chaotic system. We rigorously prove that, likewise the drive, the response impulsive system is also chaotic. Our results are based on the presence of sensitivity and infinitely many unstable periodic motions. The theoretical results are supported by simulations.

nlin.CD↗

arXiv2016

Homoclinical structure of SICNNs with rectangular input currents

Shunting inhibitory cellular neural networks (SICNNs) with continuous as well as discontinuous external inputs are investigated. The descriptions of homoclinic and heteroclinic motions are provided in the functional sense for the multidimensional dynamics of SICNNs, and it is demonstrated that the networks under investigation exhibit such motions. Homoclinic and heteroclinic outputs that are asymptotic to quasi-periodic outputs are illustrated.

nlin.CD↗

arXiv2016

Homoclinic and Heteroclinic Motions in Hybrid Systems with Impacts

In this paper, we present a method to generate homoclinic and heteroclinic motions in impulsive systems. We rigorously prove the presence of such motions in the case that the systems are under the influence of a discrete map that possesses homoclinic and heteroclinic orbits. Simulations that support the theoretical results are represented by means of a Duffing equation with impacts.

nlin.CD↗

arXiv2015

SICNNs with Li-Yorke Chaotic Outputs on a Time Scale

In the present study, we investigate the existence of Li-Yorke chaos in the dynamics of shunting inhibitory cellular neural networks (SICNNs) on time scales. It is rigorously proved by taking advantage of external inputs that the outputs of SICNNs exhibit Li-Yorke chaos. The theoretical results are supported by simulations, and the controllability of chaos on the time scale is demonstrated by means of the Pyragas control technique. This is the first time in the literature that the existence as well as the control of chaos are provided for neural networks on time scales.

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