Editorial Complex Discrete Nonlinear Dynamics: Chaos and ...downloads.hindawi.com/journals/ddns/2013/806876.pdf · clinic orbits and chaotic dynamics of the laminated composite piezoelectric

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2013, Article ID 806876, 2 pageshttp://dx.doi.org/10.1155/2013/806876

EditorialComplex Discrete Nonlinear Dynamics: Chaos and ItsApplications

Qingdu Li,1 Timothy J. Healey,2 Xiao-Song Yang,3 and Xiaofeng Liao4

1 Key Laboratory of Network Control & Intelligent Instrument, Ministry of Education, Chongqing University of Posts andTelecommunications, Chongqing 400065, China

2Department of Mathematics, Cornell University, Ithaca, NY 14850, USA3Department of Mathematics, Huazhong University of Science and Technology, Wuhan 430074, China4 School of Electronic and Information Engineering, Southwest University, Chongqing 400715, China

Correspondence should be addressed to Qingdu Li; [email protected]

Received 31 July 2013; Accepted 31 July 2013

Copyright © 2013 Qingdu Li et al. This is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Nonlinear and chaotic behavior appears widely and naturallyin many discrete dynamical systems; its study has led to vastmultidisciplinary research in fields ranging from the naturalsciences (mechanics, chemistry, biology, ecology, etc.) to thesocial sciences (economics, sociology, etc.) and engineering(electronics, control, communication, security, etc.). Recentadvances in dynamical systems theory and computationalmethodologies have provided new ways of understandingsuch complex phenomena via novel tools and applications ofdiscrete nonlinear dynamics.

This special issue contains ten papers, of which eightstudy chaotic behavior in some practical models from inter-esting fields, such as a Cournot-Bertrand duopoly gamemodel, a price game model of irrigation water, a susceptible-infected-susceptible (SIS) model, the credit risk transfer(CRT) market model, a tumor growth model, a fractional-order system, a small neural network, and rectangular plates.There are also two theoretical papers focusing on the bifur-cation of limit cycles from periodic orbits and the existenceand uniqueness of the global solution for the stochasticBoussinesq equations, respectively.

Two game models are studied in detail in this specialissue as typical discrete-time systems. “Complexity analysis ofa Cournot-Bertrand duopoly game model with limited infor-mation” by H. Wang and J. Ma proposes a Cournot-Bertrandmixed game model, where the market has linear demandand two firms have the same fixed marginal cost, and shows

complex dynamics, such as bifurcation scenarios and route tochaos, as well as parameter basin by numerical experiment.“Bifurcation and chaos in a price game of irrigation water ina coastal irrigation district” by B. Xin and Y. Li emphaticallydiscusses the stability and codimension-two period-doubling(flip) bifurcation in a price game model with numericalbifurcation-based continuation methods, and the 0-1 testalgorithm is used to compute the median value of correlationcoefficient to indicate chaotic dynamics.

Mathematical models in the real world are often time-continuous systems or hybrid systems. To study their non-linear dynamics, one frequently converts them to discretesystems. A common method is discretizing time with awell-chosen stepsize. “Bifurcation and chaotic behavior of adiscrete-time SIS model” by J. Li andN. Cui obtains a discrete-time epidemic model with the Euler method, and showsrich and complex dynamical behaviors, such as transcriticalbifurcation, flip bifurcation, Hopf bifurcation, and chaos.“Dynamics evolution of credit risk contagion in the CRTmarket” by T. Chen et al. introduces a nonlinear dynam-ics model to depict the dynamic behavior characteristicsof evolution of credit risk contagion through numericalsimulation and reports a series of Hopf bifurcations andchaotic phenomena in the process of credit risk contagion.“Modeling a tumor growth with piecewise constant arguments”by F. Bozkurt constructs two models of a tumor growthand shows that the increase of the population growth rate

2 Discrete Dynamics in Nature and Society

decreases the local stability of the positive equilibrium point.“A fractional-order chaotic system with an infinite number ofequilibrium points” by P. Zhou et al. studies chaos in a new4-dimensional fractional-order system, which has an infinitenumber of equilibrium points, and also presents a chaoticsynchronization scheme.

Another common method to discretize time-continuoussystems is using Poincare return maps. “Multipulse hetero-clinic orbits and chaotic dynamics of the laminated compositepiezoelectric rectangular plate” by M. Yao and W. Zhangdemonstrates how to employ the energy phase method toanalyze the Shilnikov type multipulse heteroclinic bifurca-tions and chaotic dynamics of high-dimensional nonlinearsystems in engineering applications. “Horseshoe chaos in a3d neural network with different activation functions” by F.Yang et al. studies a small neural network with three neurons.Although the network is a simple hybrid system with allsubsystems being exponentially stable, they find that thenetwork can exhibit very complex dynamics such as limitcycles and chaos, which implies that chaos is more related toits weight matrix than the type of activation functions.

Limit cycles, as well as their number and distribution, aresignificant for understanding the geometrical structures ofthe state space. “Bifurcation of limit cycles of a class of piecewiselinear differential systems in R4 with three zones” by Y. Chengstudies the bifurcation of limit cycles from periodic orbitsof a four-dimensional system by the averaging method, andshows that when the parameter is sufficiently small, at mostsix limit cycles can bifurcate from periodic orbits in a class ofasymmetric piecewise linear perturbed systems.

The existence and uniqueness of the solution are a fun-damental problem for each dynamical system. “Perturbationof stochastic Boussinesq equations with multiplicative whitenoise” by C. Yang and X. Zhao studies the Boussinesqequations perturbed bymultiplicative white noise, and showsthe existence and uniqueness of the global solution.They alsoobtain some regularity results for the unique solution.

The topics of the selected papers in this issue do not ofcourse cover this entire multidisciplinary field. Rather theyrepresent rich and many-faceted results that we have thepleasure of sharing with the readers.

Since most dynamical systems have no explicit solution,it is often a hard task to theoretically analyze complex discretenonlinear behavior. Although numerical methods works formost systems, it is often criticized because of numerical errorsand increasing computational cost. Combining theoreticaland numerical studies, like some of the aforementionedpapers is perhaps a powerful approach. With interval algo-rithms on massive computing devices, nonlinear behavior,such as limit cycles, homoclinic/heteroclinic orbits, and evenchaos, can be studied rigorously, and the complex geometricalstructure of the state space and the parameter space mayalso be analyzed to reveal mechanisms behind the complexbehavior, which are important for scientific and engineeringpurposes.

Acknowledgments

We would like to express our appreciation to the authorsfor their excellent contributions and patience in assisting us.The fundamental work of all reviewers on these papers isalso very greatly acknowledged. In addition, Q. Li also wouldlike to thank the National Natural Science Foundation ofChina (61104150) and Science Fund for Distinguished YoungScholars of Chongqing (cstc2013jcyjjq40001).

Qingdu LiTimothy J. HealeyXiao-Song YangXiaofeng Liao

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Editorial Complex Discrete Nonlinear Dynamics: Chaos and ...downloads.hindawi.com/journals/ddns/2013/806876.pdf · clinic orbits and chaotic dynamics of the laminated composite piezoelectric