Interruption of torus doubling bifurcation and genesis of strange nonchaotic attractors in a quasiperiodically forced map: Mechanisms and their characterizations

Abstract

A simple quasiperiodically forced one-dimensional cubic map is shown to exhibit very many types of routes to chaos via strange nonchaotic attractors (SNAs) with reference to a two-parameter (A−f ) space. The routes include transitions to chaos via SNAs from both one frequency torus and period doubled torus. In the former case, we identify the fractalization and type I intermittency routes. In the latter case, we point out that atleast four distinct routes through which the truncation of torus doubling bifurcation and the birth of SNAs take place in this model. In particular, the formation of SNAs through Heagy-Hammel, fractalization and type-III intermittent mechanisms are described. In addition, it has been found that in this system there are some regions in the parameter space where a novel dynamics involving a sudden expansion of the attractor which tames the growth of period-doubling bifurcation takes place, giving birth to SNA. The SNAs created through different mechanisms are characterized by the behaviour of the Lyapunov exponents and their variance, by the estimation of phase sensitivity exponent as well as through the distribution of finite-time Lyapunov exponents. PACS number(s): 05.45.+b

Critical curves and coexisting attractors in a quasiperiodically forced delayed system

“…A central problem in the study of such deterministic dynamical systems is to identify different types of attractors and understand how the behavior changes as a system parameter changes. Besides regular attractors and chaotic attractors, strange nonchaotic attractors (SNAs) have been the focus of considerable interest from the theoretical and experimental points of view in the past few years [1][2][3][4][5][6]. The initial work of Grebogi et al [7] showed that with quasiperiodic forcing, nonlinear systems could have SNAs.…”

Multilayered bubbling route to SNA in a quasiperiodically forced electronic circuit with a simple nonlinear element

“…But dynamically, they do not show sensitive dependence on initial conditions as seen from negative Lyapunov exponents, that is, they are strange but nonchaotic. Following the pioneering work of Grebogi et al [4], SNAs have been extensively investigated numerically in dynamical systems, such as biological oscillators [5,6], driven Duffing type oscillators [7][8][9][10][11] and in certain maps, namely driven velocitydependent systems [12,13], two dimensional maps [14], quasiperiodically forced logistic map [15][16][17][18][19][20], one dimensional cubic map [21][22][23][24], Harper map [25], map representing driven damped superconducting quantum interference device [26][27][28] and SNAs in HH-neural oscillator [29]. In some physically relevant situations, the existence of SNAs have also been demonstrated experimentally such as in electronic circuits [30][31][32][33], in neon glow discharge experiment [34] and in quasiperiodically forced, buckled, magnetoelastic ribbon [35].…”

Analytical Study and Experimental Confirmation of SNA Through Poincaré Maps in a Quasiperiodically Forced Electronic Circuit

“…However, dynamically, they do not show sensitive dependence on initial conditions as seen from their negative Lyapunov exponents, that is, they are nonchaotic. In the past two decades, SNAs have been identified through numerical investigation in a number of dynamical systems, such as damped pendulum [Romeiras & Ott, 1987], biological oscillators [Ding & Scott Kelso, 1994], driven Duffing-type oscillators [Heagy & Ditto, 1991;Staglino et al, 1996;Yalcinkaya & Lai, 1996;Kapitanik & Wojewoda, 1993;Venkatesan et al, 2000] and in certain maps, namely driven velocity-dependent system [Venkatesan & Lakshmanan, 1998a, 1998b, circle map [Ding et al, 1989], two-dimensional map [Pikovsky et al, 1995], quasiperiodically forced logistic map [Prasad et al, 1997[Prasad et al, , 1998Anishchenko et al, 1996;Sosnovtseva et al, 1996;Kuznetsov et al, 1998;Nishikawa & Kaneko, 1996;Kaneko, 1984], one-dimensional cubic maps [Venkatesan & Lakshmanan, 1998a, 1998b, 2001Yalcinkaya & Lai, 1997;Heagy & Hammel, 1994], Harper map [Prasad et al, 1999], map representing driven damped superconducting quantum interference devices [Zhou & Chen, 1997;Ramaswamy, 1997;Chacon & Gracia-Hoz, 2002] and Heagy-Hammel (HH) neural oscillator [Kim & Lim, 2004]. When these strange nonchaotic attractors are reported in quantum systems [Bondeson et al, 1985], it further increased the interest among the researchers to probe in various fields.…”