controlling chaos and bifurcations in discrete-time...

EditorialControlling Chaos and Bifurcations in Discrete-TimePopulation Models

Qamar Din,1 A. A. Elsadany,2 and Hammad Khalil3

1Department of Mathematics, The University of Poonch, Rawalakot, Pakistan2Basic Science Department, Faculty of Computers and Informatics, Suez Canal University, New Campus, Ismailia 41522, Egypt3Department of Mathematics, University of Education, Attock Campus, Lahore, Pakistan

Correspondence should be addressed to Qamar Din; [email protected]

Received 23 August 2017; Accepted 23 August 2017; Published 25 October 2017

Copyright © 2017 Qamar Din et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The theory of discrete dynamical systems has many appli-cations in almost all branches of the applied sciences. Wecan understand the nonlinear phenomena and complexityby studying the qualitative behavior of discrete dynamicalsystems. To a certain extent, the growing interest in differenceequationsmay be also attributed to their simplicity. Althoughonly quite simple computational and graphical representationtools are necessary to study the behavior of the solutionsof difference equations and their bifurcations for varyingspecific parameters in suitable intervals, it is possible toappreciate the complicated and surprisingly diverse dynamicsof difference equations. Discrete-time models can be used tostudy the population dynamics in which time is taken fromone generation to the next. These discrete-time populationmodels are suitable for nonoverlapping generations and areappropriate to describe the nonlinear dynamics and theirchaotic behavior. Controlling chaos and bifurcations refersto the task of designing a controller to modify the chaoticbehaviors and bifurcation properties of a given nonlinearsystem and thereby achieving some desirable dynamicalbehaviors.

In this editorial, we present published papers whichanalyze various aspects and applications related to this specialissue.

The research article entitled “Nonstationary and ChaoticDynamics in Age-Structured Population Models” by A.Wikan and Ø. Kristensen deals with the dynamics of nonlin-ear discrete age-structured population models. The authors

focus on bifurcations, as well as nonstationary and chaoticdynamics. Moreover, different mechanisms which may leadto periodic phenomena are discussed. Some new resultsare also presented, in particular from models where bothfecundity and survival terms contain nonlinear elements.

In “Complexity and Application of Tobacco Manufac-turer Pricing Game considering Market Segments,” S. Guoet al. consider a strategy using two methods of marketsegmentation through game theory based on the collecteddata of several typical cities in Shandong province. Threeoligarchs are discussed to analyze the complexity of themarket in tobacco system. The effect of different factorson the equilibrium price of Nash equilibrium is discussed.Furthermore, the variation of parameters drives the system tochange to uncertain status, even to bifurcation or chaos. Theauthors also discuss the model under normal circumstances,when the three parties jointly take chaos control measures,and its impact on the system. Finally, the chaos control ofeconomic significance is discussed. These results indicatethat only reasonable pricing strategy is a critical point toachieve maximized profit and maintain orderly operation atthe same time, which is a constructive suggestion for tobaccocompanies for their operational strategy.

The paper entitled “Bifurcation Analysis and ChaosControl in a Discrete-Time Predator-Prey System of LeslieType with Simplified Holling Type IV Functional Response”by S. M. S. Rana and U. Kulsum deals with the dynamicalbehavior of a discrete-time predator-prey system of Leslie

HindawiDiscrete Dynamics in Nature and SocietyVolume 2017, Article ID 3476913, 3 pageshttps://doi.org/10.1155/2017/3476913

https://doi.org/10.1155/2017/3476913

2 Discrete Dynamics in Nature and Society

type with simplified Holling type IV functional response. It ismathematically proved that the system undergoes Neimark-Sacker at its positive steady-state. Numerical simulations arepresented not only to validate analytical results but also toshow chaotic behaviors which include bifurcations, phaseportraits, periods 2, 4, 6, 8, 10, and 20 orbits, invariant closedcycle, and attracting chaotic sets. Furthermore, numericallymaximum Lyapunov exponents and fractal dimension arecomputed to justify the chaotic behaviors of the system.Furthermore, a strategy of feedback control is applied tostabilize chaos existing in the system.

By the paper entitled “Chaotic Dynamics and Control ofDiscrete Ratio-Dependent Predator-Prey System,” S. M. S.Rana examined the complexity of a discrete-time predator-prey system with ratio-dependent functional response. Theparametric conditions for existence of fixed points andtheir stability are demonstrated. Moreover, the parametricconditions for existence and direction of Neimark-Sacker areinvestigated. Numerical simulations are presented not onlyto justify theoretical results but also to exhibit new complexbehaviors which include phase portraits, orbits of periods 9,19, and 26, invariant closed circle, and attracting chaotic sets.Furthermore, maximum Lyapunov exponents and fractaldimension are computed to confirm the chaotic dynamics ofthe system. Finally, a state feedback control method is appliedto control chaos which exists in the system.

In “Analysis of Nonlinear Duopoly Games with ProductDifferentiation: Stability, Global Dynamics, and Control,” S.S. Askar and A. Al-khedhairi consider a cubic utility functionthat is derived from a constant elasticity of substitution pro-duction function (CES).This cubic function ismore desirablethan the quadratic one besides its amenability to efficiencyanalysis. Based on that utility, a two-dimensional Cournotduopoly game with horizontal product differentiation ismodeled using a discrete-time scale. Two different types ofgames are studied. In the first game, firms are updating theiroutput production using the traditional bounded rationalityapproach. In the second game, firms adopt Puu’s mechanismto update their productions. Puu’s mechanism does notrequire any information about the profit function; instead, itneeds both firms to know their production and their profitsin the past time periods. In both scenarios, an explicit formfor the Nash equilibrium point is obtained under certainconditions.The stability analysis of Nash point is considered.Furthermore, some numerical simulations are carried out toconfirm the chaotic behavior of Nash equilibrium point.Thisanalysis includes bifurcation, attractor, maximum Lyapunovexponent, and sensitivity to initial conditions.

The paper entitled “Robust Active MPC Synchronizationfor Two Discrete-Time Chaotic Systems with Bounded Dis-turbance” by L. Zhang deals with a synchronization schemefor two discrete-time chaotic systems with bounded distur-bance. By using active control method and imposing somerestriction on the error state, the computation of controller’sfeedback matrix is converted to the min-max optimizationproblem. The theoretical results are derived with the helpof predictive model predictive paradigm and linear matrix

inequality technique. Numerical examples are provided toshow the effectiveness of the designed control method.

In “An Analysis of Discrete Stage-Structured Prey andPrey-Predator Population Models,” A. Wikan studied dis-crete stage-structured prey and prey-predator models. It isproved that the models under consideration are permanent(i.e., the population will neither go extinct nor exhibitexplosive oscillations) and, moreover, that the transfer fromstability to nonstationary behavior always goes through asupercritical Neimark-Sacker bifurcation. The prey modelcovers species that possess a wide range of different life his-tories. Predation pressure may both stabilize and destabilizethe prey dynamics but the strength of impact is closely relatedto life history. Indeed, if the prey possesses a precocioussemelparous life history and exhibits chaotic oscillations, it isshown that increased predation may stabilize the dynamicsand also, in case of large predation pressure, transfer thepopulation to another chaotic regime.

The research article entitled “Neimark-Sacker Bifurcationand Chaos Control in a Fractional-Order Plant-HerbivoreModel” by Q. Din et al. deals with the dynamics of a discrete-time 3-dimensional plant-herbivore model. Moreover, theexistence and uniqueness of positive equilibrium and para-metric conditions for local asymptotic stability of positiveequilibrium point of the model are investigated. Moreover,it is also proved that the system undergoes Neimark-Sackerbifurcation for positive equilibrium with the help of anexplicit criterion for Neimark-Sacker bifurcation. The chaoscontrol in the model is discussed through implementationof two feedback control strategies, that is, pole-placementtechnique and hybrid control methodology.

In “Complex Dynamics and Chaos Control on a kind ofBertrand Duopoly Game Model Considering R&D Activi-ties,” H. Tu et al. study the dynamics and development oftwo-stage input competition game model in the Bertrandduopoly oligopoly market with spillover effects on costreduction. The stability of the Nash equilibrium point andlocal stable conditions and stability region of the Nashequilibrium point by the bifurcation theory are investigated.The complex dynamic behaviors of the system are shown bynumerical simulations. It is demonstrated that chaos occursfor a range of managerial policies, and the associated unpre-dictability is solely due to the dynamics of the interaction.It is investigated that the straight line stabilization methodis the appropriate management measure to control thechaos.

The research paper entitled “Decomposition Techniqueand a Family of Efficient Schemes for Nonlinear Equations”by F.A. Shah et al. dealswith a new family of iterativemethodsfor solving nonlinear equations which is developed by usinga new decomposition technique.The convergence of the newmethods is proved. For the implementation and performanceof the newmethods, some examples are solved and the resultsare compared with some existing methods.

The guest editors of this special issue hope that problemsdiscussed and investigated in the papers by the authors of thisissue can inspire and motivate researchers in these fields todiscover new, innovative, and novel applications in all areasof pure and applied mathematics.

Discrete Dynamics in Nature and Society 3

Acknowledgments

We would like to express our great gratitude to all ofthe authors for their contributions and the reviewers fortheir serious evaluation of the papers submitted by valuablecomments and timely feedback.

Qamar DinA. A. Elsadany

Hammad Khalil

Submit your manuscripts athttps://www.hindawi.com

Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttp://www.hindawi.com

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttp://www.hindawi.com

Volume 2014 Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Stochastic AnalysisInternational Journal of

controlling chaos and bifurcations in discrete-time...

Documents