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Nonlinear Systems and Complexity Series Editor: Albert C.J. Luo Albert C.J. Luo Hüseyin Merdan Editors Mathematical Modeling and Applications in Nonlinear Dynamics Nonlinear Systems and Complexity SeriesEditor AlbertC.J.Luo SouthernIllinoisUniversityEdwardsville Edwardsville,IL,USA Moreinformationaboutthisseriesathttp://www.springer.com/series/11433 Albert C.J. Luo • Hüseyin Merdan Editors Mathematical Modeling and Applications in Nonlinear Dynamics 123 Editors AlbertC.J.Luo HüseyinMerdan DepartmentofMechanicalandIndustrial DepartmentofMathematics Engineering TOBBUniversityofEconomics SouthernIllinoisUniversityEdwardsville andTechnology Edwardsville,IL,USA Ankara,TURKEY ISSN2195-9994 ISSN2196-0003 (electronic) NonlinearSystemsandComplexity ISBN978-3-319-26628-2 ISBN978-3-319-26630-5 (eBook) DOI10.1007/978-3-319-26630-5 LibraryofCongressControlNumber:2015960740 SpringerChamHeidelbergNewYorkDordrechtLondon ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper SpringerInternationalPublishingAGSwitzerlandispartofSpringerScience+BusinessMedia(www. springer.com) Preface Thiseditedbookcollectssevenchaptersonmathematicalmodelingandapplications in nonlinear dynamics for a deeper understanding of complex phenomena in nonlinear systems. The chapters of this edited book are selected from the 3rd International Conference on Complex Dynamical Systems: New Mathematical ConceptsandApplicationsinLifeSciences(CDSC2014),heldatAnkara,Turkey, 24–26 November 2014. The aim of this conference was to promote research on differentialequationsanddiscreteandhybridequations,especiallyinlifesciences and chemistry. This conference was for the 60th birthday celebration of Professor MaratAkhmet,whoisafacultymemberoftheMathematicsDepartmentatMiddle East Technical University, Turkey. After peer review, 54 papers were accepted for presentation from 17 countries. The chapters of this edited book are based on the invitedlectureswithextendedresultsinnonlineardynamicalsystems,andtheedited bookisdedicatedtoProf.Akhmet’s60thbirthday.Theeditedchaptersincludethe followingtopics: (cid:129) Integrate-and-firebiologicalmodelswithcontinuous/discontinuouscouplings (cid:129) Analyticalperiodicsolutionsinnonlineardynamicalsystems (cid:129) Dynamicsofhematopoieticstemcells (cid:129) Dynamicsofperiodicevolutionprocessesinpharmacotherapy (cid:129) Ultimatesolutionboundednessfordifferentialequationswithseveraldelays (cid:129) DelayeffectsonthedynamicsoftheLengyel–Epsteinreaction-diffusionmodel (cid:129) SemilinearimpulsivedifferentialequationinanabstractBanachspace During this conference, comprehensive discussions on the above topics were made,ledbyinvitedrecognizedscientists.Fromsuchdiscussions,youngscientists andstudentslearnednewmethods,ideas,andresults. The editors would like to thank TÜBPITAK (The Scientific and Technological Research Council of Turkey), TOBB University of Economics and Technology, Ankara, Turkey, and the Institute of Informatics and Control Problems, Almaty, v vi Preface Kazakhstan,forallfinancialsupport,andtheauthorsandreviewersforsupporting theconferenceandcollection.Wehopetheresultspresentedinthiseditedbookwill beusefulforotherspecialistsincomplexdynamicalsystems. Ankara,TURKEY HüseyinMerdan Edwardsville,IL,USA AlbertC.J.Luo Contents 1 TheSolutionoftheSecondPeskinConjectureandDevelopments..... 1 M.U.Akhmet 2 On Periodic Motions in a Time-Delayed, Quadratic NonlinearOscillatorwithExcitation ...................................... 47 AlbertC.J.LuoandHanxiangJin 3 MathematicalAnalysisofaDelayedHematopoieticStem CellModelwithWazewska–LasotaFunctionalProductionType...... 63 RadouaneYafia,M.A.AzizAlaoui,AbdessamadTridane, andAliMoussaoui 4 RandomNoninstantaneousImpulsiveModelsforStudying PeriodicEvolutionProcessesinPharmacotherapy ...................... 87 JinRongWang,MichalFecˇkan,andYongZhou 5 Boundedness of Solutions to a Certain System ofDifferentialEquationswithMultipleDelays........................... 109 CemilTunç 6 DelayEffectsontheDynamics oftheLengyel–Epstein Reaction-DiffusionModel................................................... 125 HüseyinMerdanandS¸eymaKayan 7 Almost Periodic Solutions of Evolution Differential EquationswithImpulsiveAction........................................... 161 ViktorTkachenko vii Chapter 1 The Solution of the Second Peskin Conjecture and Developments M.U.Akhmet Abstract The integrate-and-fire cardiac pacemaker model of pulse-coupled oscillators was introduced by C. Peskin. Because of the pacemaker’s function, two famous synchronization conjectures for identical and nonidentical oscillators wereformulated.ThefirstofPeskin’sconjectureswassolvedinthepaper(J.Phys. A 21:L699–L705, 1988) by S. Strogatz and R. Mirollo. The second conjecture was solved in the paper by Akhmet (Nonlinear Stud. 18:313–327, 2011). There are still many issues related to the nature and types of couplings. The couplings may be impulsive, continuous, delayed, or advanced, and oscillators may be locally or globally connected. Consequently, it is reasonable to consider various ways of synchronization if one wants the biological and mathematical analyses to interact productively. We investigate the integrate-and-fire model in both cases— one with identical and another with not-quite-identical oscillators. A combination of continuous and pulse couplings that sustain the firing in unison is carefully constructed. Moreover, we obtain conditions on the parameters of continuous couplingsthatmakepossiblearigorousmathematicalinvestigationoftheproblem. Thetechniquedevelopedfordifferentialequationswithdiscontinuitiesatnonfixed moments (Akhmet, Principles of Discontinuous Dynamical Systems, Springer, NewYork,2010)andaspecialcontinuousmapformthebasisoftheanalysis.We consider Peskin’s model of the cardiac pacemaker with delayed pulse couplings aswellaswithcontinuouscouplings.Sufficientconditionsforthesynchronization of identical and nonidentical oscillators are obtained. The bifurcation of periodic motionisobserved.Theresultsaredemonstratedwithnumericalsimulations. 1.1 Introductionand Preliminaries In the paper [50], C. Peskin developed the integrate-and-fire model of the cardiac pacemaker [32] to a population of identical pulse-coupled oscillators. Thus, a cardiacpacemakermodelwasproposedwherethesignaltofirearisesnotfroman M.U.Akhmet((cid:2)) DepartmentofMathematics,MiddleEastTechnicalUniversity,06800Ankara,Turkey e-mail:[email protected] ©SpringerInternationalPublishingSwitzerland2016 1 A.C.J.Luo,H.Merdan(eds.),MathematicalModelingandApplications inNonlinearDynamics,NonlinearSystemsandComplexity14, DOI10.1007/978-3-319-26630-5_1 2 M.U.Akhmet outsidestimuli,butinthepopulationofcellsitself.Well-knownconjecturesofself- synchronization were formulated and solutions of these conjectures for identical oscillators[45,50]stimulatedmathematiciansaswellasbiologistsfortheintensive investigationsinthefield[7,16,19,25,33,36,44,47,52,58,60,62]. A specialized bundle of about 10,000 neurons located in the upper part of the rightatriumoftheheartisknownasthesinoatrialnode.Itfiresatregularintervals to cause the heart to beat, with a rhythm of about 60 to 70 beats per minute for a healthy, resting heart. The electrical impulse from the pacemaker triggers a sequenceofelectricaleventsinthehearttocontroltheorderlysequenceofmuscle contractionsthatpumpthebloodoutoftheheart.Thatiswhyitiscalledthecardiac pacemaker in the literature. The cells of the sinoatrial node are able to depolarize spontaneously toward the threshold firing and then recover [9]. The electrical activityofthecardiacpacemakerproducesastrongpatternofvoltagechange.While the nerve cells require a stimulus to fire, cells of the cardiac pacemaker can be consideredtobe“self-firing.”Theyrepetitivelygothroughadepolarizingdischarge andthenrecovertofireagain.Thisactionisanalogoustoarelaxationoscillatorin electronics. The circuit involves a capacitor, which is charged by the energy of a battery(themembranesofthesinoatrialnodeandtheiontransportprocessesplay the role), and a resistor, which controls the flashing rate of the light. In the case of the sinoatrial node, there is input from the physiology of the body related to oxygen demand and other factors that control the rate of firing of the sinoatrial node and hence the heart rate. The question naturally arises of how the neurons organize their firing in unison. The simplest explanation is that the fastest neuron drivesalltheothers,bringingthemtothethreshold.Ifthatwerethecase,thenthe injury of a single cell could significantly change the frequency of the heartbeat. To avoid this important shortcoming, in the paper [50], Peskin proposed a cardiac pacemaker model where signals to fire do not arise from an outside stimuli but instead originate in the population of cells itself. Moreover, the paper proposed thatacardiacpacemakerisapopulationofneuronswithweakcouplingssuchthat synchronyemergesasaresultoftheinteractionofallcells,ratherthanasinglecell dominating. In the papers [3–6], we introduced a new method for the investigation of biological oscillators. The method seems to be universal to analyze integrate-and- fireoscillators.Inparticular,wesolvedthesecondPeskinconjecturein[3,5].Itwas provedthatanensembleofanarbitrarynumberofoscillatorssynchronizesevenif theyarenotquiteidentical. Inthischapterweextendtheapproachtothemodelwithdelayedpulsecoupling. Conditionsthatguaranteethesynchronizationofthemodelarefound.Oursystemis differentthanthatin[16]sincewesupposethatthepulsecouplingisinstantaneous if oscillators are close to each other and are near threshold. In upcoming papers, we plan to consider other models, varying types of the delay involvement, as well as inhibitory models such thatanalogs of resultsin [16] and [62]can be obtained. Moreover, we plan to develop for these systems the theory of the bifurcation of periodic solutions. Some open problems are discussed in Sect.1.5. The method of theanalysisofnonidenticaloscillatorsisbasedonresultsofthetheoryofdifferential equationswithdiscontinuitiesatnonfixedmoments[2].