theoretical and computational advances in nonlinear dynamical...

EditorialTheoretical and Computational Advances in NonlinearDynamical Systems 2018

Zhi-Yuan Sun 12 K Nakkeeran3 Christos Volos4 and Xin Yu5

1 Institute of Fluid Mechanics Beihang University Beijing 100191 China2International Research Institute for Multidisciplinary Science Beihang University Beijing 100191 China3School of Engineering University of Aberdeen Aberdeen AB24 3UE UK4Department of Physics Aristotle University of Thessaloniki 54124Thessaloniki Greece5School of Aeronautic Science and Engineering Beihang University Beijing 100191 China

Correspondence should be addressed to Zhi-Yuan Sun sunzhiyuan137aliyuncom

Received 25 September 2018 Accepted 25 September 2018 Published 18 October 2018

Copyright copy 2018 Zhi-Yuan Sun et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

After the success of last yearrsquos special issue we have receivedmore than 30 submissions for TCAN 2018 Twelve articlescontaining the studies in chaos and synchronization non-linear evolution equations and applied dynamical systemsare accepted after strict review process A new Guest EditorChristos Volos from Aristotle University of Thessalonikiwas invited to serve in the areas of chaotic systems andsynchronization We believe these articles including theirbibliographic resources will substantially improve the qualityof our special issue and show wide interest to the readers innonlinear communities

Chaos belongs to the field of ldquoNonlinear OscillationsTheoryrdquo which was initiated in the previous century Theexperiment that boosted the consideration of chaotic behav-ior was due to Lorenz [1] In 1961 working in a simplifiedmodel of atmospheric transfer with three nonlinear differ-ential equations he observed numerically that when makingvery small changes in the initial conditions he got a hugeeffect on their solutions It was one evidence of the mainproperties of chaotic dynamics which was later known assensitive dependence on initial conditions also known asldquoButterfly Effectrdquo This property in fact had already beeninvestigated from the topological point of view by Poincarewhodescribed it in hismonograph ldquoScience andMethodrdquo [2]

Formany years the property of chaos becameundesirablesince it reduced the predictability of the chaotic systemover long time periods However the scientific communitywas gradually becoming aware of this type of dynamicalbehavior Some experiments where abnormal results had

been previously explained in terms of experimental error oradditional noise were evaluated for an explanation in termsof chaos In the mid-70s the term deterministic chaos wasintroduced by Li andYorke in a famous paper entitled ldquoPeriodThree Implies Chaosrdquo [3] Since then a huge number ofstudies in chaotic phenomena and dynamical systems thatproduce chaos have been published

The dynamics of a system displays chaotic behaviorwhen it never repeats itself and even if initial conditionsare correlated by proximity the corresponding trajectoriesquickly become uncorrelated As such the possibility oftwo (or more) chaotic systems oscillating in a coherent andsynchronized way seems to be not an obvious phenomenonHowever there are sets of coupled chaotic oscillators inwhichthe attractive effect of a sufficiently strong coupling can coun-terbalance the trend of the trajectories to diverge As a resultit is possible to reach full synchronization in chaotic systemssince they are coupled by a suitable dissipative couplingChaos synchronization began in the mid-80s about couplingof discrete and continuous identical systems evolving fromdifferent initial conditions [4ndash8] These works immediatelyreceived a great deal of attention from the scientific commu-nity and opened up a wide range of applications outside thetraditional scope of chaos and nonlinear dynamics researchSince then various synchronizationmethods and several newconcepts necessary for analyzing synchronization have beendeveloped

In this special issue three articles are dedicated to theinvestigation of chaotic systems and their synchronization

HindawiAdvances in Mathematical PhysicsVolume 2018 Article ID 4732167 3 pageshttpsdoiorg10115520184732167

2 Advances in Mathematical Physics

In ldquoFractional-Order Sliding Mode Synchronization forFractional-Order Chaotic Systemsrdquo by C Wang some suf-ficient conditions which are valid for the stability checkof fractional-order nonlinear systems are presented Basedon the aforementioned conditions the synchronization oftwo fractional-order chaotic systems is investigated Theasymptotical stability of the synchronization error can beguaranteed by a proposed fractional-order sliding modecontroller The numerical examples show the feasibility of theproposed method

In ldquoAdaptive Fuzzy Synchronization of Fractional-OrderChaotic Neural Networks with Backlash-Like Hysteresisrdquoby W Fu and H Liu an adaptive fuzzy synchronizationcontroller is designed for a class of fractional-order neuralnetworks subjected to backlash-like hysteresis input Thestability of the closed-loop system under the influence ofthe adaptive fuzzy controller is rigorously analyzed basedon the fractional Lyapunov stability criterion Furthermorefractional adaptation laws are established to update the fuzzyparameters The simulation examples indicate the effective-ness and the robustness of the proposed control method

In ldquoChaos in a System with an Absolute Nonlinearity andChaos Synchronizationrdquo by V K Tamba et al a system withan absolute nonlinearity is studied The system is shown tobe chaotic and has an adjustable amplitude variable whichis suitable for practical uses Circuit design of such a systemhas been realized without any multiplier and experimentalmeasurements have been reported In addition an adaptivecontrol has been applied to reach the synchronization of thesystem

Nonlinear evolution equations (NEEs) play importantroles in simulating the real dynamical behaviors that appearin various scientific and engineering fields Analysis of theNEEs especially for finding their solutions is one of themain tasks in nonlinear communities For integrable NEEsthere exist several effective methods such as the inversescattering transformation and the Hirota method in derivingcertain types of localized wave solutions eg the soliton andbreather solutions [9 10] For nonintegrable NEEs multipletools are employed to analyze their properties among whichthe numerical methods are becoming powerful with the rapiddevelopment of computational resources

In this special issue six articles are included to demon-strate advances relating to theNEEs fromboth of the analyticand numerical aspects

In ldquoTraveling Wave Solutions of Two Nonlinear WaveEquations by (G1015840G)-Expansion Methodrdquo by Y Shi et al the(G1015840G)-expansionmethod is employed to seek exact travelingwave solutions of two nonlinear wave equations Pade-II equation and Drinfelrsquod-Sokolov-Wilson equation Hyper-bolic function solution trigonometric function solution andrational solution with general parameters are obtained Thesolitary wave solutions and new traveling wave solutions canbe derived when special values of the parameters are taken

The advantageous Greenrsquos function method that origi-nally has been developed for nonhomogeneous linear equa-tions has been recently extended to nonlinear equations byFrasca In ldquoNonlinear Greenrsquos Functions for Wave Equationwith Quadratic and Hyperbolic Potentialsrdquo the author A

Zh Khurshudyan developed a rigorous numerical analysis ofsome second order one-dimensional wave differential equa-tions with quadratic and hyperbolic nonlinearities by meansof Frascarsquos method Numerical error analysis in both casesof nonlinearity is carried out for various source functionssupporting the advantage of the method

In ldquoThe Global Existence and Uniqueness of the ClassicalSolution with the Periodic Initial Value Problem for One-Dimension Klein-Gordon-Zakharov Equationsrdquo by C Sunand L Li the Galerkin method is applied to establish theapproximate solutions for the one-dimension Klein-Gordon-Zakharov (KGZ) equations and the local classical solutionsare obtained The authors also derive the existence anduniqueness of the global classical solutions of the KGZequations by integral estimates

In ldquoCIP Method of Characteristics for the Solution ofTide Wave Equationsrdquo by Y Nie et al the ConstrainedInterpolation ProfileMethod of Characteristics (CIP-MOC)is proposed to solve the tide wave equations with largetime step size The bottom topography and bottom frictionare included to the equation of Riemann invariants as thesource term Numerical experiments demonstrate the goodperformance of the scheme In addition numerical tests withreflective boundary conditions are carried out by CIP-MOCwith large time step size and good results are obtained aswell

In ldquoReachable Set Bounding for a Class of NonlinearTime-Varying Systems with Delayrdquo by X Zhu et al theauthors investigate the problem of reachable set boundingfor a class of continuous-time and discrete-time nonlineartime-varying systems with time-varying delay They use anapproach which does not involve the conventional Lyapunov-Krasovskii functional and propose new conditions such thatall the state trajectories of the system converge asymptoticallywithin a ball

In ldquoOn the Convergence Ball and Error Analysis ofthe Modified Secant Methodrdquo by R Lin et al the authorshave studied the convergence properties of a modification ofsecant iteration methods This work introduced the conver-gence ball and error estimate of the modified secant methodusing a technique based on Fibonacci series

In this special issue we also include three articles focusingon application of the nonlinear dynamical systems in diffu-sion behavior fluid and solid mechanics

In ldquoMotion of a Spot in aReactionDiffusion Systemunderthe Influence of Chemotaxisrdquo by S Kawaguchi the motionof a spot under the influence of chemotaxis is considered Forthis reason a two-component reaction diffusion system witha global coupling term and a Keller Segel type chemotaxisterm is presented The existence of an upper limit for thevelocity and a critical intensity for the chemotaxis over whichthere is no circular motion is proved As a consequence thechemotaxis suppresses the range of velocity for the circularmotion This braking effect on velocity originates from therefractory period behind the rear interface of the spot and thenegative chemotactic velocity

In ldquoA Study of the Transport of Marine PollutantsUsing Adjoint Method of Data Assimilation with Method ofCharacteristicsrdquo by X Li et al the authors apply an adjointmethod of data assimilation with the characteristic finite

Advances in Mathematical Physics 3

difference scheme to marine pollutant transport problemsand simulate the temporal and spatial distribution of marinepollutants They show that their method can not only reducesimulation error to get a good inversion but also enable largertime step size to decrease computation time and improve thecalculation efficiency

In ldquoDynamic Characteristics of Deeply-Buried SphericalBiogas Digesters in Viscoelastic Soilrdquo by H Hou et althe vibration characteristics of a deeply-buried sphericalmethane tank in viscoelastic soil subjected to cyclic loadingin the frequency domain are investigated By introducingpotential functions the closed-form expressions for thedisplacement and the stress of the soil surrounding the tankare obtained In addition the effects of relative physicalproperties and geometrical parameters on the dynamics ofthe system are discussed

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

Wewould like to express our sincere thanks to all the authorsand reviewers for their contributions to this special issue

Zhi-Yuan SunK NakkeeranChristos Volos

Xin Yu

References

[1] E N Lorenz ldquoDeterministic nonperiodic flowrdquo Journal of theAtmospheric Sciences vol 20 no 2 pp 130ndash141 1963

[2] H Poincare and F Maitland Science and method CourierCorporation 2003

[3] T Y Li and J A Yorke ldquoPeriod three implies chaosrdquo TheAmerican Mathematical Monthly vol 82 no 10 pp 985ndash9921975

[4] K Kaneko ldquoLyapunov analysis and information flow in coupledmap latticesrdquo Physica D Nonlinear Phenomena vol 23 no 1-3pp 436ndash447 1986

[5] V S Afraimovichh N N Verichev and M I RabinovichldquoStochastic synchronization of oscillations in dissipative sys-temsrdquo Izvestiya Vysshikh Uchebnykh Zavedeniui Radiofizikavol 29 no 9 pp 1050ndash1060 1986

[6] H Fujisaka and T Yamada ldquoStability theory of synchronizedmotion in coupled-oscillator systemsrdquo Progress of Theoreticaland Experimental Physics vol 69 no 1 pp 32ndash47 1983

[7] T Yamada and H Fujisaka ldquoStability theory of synchro-nized motion in coupled-oscillator systems II The mappingapproachrdquo Progress ofTheoretical and Experimental Physics vol70 no 5 pp 1240ndash1248 1983

[8] A S Pikovsky ldquoSynchronization and stochastization of theensemble of autogeneratorsby external noiserdquoRadiophysics andQuantum Electronics vol 27 pp 576ndash581 1984

[9] M J Ablowitz and P A Clarkson Solitons Nonlinear EvolutionEquations and Inverse Scattering Cambridge University PressNew York NY USA 1991

[10] R Hirota The Direct Method in Soliton Theory CambridgeUniversity Press Cambridge UK 2004

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2 Advances in Mathematical Physics

In ldquoFractional-Order Sliding Mode Synchronization forFractional-Order Chaotic Systemsrdquo by C Wang some suf-ficient conditions which are valid for the stability checkof fractional-order nonlinear systems are presented Basedon the aforementioned conditions the synchronization oftwo fractional-order chaotic systems is investigated Theasymptotical stability of the synchronization error can beguaranteed by a proposed fractional-order sliding modecontroller The numerical examples show the feasibility of theproposed method

In ldquoAdaptive Fuzzy Synchronization of Fractional-OrderChaotic Neural Networks with Backlash-Like Hysteresisrdquoby W Fu and H Liu an adaptive fuzzy synchronizationcontroller is designed for a class of fractional-order neuralnetworks subjected to backlash-like hysteresis input Thestability of the closed-loop system under the influence ofthe adaptive fuzzy controller is rigorously analyzed basedon the fractional Lyapunov stability criterion Furthermorefractional adaptation laws are established to update the fuzzyparameters The simulation examples indicate the effective-ness and the robustness of the proposed control method

In ldquoChaos in a System with an Absolute Nonlinearity andChaos Synchronizationrdquo by V K Tamba et al a system withan absolute nonlinearity is studied The system is shown tobe chaotic and has an adjustable amplitude variable whichis suitable for practical uses Circuit design of such a systemhas been realized without any multiplier and experimentalmeasurements have been reported In addition an adaptivecontrol has been applied to reach the synchronization of thesystem

Nonlinear evolution equations (NEEs) play importantroles in simulating the real dynamical behaviors that appearin various scientific and engineering fields Analysis of theNEEs especially for finding their solutions is one of themain tasks in nonlinear communities For integrable NEEsthere exist several effective methods such as the inversescattering transformation and the Hirota method in derivingcertain types of localized wave solutions eg the soliton andbreather solutions [9 10] For nonintegrable NEEs multipletools are employed to analyze their properties among whichthe numerical methods are becoming powerful with the rapiddevelopment of computational resources

In this special issue six articles are included to demon-strate advances relating to theNEEs fromboth of the analyticand numerical aspects

In ldquoTraveling Wave Solutions of Two Nonlinear WaveEquations by (G1015840G)-Expansion Methodrdquo by Y Shi et al the(G1015840G)-expansionmethod is employed to seek exact travelingwave solutions of two nonlinear wave equations Pade-II equation and Drinfelrsquod-Sokolov-Wilson equation Hyper-bolic function solution trigonometric function solution andrational solution with general parameters are obtained Thesolitary wave solutions and new traveling wave solutions canbe derived when special values of the parameters are taken

The advantageous Greenrsquos function method that origi-nally has been developed for nonhomogeneous linear equa-tions has been recently extended to nonlinear equations byFrasca In ldquoNonlinear Greenrsquos Functions for Wave Equationwith Quadratic and Hyperbolic Potentialsrdquo the author A

Zh Khurshudyan developed a rigorous numerical analysis ofsome second order one-dimensional wave differential equa-tions with quadratic and hyperbolic nonlinearities by meansof Frascarsquos method Numerical error analysis in both casesof nonlinearity is carried out for various source functionssupporting the advantage of the method

In ldquoThe Global Existence and Uniqueness of the ClassicalSolution with the Periodic Initial Value Problem for One-Dimension Klein-Gordon-Zakharov Equationsrdquo by C Sunand L Li the Galerkin method is applied to establish theapproximate solutions for the one-dimension Klein-Gordon-Zakharov (KGZ) equations and the local classical solutionsare obtained The authors also derive the existence anduniqueness of the global classical solutions of the KGZequations by integral estimates

In ldquoCIP Method of Characteristics for the Solution ofTide Wave Equationsrdquo by Y Nie et al the ConstrainedInterpolation ProfileMethod of Characteristics (CIP-MOC)is proposed to solve the tide wave equations with largetime step size The bottom topography and bottom frictionare included to the equation of Riemann invariants as thesource term Numerical experiments demonstrate the goodperformance of the scheme In addition numerical tests withreflective boundary conditions are carried out by CIP-MOCwith large time step size and good results are obtained aswell

In ldquoReachable Set Bounding for a Class of NonlinearTime-Varying Systems with Delayrdquo by X Zhu et al theauthors investigate the problem of reachable set boundingfor a class of continuous-time and discrete-time nonlineartime-varying systems with time-varying delay They use anapproach which does not involve the conventional Lyapunov-Krasovskii functional and propose new conditions such thatall the state trajectories of the system converge asymptoticallywithin a ball

In ldquoOn the Convergence Ball and Error Analysis ofthe Modified Secant Methodrdquo by R Lin et al the authorshave studied the convergence properties of a modification ofsecant iteration methods This work introduced the conver-gence ball and error estimate of the modified secant methodusing a technique based on Fibonacci series

In this special issue we also include three articles focusingon application of the nonlinear dynamical systems in diffu-sion behavior fluid and solid mechanics

In ldquoMotion of a Spot in aReactionDiffusion Systemunderthe Influence of Chemotaxisrdquo by S Kawaguchi the motionof a spot under the influence of chemotaxis is considered Forthis reason a two-component reaction diffusion system witha global coupling term and a Keller Segel type chemotaxisterm is presented The existence of an upper limit for thevelocity and a critical intensity for the chemotaxis over whichthere is no circular motion is proved As a consequence thechemotaxis suppresses the range of velocity for the circularmotion This braking effect on velocity originates from therefractory period behind the rear interface of the spot and thenegative chemotactic velocity

In ldquoA Study of the Transport of Marine PollutantsUsing Adjoint Method of Data Assimilation with Method ofCharacteristicsrdquo by X Li et al the authors apply an adjointmethod of data assimilation with the characteristic finite






Acknowledgments



Xin Yu

References


















Journal of












Journal of







Volume 2018






Volume 2018













Acknowledgments



Xin Yu

References


















Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








theoretical and computational advances in nonlinear dynamical...

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