International Journal of Bifurcation and Chaos, Vol. 23, No. 3 (2013) 1350045 (7 pages) c  World Scientific Publishing Company DOI: 10.1142/S0218127413500454 THE GENERALIZED TIME-DELAYED H ´ ENON MAP: BIFURCATIONS AND DYNAMICS SHAKIR BILAL School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110 067, India shakir.bilal@gmail.com RAMAKRISHNA RAMASWAMY University of Hyderabad, Hyderabad 500 046, India r.ramaswamy@gmail.com Received January 27, 2012; Revised May 23, 2012 We analyze the bifurcations of a family of time-delayed H´ enon maps of increasing dimension and determine the regions where the motion is attracted to different dynamical states. As a function of parameters that govern nonlinearity and dissipation, boundaries that confine asymptotic periodic motion are determined analytically, and we examine their dependence on the dimension d. For large d these boundaries converge. In low dimensions both the period-doubling and quasiperiodic routes to chaos coexist in the parameter space, but for high dimensions the latter predominates and prior to the onset of chaos, the systems exhibit multistability. When the nonlinearity parameter is varied, the dimension of chaotic attractors in the systems changes smoothly with increasing number of non-negative Lyapunov exponents. Keywords : High dimension; diffeomorphism; dissipative map; normal form coefficient; limiting curves; hyperchaos. 1. Introduction Over the past few decades, the behavior of low- dimensional nonlinear iterative maps and flows has been extensively studied and characterized, in par- ticular with reference to the creation of chaotic dynamics [May, 1976; Li & Yorke, 1975; H´ enon, 1976]. The various scenarios or routes to chaos in such systems are by now fairly well known [May, 1976; Arnold, 1965; Newhouse et al., 1978; Rand et al., 1982]. The motion in higher-dimensional systems — for instance, the dynamics of attractors with more than one positive Lyapunov exponent and the bifurcations through which they have been created — has not been studied in as much detail even in relatively “simple” systems [Albers & Sprott, 2006; Sprott, 2006; Baier & Klein, 1990]. Our goal in the present work is to explore the transition from low- to high-dimensional dynamics in a generalized H´ enon map [H´ enon, 1976]. This time-delayed iterative map with a single quadratic nonlinear term also contains the time-delayed H´ enon maps introduced by Sprott [2006] and Baier [Baier & Klein, 1990] as special cases. The phe- nomenon of multistability in high-dimensional maps has been addressed in recent studies [Richter, 2008; Sprott, 2006] and in particular, multistability near the onset of chaos was studied by Sprott [2006]. While the principal focus in the work of Baier [Baier & Klein, 1990] was the study of hyper- chaos. We find that the present generalization of the quadratic map also results in a system that is hyperchaotic, while exhibiting multistability. 1350045-1