Table Of Content⋅
Viet-Thanh Pham Sundarapandian Vaidyanathan
⋅
Christos Volos Tomasz Kapitaniak
Editors
Nonlinear Dynamical
Systems with Self-Excited
and Hidden Attractors
123
Editors
Viet-Thanh Pham Christos Volos
Schoolof Electronics and Department ofPhysics
Telecommunications Aristotle University of Thessaloniki
HanoiUniversityofScienceandTechnology Thessaloniki
Hanoi Greece
Vietnam
Tomasz Kapitaniak
Sundarapandian Vaidyanathan Division of Dynamics
Research andDevelopment Centre Faculty of MechanicalEngineering
VelTech University Lodz University of Technology
Chennai, Tamil Nadu Łódź
India Poland
ISSN 2198-4182 ISSN 2198-4190 (electronic)
Studies in Systems,DecisionandControl
ISBN978-3-319-71242-0 ISBN978-3-319-71243-7 (eBook)
https://doi.org/10.1007/978-3-319-71243-7
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Preface
Recently, there has been an increasing interest in a new classification of nonlinear
dynamical systems including two kinds of attractors: self-excited attractor and
hiddenattractor.Previousresearchhasestablishedthataself-excitedattractorhasa
basin of attraction which is excited from unstable equilibrium point. As a result,
classical nonlinear systems such as Lorenz’s system, Rössler’s system, Chen’s
system,Lü’ssystem,orSprott’ssystemareconsideredassystemswithself-excited
attractors. Several attempts have been made to study systems with self-excited
attractors, which appear in various fields from computer sciences, physics, com-
munications, biology, mechanics, chemistry, to economics and finance. However,
there are still different questions which invite more investigation in such systems
with self-excited attractors.
Inrecentyears,systemswithhiddenattractorshavereceivedgreatattentionfrom
both a theoretical and a practical viewpoint. There are a number of important
differences between self-excited attractors and hidden attractors. Self-excited
attractor can be localized straightforwardly by applying a standard computational
procedure. By contrast, we have to develop a specific computational procedure to
identifyahiddenattractor duetothefactthattheequilibriumpointsdonothelpin
their localization. There is evidence that hidden attractors play a crucial role inthe
fieldsofoscillators,describingconvectivefluidmotion,modelofdrillingsystem,or
multilevel DC/DC converter. In addition, hidden attractors are attracting wide-
spread interest because they may lead to unexpected and disastrous responses, for
example,inastructure like abridgeoranairplane wing.Therefore,it isusefulfor
engineering students and researchers to know emergent topics of this new classi-
ficationofattractors.Forthepast5years,althoughtherehasbeenarapidriseinthe
discovery of systems with hidden attractors, there is still very little scientific
understanding of hidden attractors. For example, to date there has been little dis-
cussion on the existence of systems with different families of hidden attractors.
Further studies need to be carried out in order to provide insights for hidden
attractor.
v
vi Preface
The aim of this book, Nonlinear Dynamical Systems with Self-Excited and
Hidden Attractors, is to report the latest advances, developments, research trends,
design, and realization as well as practical applications of nonlinear systems with
self-excited attractors and hidden attractors. The book consists of 20 contributed
chaptersofexpertswhoarespecializedinthese areas. Wehopethatthisbookwill
serve as a reference book about nonlinear systems with self-excited and hidden
attractors for researchers and graduate students.
We would like to thank the authors of all chapters submitted to our book. We
alsowishtothankthereviewersfortheircontributionsinreviewingthechapters.In
addition,wewouldliketoexpressourgratitudetoSpringer,especiallytothebook
editorial team.
Hanoi, Vietnam Viet-Thanh Pham
Chennai, India Sundarapandian Vaidyanathan
Thessaloniki, Greece Christos Volos
Łódź, Poland Tomasz Kapitaniak
October 2017
Contents
Part I Nonlinear Dynamical Systems with Self-Excited Attractors
Bifurcation Analysis and Chaotic Behaviors of Fractional-Order
Singular Biological Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Komeil Nosrati and Christos Volos
Chaos and Bifurcation in Controllable Jerk-Based Self-Excited
Attractors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Wafaa S. Sayed, Ahmed G. Radwan and Hossam A. H. Fahmy
Self-Excited Attractors in Jerk Systems: Overview and Numerical
Investigation of Chaos Production . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Wafaa S. Sayed, Ahmed G. Radwan and Salwa K. Abd-El-Hafiz
Synchronization Properties in Coupled Dry Friction Oscillators . . . . . . 87
Michał Marszal and Andrzej Stefański
Backstepping Control for Combined Function Projective
Synchronization Among Fractional Order Chaotic Systems with
Uncertainties and External Disturbances . . . . . . . . . . . . . . . . . . . . . . . . 115
Vijay K. Yadav, Mayank Srivastava and Subir Das
Chaotic Business Cycles within a Kaldor-Kalecki Framework. . . . . . . . 133
Giuseppe Orlando
Analysis of Three-Dimensional Autonomous Van der Pol–Duffing
Type Oscillator and Its Synchronization in Bistable Regime . . . . . . . . . 163
Gaetan Fautso Kuiate, Victor Kamdoum Tamba and
Sifeu Takougang Kingni
Dynamic Analysis, Electronic Circuit Realization of Mathieu-Duffing
Oscillator and Its Synchronization with Unknown Parameters
and External Disturbances. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
Victor Kamdoum Tamba, François Kapche Tagne,
Elie Bertrand Megam Ngouonkadi and Hilaire Bertrand Fotsin
vii
viii Contents
An Autonomous Helmholtz Like-Jerk Oscillator: Analysis, Electronic
Circuit Realization and Synchronization Issues . . . . . . . . . . . . . . . . . . . 203
VictorKamdoumTamba,GaetanFautsoKuiate,SifeuTakougangKingni
and Pierre Kisito Talla
Synchronization in Kuramoto Oscillators Under Single
External Oscillator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
Gokul P. M., V. K. Chandrasekar and Tomasz Kapitaniak
Analysis, Circuit Design and Synchronization of a New Hyperchaotic
System with Three Quadratic Nonlinearities . . . . . . . . . . . . . . . . . . . . . 251
A. A. Oumate, S. Vaidyanathan, K. Zourmba, B. Gambo
and A. Mohamadou
A New Chaotic Finance System: Its Analysis, Control,
Synchronization and Circuit Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . 271
Babatunde A. Idowu, Sundarapandian Vaidyanathan, Aceng Sambas,
Olasunkanmi I. Olusola and O. S. Onma
Part II Nonlinear Dynamical Systems with Hidden Attractors
Periodic Orbits, Invariant Tori and Chaotic Behavior in Certain
Nonequilibrium Quadratic Three-Dimensional
Differential Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
Alisson C. Reinol and Marcelo Messias
Existence and Control of Hidden Oscillations in a Memristive
Autonomous Duffing Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
Vaibhav Varshney, S. Sabarathinam, K. Thamilmaran, M. D. Shrimali
and Awadhesh Prasad
A Novel 4-D Hyperchaotic Rikitake Dynamo System with Hidden
Attractor, its Properties, Synchronization and Circuit Design . . . . . . . . 345
Sundarapandian Vaidyanathan, Viet-Thanh Pham, Christos Volos
and Aceng Sambas
A Six-Term Novel Chaotic System with Hidden Attractor and
Its Circuit Design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
Aceng Sambas, Sundarapandian Vaidyanathan, Mustafa Mamat
and W. S. Mada Sanjaya
Synchronization Phenomena in Coupled Dynamical Systems
with Hidden Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
C. K. Volos, Viet-Thanh Pham, Ahmad Taher Azar, I. N. Stouboulos
and I. M. Kyprianidis
Contents ix
4-D Memristive Chaotic System with Different Families of
Hidden Attractors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
Dimitrios A. Prousalis, Christos K. Volos, Viet-Thanh Pham,
Ioannis N. Stouboulos and Ioannis M. Kyprianidis
Hidden Chaotic Path Planning and Control of a Two-Link
Flexible Robot Manipulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433
Kshetrimayum Lochan, Jay Prakash Singh, Binoy Krishna Roy
and Bidyadhar Subudhi
5-D Hyperchaotic and Chaotic Systems with Non-hyperbolic
Equilibria and Many Equilibria. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
Jay Prakash Singh and Binoy Krishna Roy
Part I
Nonlinear Dynamical Systems with
Self-Excited Attractors
Bifurcation Analysis and Chaotic
Behaviors of Fractional-Order Singular
Biological Systems
Komeil Nosrati and Christos Volos
Abstract Inthischapter,singularsystemtheoryandfractionalcalculusareutilized
to model the biological systems in the real world, some fractional-order singular
(FOS) biological systems are established, and some qualitative analyses of pro-
posedmodelsareperformed.Throughthefractionalcalculusandeconomictheory,
a new and more realistic model of biological systems predator-prey, logistic map
and SEIR epidemic system have been extended, and besides some mathematical
analysis, the numerical simulations are considered to illustrate the effectiveness of
the numerical method to explore the impacts of fractional-order and economic
interestonthepresentedsystemsinbiologicalcontexts.Itwillbedemonstratedthat
thepresenceoffractional-orderchangesthestabilityofthesolutionsandenrichthe
dynamics of system. In addition, singular models exhibit more complicated
dynamics rather than standard models, especially the bifurcation phenomena and
chaotic behaviors, which can reveal the instability mechanism of systems. Toward
this aim, some materials including several definitions and existence theorems of
uniqueness of solution, stability conditions and bifurcation phenomena in FOS
systems and detailed introductions to fundamental tools for discussing complex
dynamical behavior, such as chaotic behavior have been added.
⋅
Keywords Fractional-Order singular system Bifurcation and chaos
⋅
Biological systems Qualitative analysis
K.Nosrati(✉)
DepartmentofElectricalEngineering,AmirkabirUniversityofTechnology,
424HafezAve,15875-4413,Tehran,Iran
e-mail:[email protected]
C.Volos
PhysicsDepartment,AristotleUniversityofThessaloniki,
54124Thessaloniki,Greece
e-mail:[email protected]
©SpringerInternationalPublishingAG2018 3
V.-T.Phametal.(eds.),NonlinearDynamicalSystemswithSelf-Excited
andHiddenAttractors,StudiesinSystems,DecisionandControl133,
https://doi.org/10.1007/978-3-319-71243-7_1
4 K.NosratiandC.Volos
1 Introduction
Singular systems (differential-algebraic systems, descriptor systems, generalized
state space systems, semi-state systems, singular singularly perturbed systems,
degenerate systems, constrained systems, etc.), more general kind of equations
whichhavebeeninvestigatedoverthepastthreedecades,areestablishedaccording
to relationships among the variables (Dai 1989). As a valuable tool for system
modelingandanalysis,singularsystemtheoryhasbeenwidelyutilizedindifferent
fields including nonlinear electric and electronic circuits, constrained mechanics,
networks and economy (Lewis 1986).
Thisclassofsystems,whichwasintroducedfirstbyLuenbergerin1977,canbe
described as the following form.
EðtÞẋðtÞ=HðxðtÞ,uðtÞ,tÞ,
ð1Þ
yðtÞ=JðxðtÞ,uðtÞ,tÞ,
where H and J are appropriate dimensional vector functions, and the matrix EðtÞ
may be singular.
In1954,Gordoninvestigatedtheeconomictheoryofnaturalresourceutilization
in fishing industry and discussed the effects of harvest effort on its ecosystem
(Gordon 1954). To study the economic interest of the yield of harvest effort in his
theory of a common-property resource, Gordon proposed an algebraic equation to
put his idea into practice. Recently, by using this theory of natural resource uti-
lizationinindustry,theeffectsofharvesteffortonbiologicalsystemswerestudied,
and some singular model of these ecosystems were investigated to study the eco-
nomicinterestoftheyieldofharvesteffort.Besides,manyqualitativeanalysessuch
as stabilityanalysis, presence of bifurcations and chaos and controller design were
investigated(Zhangetal.2010;Chakrabortyetal.2011;Zhangetal.2012,2014).
The majority of these works has been carried out in dynamical modeling of
biological systems using integer-order differential equations which are valuable in
understandingthedynamicsbehavior.However,theeffectsoflong-rangetemporal
memoryandlong-rangespaceinteractionsinthesesystemsareneglected.Duetoits
ability to provide an exact description of different nonlinear phenomena, inherent
relation to various materials and processes with memory and hereditary properties
and greater degrees offreedom, fractional-order modeling has recently garnered a
lot of attention and gained popularity in the evaluation of dynamical systems
(Podlubny 1998; Diethelm 2010; Petras 2011). According to these reasons,
fractional-order modeling of many real phenomena such as biological systems has
more advantages and consistency rather than classical integer-order mathematical
modeling (Rivero et al. 2011).
In this chapter, singular system theory besides fractional calculus is utilized to
model the biological systems in the real world which takes the general form
