ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.11(2011) No.1,pp.121-128 Control and Synchronization of Chaos in Biological Systems Via Backsteping Design Olasunkanmi I. Olusola 1 ∗ , Uchechukwu E. Vincent 2,3,4 ,Abdulahi N. Njah 1 ,Emad Ali 2 1 Nonlinear Dynamics Research Group, Department of Physics, University of Agriculture, P.M.B 2240, Abeokuta, Nigeria. 2 Department of Nonlinear Dynamics and Statistical Physics, Institute of Theoretical Physics, Technical University of Clausthal, Arnold-Sommer Str. 6, 38678 Clausthal-Zellerfeld, Germany. 3 Department of Physics, Olabisi Onabanjo University, P.M.B. 2002, Ago-Iwoye, Nigeria. 4 Nonlinear Biomedical Physics, Department of Physics, Lancaster University, LA1, 4YB Lncaster, UK. (Received 31 March 2010 , accepted 10 January 2011) Abstract: In this study, recursive and adaptive backstepping nonlinear controllers are proposed to, respec- tively , control and synchronize the biological system. The designed recursive nonlinear backstepping con- troller is capable of stabilizing the biological system at any position as well as controlling it to track any trajectory that is a smooth function of time. The designed adaptive nonlinear controller globally synchronizes two identical biological systems evolving from different initial conditions. Numerical simulations are given to validate the effectiveness of the proposed controllers and show the robustness against noise. Keywords:control; synchronization; chaos; secursive Backstepping; adaptive Backstepping; biological sys- tems 1 Introduction The seminar work of Pecora and Carroll [1] has motivated many researchers to investigate the phenomena of synchroniza- tion in several nonlinear chaotic systems; and thus this phenomena had been extensively studied using both theoretical and experimental approaches (see for example [2, 3] and the refs. therein). The phenomena of synchronization is a uni- versal concept that can occur when two or more systems are either coupled or forced. The ability of nonlinear systems to synchronize with each other is a basis for many processes in nature and therefore, synchronization plays a very important role in several branches of science. Its numerous applications in mechanics, electronics, measurement and in many other fields have shown that synchronization is extremely important in technological studies. Furthermore, the study of syn- chronized dynamics derived its motivations from its potential applications in secure communications, time series analysis, modelling brain and cardiac rhythmic activity as well as earthquake dynamics [4]. In view of its practical applications, many types of synchronization have been reported in the literature; these include Complete, Generalized, Anticipated, Lag, Measure, Projective, Phase, Reduced order and Adaptive Synchronizations. These concepts of synchronization have led to several effective chaos control and synchronization methods including backstepping nonlinear control approach [5– 7], sliding mode control approach [8], linear state error feedback control techniques [10? , 11], and active control method [12, 13]. Out of these control/synchronization schemes, the backstepping nonlinear control technique has been widely employed due to the following reasons. It has the ability to achieve global stability, tracking and transient performance for a broad class of strict-feedback nonlinear systems [7-13]. In addition, the backstepping technique has the advantages of applicability to a variety of chaotic systems including excited and non-excited ones; needs only one control to achieve sychronization between chaotic systems, thereby reducing controller complexity; there are no derivatives in the controller [17, 18]; the controller is singularity free from the nonlinear term of quadratic type; gives flexibility to construct a control law which can be extended to higher dimensional hyperchaotic systems and the closed-loop system is globally stable [21]; it requires less control effort in comparism with the differential geometric method [19]. ∗ Corresponding author. E-mail address: 2000@ℎ. Copyright c ⃝World Academic Press, World Academic Union IJNS.2011.02.15/458