International Journal of Advanced Scientific and Technical Research Issue 3 volume 6, Nov.-Dec. 2013 Available online on http://www.rspublication.com/ijst/index.html ISSN 2249-9954 R S. Publication (http://rspublication.com), rspublicationhouse@gmail.com Page 37 Quasi-periodicity and mode-locking in Maynard Smith map 1 Hemanta Kumar Sarmah, 2 Tapan Kumar Baishya, 3 Debasish Bhattacharjee, 4 Mridul Chandra Das 1,4 Department of Mathematics, Gauhati University, Assam 2 Department of Mathematics, Debraj Roy College, Golaghat, Assam 3 Department of Mathematics, B.Borooah College, Guwahati, Assam Abstract: This paper investigates the emergence of quasiperiodic and mode-locked states which arises from Neimark-Sacker(NS) bifurcation in Maynard Smith map which is given by ,  = ,  + − 2 , where  and  are real parameters. Analytical results are obtained near NS bifurcation using normal forms. In our investigation, we have used the techniques of Lyapunov exponent, bifurcation diagram and phase portrait to show transition from quasiperiodic and from mode-locked states into chaotic states. 1. Introduction: In the last three decades, the phenomenon of chaos has been studied extensively and it has attracted increasing interests from mathematicians, physicists, engineers, and so on. Since chaotic systems not only admit abundant complex and interesting dynamical behaviours [such as bifurcations, chaos and strange attractors] but also have many potential practical applications, great efforts have been devoted to investigation related to these systems. Research on bifurcation, such as Hopf bifurcation, Homoclinic bifurcation and Period Doubling bifurcation is one of the most hot topics in the field of nonlinear science. It has been found that bifurcation frequently leads to chaos in nonlinear systems. So, it is necessary to explore the bifurcation of dynamical systems so as to understand the complex dynamical behaviours. Recently, Hopf bifurcation of some famous chaotic systems have been investigated and it has been becoming one of the most active topics in the field of chaotic systems. The Neimark-Sacker (NS) bifurcation occurs for a discrete system depending on parameter, with a fixed point whose Jacobian has a pair of complex conjugate eigenvalues which cross the unit circletransversally. The NS bifurcation in case of maps is equivalent to the Hopf bifurcation for flows [26, 27, 34, 41]. For instance, in the case of a supercritical NS bifurcation, a stable focus loses its stability as a parameter is varied with the consequent birth of a stable cycle or quasi-cycle which is known as closed invariant curve. In the case of a subcritical NS bifurcation, a stable focus enclosed by an unstable closed curve loses its stability with the consequent disappearance of the closed invariant curve as a parameter is varied. The most probable route to chaos, for high dimensional discrete time maps, from a fixed point is via at least one Neimark-Sacker(NS) bifurcation, followed by persistent zero Lyapunov exponent, and finally a bifurcation into Chaos[1]. Orbits that are not periodic and have the Lyapunov exponent equal to zero are said to be quasiperiodic[5, 7, 8, 28, 35]. In 1971, Rullle and Takens [30] first proposed the quasi-periodic scenario. It is observed that in both the cases of NS bifurcation for maps and Hopf bifurcation for flows quasi-periodic scenario come into picture. Hopf bifurcation in fact is related to the birth and death of limit cycles in a system. The existence of the limit cycles can be observed in fluid dynamics where vortex structures appear [19,32]. The theory underlying the quasiperiodic route to chaos tells us only that this scenario may lead to chaotic behaviour. In 1978, Newhouse, Rulle and Takens [24] proved more rigorously in case of flows that if the state space trajectories of a system are confined to a three dimensional torus, then