Table Of ContentNonlinear 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
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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].
