editorial differential games and discrete dynamics in applied...

EditorialDifferential Games and Discrete Dynamics in Applied Sciences

Massimiliano Ferrara,1,2 Gafurjan Ibragimov,3 and Vincenzo Scalzo4

1Department of Law and Economics, University Mediterranea of Reggio Calabria, Via dei Bianchi 2, Palazzo Zani,89127 Reggio Calabria, Italy2The Invernizzi Centre for Research in Innovation, Organization, Strategy and Entrepreneurship (ICRIOS), Bocconi University,Via Sarfatti 25, 20136 Milan, Italy3INSPEM and Department of Mathematics, Faculty of Science, Universiti Putra Malaysia, 43400 Serdang, Selangor, Malaysia4Department of Economics and Statistics, University of Naples Federico II, Complesso Monte S. Angelo, Via Cintia, 80126 Naples, Italy

Correspondence should be addressed to Massimiliano Ferrara; massimiliano.ferrara@unirc.it

Received 29 December 2016; Accepted 4 January 2017; Published 24 January 2017

Copyright © 2017 Massimiliano Ferrara et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

In many scientific fields, differential games are problemsrelated to the analysis of conflict and strategical interactionsin the model of a dynamical system. In particular, in apursuit-evasion game, one or more pursuers try to captureone or more evaders that try to avoid capture. Contests ofpursuit and evasion are among the most widespread, chal-lenging, and important optimization problems that confrontmobile agents. But dynamic, stochastic, continuous-space,continuous-time, or discrete-time discrete-space games areusually difficult to handle. Agents that pursue or evade mustmaintain complex sensory-motor coordination with respectto both a physical environment and a hostile opponent.Because of its extensive applications, such as searching build-ings for intruders, traffic control, military strategy, surgicaloperation, and industrial management, a lot of research hasbeen developed in various directions during the last threedecades, not only for that concerns the specific fields ofapplication. In time of these methods, related mathematicaltools and models have been intensively studied focusingon all the aspects concerning “dynamics” in a broad sense.In compliance of this aim, by this special issue, we havebeen interested in articles that explored various aspectsof differentiable games and also in dynamics models. Thiseditorial provides a brief review of some concepts related tothe subject of the papers published in this collectanea devotedto the onset of differential games and discrete dynamics withapplications to various branches of research. “Dynamics” iscurrently an active and fashionable discipline that is havinga profound effect on a wide variety of fields, including

engineering, physics, biology, economics, finance, and socialsciences.

By the present work we are going to present some selectedpapers which analyze various aspects and applications fall inthe “dynamics” following a “discrete”mathematical approach.

The paper entitled “Linear Pursuit Differential Gameunder Phase Constraint on the State of Evader” by M.Ferrara et al. deals with the linear pursuit differential gameswhere pursuit control is subjected to integral constraints andevasion control is subjected to geometric constraint, withphase constraint being imposed on the state of the evader.For such constraints on controls of players, to obtain solutionof pursuit problem for all initial states of phase space iscomplicated because of boundedness of pursuit resources. Bythis work, the following technique is proposed to control thephase vector of the pursuer: first-phase vector of the pursueris brought to a given bounded, convex set, regardless of thebehavior of evading player and then, using the evasion controland initial positions of the players, the pursuit control thatensures the completion of pursuit is constructed.

In “Simple Motion Pursuit and Evasion DifferentialGames with Many Pursuers on Manifolds with EuclideanMetric,” M. Ferrara et al. consider pursuit and evasiondifferential games of a group of pursuers and one evaderon manifolds with Euclidean metric. The authors establishthat each of the differential games (pursuit or evasion) isequivalent to a differential game of groups of countablymany pursuers and one group of countably many evaders inEuclidean space. All the players in any of these groups are

HindawiDiscrete Dynamics in Nature and SocietyVolume 2017, Article ID 6023140, 3 pageshttps://doi.org/10.1155/2017/6023140

2 Discrete Dynamics in Nature and Society

controlled by one controlled parameter. It has been founda condition under which pursuit can be completed, and ifthis condition is not satisfied, then evasion is possible. Somestrategies for the pursuers in pursuit game which ensurecompletion of the game for a finite time have been given. Inthe specific case of evasion game, it was given a strategy forthe evader.

By the paper entitled “New JLS-Factor Model versusthe Standard JLS Model: A Case Study on Chinese StockBubbles” Z. Hu and C. Li extend the Johansen-Ledoit-Sornette(JLS) model by introducing fundamental economic factors inChina (including the interest rate and deposit reserve rate)and the historical volatilities of targeted andUS equity indicesinto the original model, which is a flexible tool to detectbubbles and predict regime changes in financial markets. Ageneral method has been introduced to incorporate theseselected factors in addition to the log-periodic power lawsignature of herding and compare the prediction accuracyof the critical time between the original and the new JLSmodels (termed the JLS-factor model) by applying thesetwo models to fit two well-known Chinese stock indices inthree bubble periods. The results showed that the JLS-factormodel with Chinese characteristics successfully depicts theevolutions of bubbles and “antibubbles” as well as constructsefficient end-of-bubble signals for all bubbles in Chinesestock markets. In addition, the results of standard statisticaltests demonstrate the excellent explanatory power of theseadditive factors and confirm that the new JLSmodel providesuseful improvements over the standard JLS model.

X. Meng et al. in the work entitled “Worst-Case Invest-ment and Reinsurance Optimization for an Insurer underModel Uncertainty” study optimal investment-reinsurancestrategies for an insurer who faces model uncertainty. Theinsurer is allowed to acquire new business and invest into afinancial market which consists of one risk-free asset and onerisky asset whose price process is modeled by a geometricBrownian motion. Minimizing the expected quadratic dis-tance of the terminal wealth to a given benchmark underthe “worst-case” scenario, the authors obtain the closed-formexpressions of optimal strategies and the corresponding valuefunction by solving the Hamilton-Jacobi-Bellman (HJB)equation. Numerical examples have been presented to showthe impact of model parameters on the optimal strategies.

In “Numerical Approach Based on Two-DimensionalFractional-Order Legendre Functions for Solving FractionalDifferential Equations” a robust numerical approach is pro-posed to obtain the numerical solution of some fractionaldifferential equations. The principal characteristic of theapproach proposed by F. Zhao et al. was the new orthogonalfunctions based on shifted Legendre polynomials to the frac-tional calculus. Also the fractional differential operationalmatrix is driven. Then the matrix with the “Tau method”is utilized to transform this problem into a system of linearalgebraic equations. By solving the linear algebraic equations,the numerical solution is obtained.The approach is tested viasome examples. It is shown that the FLFs yield better results.Finally, error analysis shows that the algorithm is convergent.

In “Dynamical Analysis of a Computer Virus Modelwith Delays,” J. Liu et al. have proposed an SIQR computer

virus model with two delays. The linear stability conditionsare obtained by using characteristic root method and thedeveloped asymptotic analysis shows the onset of a Hopfbifurcation occurswhen the delay parameter reaches a criticalvalue. Moreover the direction of the Hopf bifurcation andstability of the bifurcating period solutions are investigatedby using the normal form theory and the center manifoldtheorem. Finally, numerical investigations are carried out toshow the feasibility of the theoretical results.

By “A Note on Discrete Multitime Recurrences ofSamuelson-Hicks Type” B. A. Pansera and F. Strati aim atfostering further research on the applications of multitimerecurrences in the frame of multitime theory. In particular,the authors apply thismethod by generalizing the Samuelson-Hicks model so as to make the new concept of time that thismethod proposes clear. In particular, the multitime approachdecomposes a point of time into a vector, taking into accounthow different coordinates of time referring to the same datecan affect the dynamics of a model. Since 1932, the adjectivemultitime, introduced by Paul Dirac, appears so as to copewith an environment in which there might be more than onedimension of time. The authors make some considerationsconcerning the physical idea of the multitime notion: theyrecall that a coordinate space is an index numbering degreeof freedom, and the coordinate of time is the usual physicaltime in which a system evolves. This model is satisfactory,unless the reader turns his attention to relativistic problems.Moreover, some physical phenomena (and social sciencestoo) are observed in a two-time environment, where theintrinsic time and the observer time are found. In some realphenomena, there is no reason to prefer one coordinate toanother. Following these assumptions, the authors refer tomultitime as vector parameters of evolution in multitimegeometric evolution andmultitime optimal control problems.

Constructing a Poincare map is a method that is oftenused to study high-dimensional dynamical systems. In“Topological Entropy of One Type of Nonoriented Lorenz-Type Maps,” G. Feng has proposed a geometric model ofnonoriented Lorenz-type attractor by using this method, andits dynamical property has been described. The topologicalentropy of one-dimensional nonoriented Lorenz-type mapsis also computed in terms of their kneading sequences.

In “Dynamic Complexities in 2-Dimensional Discrete-Time Predator-Prey Systems with Allee Effect in the Prey,”L. Ren et al. have presented a study by which the “Alleeeffect” is incorporated into a predator-prey model withlinear functional response. Compared with the predator-preywhich only takes the crowding effect and predator partiallydependent on prey into consideration, it has been foundthat the “Allee effect” of the prey species would increase theextinction risk of both the prey and predator. Moreover, byusing a centermanifold theoremand bifurcation theory, it hasbeen shown that the model with “Allee effect” undergoes theflip bifurcation andHopf bifurcation in the interior of𝑅2

+with

different “Allee effect” values. In the two bifurcations, theauthors came to the conclusion that different “Allee effects”have different bifurcation value and the increasing of the“Allee effect” increases the value of bifurcation, respectively.

Discrete Dynamics in Nature and Society 3

By “Existence of Nonoscillatory Solutions of Higher-Order Neutral Differential Equations with Distributed Coef-ficients andDelays,” Y. Liu et al. have considered the existenceof nonoscillatory solutions of higher-order neutral differen-tial equations with distributed coefficients. The authors haveused the contraction principle in obtaining new sufficientcondition for the existence of nonoscillatory solutions.

In the paper entitled “Modeling and Nonlinear Responseof the Cam-Follower Oblique-Impact System” by Y.-F. Yanget al., in order to quickly and accurately analyze the complexbehavior of cam-follower oblique-impact system, a mathe-matical model, which can describe separation, impact, andcontact, was established.The transient impact hypothesis wasextended, and the oblique collision model was establishedby considering the tangential slip. “Moreau time-steppingmethod” was employed to solve the linear complementarityproblem transformed by the oblique-impact equations. Thesimulation results have been organized showing that the camand follower kept permanent contactwhen the cam rotationalspeed was low.With the increase of the cam rotational speed,the cam and follower would be separated and then haveimpact under the gravity action.The system performance hasshown very complex nonlinear characteristics.

In “Risk-Averse Evolutionary Game Model of AviationJoint Emergency Response,” W. Pan et al. have studiedeffects of risk-averse attitude of both participators in aviationjoint emergency response on the coevolution of cooperationmechanisms and individual preferences between airport andnonprofit organization. First, based on the current aviationjoint emergency mechanism in China, the authors put for-ward two mechanisms to select the joint nonprofit organi-zation, including reputation cooperation and bidding com-petition. Meanwhile, they have considered two preferencesincluding altruism and selfishness.Then “replicator dynamicsequations” by using the theory of conditional value-at-risk(CVaR) taking risk aversion attitude into account were built.Finally, the authors show that the risk-averse attitude of theother game participator affects the one participator’s decisionand the effects subject to some parameters.

The guest editors of this special issue hope that problemsdiscussed and investigated in the papers by the authors of thisissue can inspire and motivate researchers in these fields todiscover new, innovative, and novel applications in all areasof pure and applied mathematics.

Acknowledgments

We would like to express our great gratitude to all ofthe authors for their contributions and the reviewers fortheir serious evaluation of the papers submitted by valuablecomments and timely feedback.

Massimiliano FerraraGafurjan Ibragimov

Vincenzo Scalzo

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