editorial control, stability, and qualitative theory of...

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013, Article ID 792518, 2 pageshttp://dx.doi.org/10.1155/2013/792518

EditorialControl, Stability, and Qualitative Theory of Dynamical Systems

Nazim Idrisoglu Mahmudov,1 Mark A. McKibben,2 Sakthivel Rathinasamy,3 and Yong Ren4

1 Department of Mathematics, Eastern Mediterranean University, Gazimagusa, North Cyprus, Mersin 10, Turkey2Department of Mathematics, West Chester University of Pennsylvania, 25 University Avenue, West Chester, PA 19383, USA3Department of Mathematics, Sungkyunkwan University, Suwon 440 746, Republic of Korea4Department of Mathematics, Anhui Normal University, Wuhu 241000, China

Correspondence should be addressed to Nazim Idrisoglu Mahmudov; [email protected]

Received 5 November 2013; Accepted 5 November 2013

Copyright © 2013 Nazim Idrisoglu Mahmudov 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.

Dynamical systems play a crucial role in the mathematicalmodeling of phenomena across disciplines. Understandingissues concerning controllability, stability, and other qual-itative aspects of such systems is important in enhancingour understanding of the mathematical models described bythese systems. This issue compiles 18 manuscripts coveringvarious aspects of deterministic and stochastic dynamicalsystems theory and its applications. Elements of the theoryand techniques from dynamical systems, control and optimalcontrol theory, stochastic analysis, and stochastic evolutionequations are used throughout these papers to establish animpressive collection of results. The results established in allarticles in this issue have application in multiple disciplines,and they often contain replicable numerical analysis compo-nents that could have broader applicability.

On the more abstract end of the spectrum, two papersexplore theoretical issues arising in deterministic andstochastic fractional differential equations. Specifically, H.Aktuglu et al. established existence results for Caputo frac-tional BVPs andG. Shen et al. studied theHolder regularity inlocal time of the fractionalOrnstein-Uhlenbeck process; bothhave applications in diffusion, finance, and econophysics.Two papers focus on abstract optimal control problems: A.R. Safari et al. establish amaximum principle for abstract sys-tems with integral boundary conditions arising in the math-ematical modeling of heat conduction and plasma physics,while S. Meherrem and R. Akbarov study the role thatexhausters and quasidifferentiability play in switching con-trol problems arising in the mathematical study of chemi-cal processes, automotive systems, and circuit theory. One

paper, by X. Qin, focuses on the stabilization of a class ofstochastic nonholonomic system, which involved construct-ing a smooth state-feedback control law that ensures certainstochastic properties of the solution. Also, Y. Li and Y.Zhao study tracking control and synchronization of thehyperchaotic Lorenz-Stenflo system, which contributes to thegrowing literature in that direction.

Three papers study control problems for very differentsystems (networks, deterministic systems, and stochasticsystems), all with time delays. B.Wang andY. Sun investigatedcontrol problems for amultiagent systemwith heterogeneousdelays in directed networks; A. E. Bashirov and M. Jneidestablished partial complete controllability results for abstractdeterministic systems; and X. Zhou et al. studied the BIBOstabilization in the mean square for discrete-time stochasticsystems. All of these results have significant applicability formathematical models involving the respective type of system.

Qualitative results involving the notion of periodicitywere established in two of the papers. D. de C. Braga et al.studied the approximation of periodic orbits for dynamicalsystems of the type arising in the circuit analysis and math-ematical ecology. G. Ge and W. Wang established period-doubling bifurcation results for feedback control systems ofdamped linear oscillators.

Finally, on the more applied end of the spectrum, nearlyhalf of the papers in this issue study more concrete systemsarising in the mathematical models from population ecology,mathematical biology, andmathematical finance. Two papersfocus on different aspects of the SIRS epidemic model:W. Liustudied the SIRS model with random perturbations, while

2 Abstract and Applied Analysis

Y. Cai et al. focused on related stochastic dynamics.Four papers studied population models of various types.C. Xu and Y. Wu studied the Lotka-Volterra modelwith time-varying delays. H. Li et al. investigated the(non)persistence of the food chain model with stochasticperturbation. L. Qi and J. Cui discussed the parasite disease,schistosomiasis, with mating structure. And, Z. Luo andL. Luo established the existence, uniqueness, and globalattractivity of positive periodic solutions for multispeciespopulation model with impulses. One paper focusedonmathematical finance; specifically, Y. Zhai et al. performeda bifurcation analysis for the Rayleigh price model with timedelay.

It is our hope that this compilation of papers will provideour readers and researchers with new ideas to continuesimilar lines of research in control, stability, and qualitativetheory of dynamical systems.

Nazim Idrisoglu MahmudovMark A. McKibben

Sakthivel RathinasamyYong Ren

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