Editorial - Hindawi Publishing Corporationdownloads.hindawi.com/journals/mpe/2017/3737681.pdf2 MathematicalProblemsinEngineering et al. have studied the stabilization problem for a

EditorialNew Trends on Modeling, Design, and Control ofChaotic Systems

Jesus M. Munoz-Pacheco,1 Christos Volos,2 Eric Campos-Canton,3,4

Ahmad Taher Azar,5,6 Viet-Thanh Pham,7 and Ahmed G. Radwan8

1Facultad de Ciencias de la Electrónica, Autonomous University of Puebla (BUAP), Av. San Claudio y 18 Sur, Edif. FCE1,72570 Puebla, PUE, Mexico2Department of Physics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece3División de Matemáticas Aplicadas, IPICYT, Camino a la Presa San José 2055, Col. Lomas 4a Sección, 78216 San Luis Potośı,SLP, Mexico4Mathematics Department, University of Houston, Houston, TX, USA5Faculty of Computers and Information, Benha University, Benha 13511, Egypt6Nile University, Giza, Egypt7School of Electronics and Telecommunications, Hanoi University of Science and Technology, Hanoi 100000, Vietnam8Engineering Mathematics and Physics Department, Faculty of Engineering, Cairo University, Giza 12613, Egypt

Correspondence should be addressed to Jesus M. Munoz-Pacheco; [email protected]

Received 6 August 2017; Accepted 6 August 2017; Published 13 September 2017

Copyright © 2017 Jesus M. Munoz-Pacheco et al. This is an open access article distributed under the Creative CommonsAttribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work isproperly cited.

Nowadays, one of the most studied phenomena is chaos intothe nonlinear dynamical systems. For deterministic chaos toexist, a nonlinear dynamical system must have a dense setof periodic orbits and be transitive and sensitive to initialconditions. Their significance has been increased during thelast decade because of several applications in diverse fieldsranging from living systems, such as synchronization inneurobiology, chemical reactions among pancreatic cells, andsocial, economical, or political events, to nonliving systemsincluding robotics, low power high-speed data transceiversfor medical applications, chaotic electrochemical oscilla-tors, encrypted communications, and control algorithms formotor drivers in electric vehicles. However, in order toexploit all the possible engineering applications, the openproblems about chaotic systems need to be addressed byproposing novel theoretical and practical approaches focusedon modeling, simulation, synthesis, design, control, andcircuit implementation.

Lately, several synchronization schemes to coupling twochaotic systems, at least, have attracted increasing attention

due to the fact that they are the core of many chaos-based secure communications technologies. Also, the currentresearch interest in proposing novel multiscroll chaotic sys-tems with self-excited or hidden attractors to increase thecomplexity has augmented considerably. Each new chaoticsystem is a potential candidate to improve chaos-based appli-cations. Additionally, chaotic systems have been recently usedto analyze financial models trying to predict the complexityof the markets with a huge effect in the global economy.

Therefore, the overall purpose of this special issue liesin gathering the latest scientific trends on the topics ofchaotic systems with emphasis on real-world engineeringapplications. We had received a total of 28 submissionswhere the authors are from geographically distributed coun-tries (China, Vietnam, Greece, Poland, Taiwan, Serbia, Iran,Tunisia, Turkey, Saudi Arabia, Egypt, Mexico, and CzechRepublic).This reflects the high impact of the proposed topicand the seniority in organization of this special issue.

In the paper “Uncertain Unified Chaotic Systems Controlwith Input Nonlinearity via Sliding Mode Control,” Z. Shen

HindawiMathematical Problems in EngineeringVolume 2017, Article ID 3737681, 3 pageshttps://doi.org/10.1155/2017/3737681

https://doi.org/10.1155/2017/3737681

2 Mathematical Problems in Engineering

et al. have studied the stabilization problem for a class ofunified chaotic systems subject to uncertainties and inputnonlinearity. Based on the sliding mode control theory,authors present a newmethod for the slidingmode controllerdesign and the control law algorithm for such systems. Inorder to achieve the goal of stabilization unified chaoticsystems, the presented controller can make the movementstarting from any point in the state space reach the slidingmode in limited time and asymptotically reach the originalong the switching surface. Compared with the existingliterature, the proposed controller has many advantages, suchas having small chattering, having good stability, and beingless conservative.

In the paper “ComplexityAnalysis of a Triopoly Coopera-tion-Competition Game Model in Convergence ProductMarket,” L. Zhao et al. consider a tripartite cooperation-competition game model for the convergence product mar-ket, whose products are compounds of two base productsor services. An early convergence product firm monopolyin this market and two potential entrants from the baseproducts decide to cooperate with another to compete withthemonopolist. Authors analyzed factors that affect existenceand local stability of the Nash equilibrium. Rich nonlineardynamic behaviors like bifurcation, chaos, and attractorsare presented to explain the complex relationships betweenthe three players. Results showed that the pulling effect onprofit for the united R&D activity can significantly enlargethe stable region. Too frequently adjusted price strategy willbring the system into chaos. A parameter feedback controlmethod is given to control the chaotic system and theauthors numerically verified its effectiveness. This study hassignificant values to understand the fluctuations in relatedconvergence product market.

In the paper “Hopf Bifurcation, Positively Invariant Set,and Physical Realization of a New Four-Dimensional Hyper-chaotic Financial System,” G. Kai et al. introduce a newfour-dimensional hyperchaotic financial system on the basisof an established three-dimensional nonlinear financial sys-tem and a dynamic model by adding a controller termto consider the effect of control on the system. In termsof the proposed financial system, the sufficient conditionsfor nonexistence of chaotic and hyperchaotic behaviors arederived theoretically. Then, the solutions of equilibria areobtained. For each equilibrium, its stability and existenceof Hopf bifurcation are validated. Based on correspondingfirst Lyapunov coefficient of each equilibrium, the analyticalproof of the existence of periodic solutions is given. Theultimate bound and positively invariant set for the financialsystem are obtained and estimated. There exists a stableperiodic solution obtained near the unstable equilibriumpoint. Finally, the dynamic behaviors of the new systemare explored by theoretical analysis by using the bifurcationdiagrams and phase portraits. Moreover, the hyperchaoticfinancial system has been simulated and implemented toconfirm the results of the numerical integrations and its realcontribution to engineering.

In the paper “Sliding Mode Control of Discrete ChaoticSystem Based on Multimodal Function Series Coupling,”F. Hu has proposed a new sliding mode control model

of discrete chaotic systems based on multimodal functionseries coupling to overcome the shortcomings of the standardPSO algorithm in multimodal function optimization. Firstly,a series coupled PSO algorithm (PP algorithm) based onmultimodal function is constructed, which is optimized bymultipeak solution on the basis of the standard PSO algo-rithm. Secondly, the improved PSO algorithm is applied tosearch all the extreme points in the feasible domain. Thirdly,the Powell method is used to perform the local optimizationof the search results, and the newly generated extreme pointsare added to the extreme point database according to thesame peak judgment operator. Finally, the long training timeof PP algorithm can be overcome by the characteristics of fastconvergence rate of the cloudmutationmodel. And also, boththe population size and the redundancy can be reduced.Then,the clonal selection algorithm is used to keep the diversityof the population effectively. Simulation results of the slidingmode control of discrete chaotic systems have shown that theimproved PSO algorithm obviously improves the responsespeed, overshoot, and so on.

In the paper “Synchronization in Coupled MultistableSystems with Hidden Attractors,” G. PM and T. Kapitaniakpresent the results of coupling multistable systems whichhave hidden attractors with each other. Three modifiedSprott systems were coupled and their synchronization wasobserved. The final state of the synchronized system changeswith the change in the coupling strength. This was observedfor two different types of coupling, one with a single variableand the other with two system variables.

In the paper “Stability and Multiscroll Attractors of Con-trol Systems via the Abscissa,” E.-C. Dı́az-González et al. haveestablished an approach to generate multiscroll attractors viadestabilization of piecewise linear systems based on Hurwitzmatrix. First the authors present some results about theabscissa of stability of characteristic polynomials from lineardifferential equations systems; that is, they consider Hurwitzpolynomials. The starting point is the Gauss-LucasTheorem,the authors provide lower bounds for Hurwitz polynomials,and by successively decreasing the order of the derivativeof the Hurwitz polynomial one obtains a sequence of lowerbounds. The results are extended in a straightforward wayto interval polynomials; then authors apply the abscissa as ameasure to destabilizeHurwitz polynomial for the generationof a family ofmultiscroll attractors based on a class of unstabledissipative systems (UDS) of type affine linear.

Acknowledgments

The guest editorial team would like to thank the authors ofall the papers submitted to this special issue. Given the spacelimitations, a number of high quality contributions couldnot be accommodated. The editors also wish to thank theanonymous reviewers, some of whom helped with multiplereview assignments. Additionally, we would like to thankthe journal’s Editorial Board for being very encouraging andaccommodative regarding this special issue. Finally, J. M.Munoz-Pacheco acknowledges CONACYT for the financialsupport (no. 258880: Proyecto Apoyado por el Fondo Secto-rial de Investigación para la Educación). We hope that you

Mathematical Problems in Engineering 3

will enjoy reading this special issue devoted to this excitingand fast-evolving field as much as we have done.

Jesus M. Munoz-PachecoChristos Volos

Eric Campos-CantonAhmad Taher AzarViet-Thanh PhamAhmed G. Radwan

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Editorial - Hindawi Publishing Corporationdownloads.hindawi.com/journals/mpe/2017/3737681.pdf2 MathematicalProblemsinEngineering et al. have studied the stabilization problem for a