PHYSICAL REVIEW E 87, 034901 (2013) Quasiperiodically driven maps in the low-dissipation limit Shakir Bilal 1 and Ramakrishna Ramaswamy 1,2 1 School of Physical Sciences, Jawaharlal Nehru University, New Delhi 110 067, India 2 University of Hyderabad, Hyderabad 500 046, India (Received 24 September 2012; revised manuscript received 18 February 2013; published 11 March 2013) We study the quasiperiodically driven H´ enon and Standard maps in the weak dissipative limit. In the absence of forcing, there are a large number of coexisting periodic attractors. Although chaotic attractors can also be found, these typically have vanishingly small basins of attraction. Quasiperiodic forcing reduces the multistability in the system, and as the bifurcation parameter is varied, strange nonchaotic attractors (SNAs) are created. The attractor basin for SNAs appears to be the largest among those of all coexisting attractors at such a transition. DOI: 10.1103/PhysRevE.87.034901 PACS number(s): 05.45.Ac, 05.45.Gg I. INTRODUCTION Both the nature of the dynamics as well as the bifurcations in simple dynamical systems are significantly modified in the presence of external forcing [1–3]. Depending on whether the unforced system is conservative or dissipative, different behaviors can result [4–6], and these issues have been studied extensively over the years with particular focus on the different dynamical attractors that can be formed, the various transi- tions that take place, and their potential applications [3,4]. Additionally, different types of forcing—most notably random and quasiperiodic modulation—have been studied extensively. Noise has been shown to be effective in controlling the dynamics in multistable systems [1,2,7,8] or in enabling escape [9]. Quasiperiodic forcing of dissipative systems also results in stabilization, but through the creation of strange nonchaotic at- tractors (SNAs), namely dynamics with no positive Lyapunov exponents and with an underlying fractal geometry [4]. How do the attractors in forced and damped systems evolve as the damping is turned off? The conservative limits of many dynamical systems have been studied in detail [6,10,11]; when this limit is Hamiltonian, the dynamics is either on n-dimensional tori in the phase space, or in the chaotic web [12]. Externally forced weakly dissipative systems have been studied earlier [4] for the case of noisy forcing [1,2,7]. A parallel investigation of the dynamics with quasiperiodic mod- ulations thus seems appropriate. Furthermore, the H´ enon map with high dissipation has also been studied with quasiperiodic forcing [13]. In the absence of forcing, it is known that for large dissipation, typically a single attractor is observed for a given value of the nonlinearity parameter. As the conservative limit is approached, however, multistability is abundant: most of the attractors are periodic orbits, and chaotic attractors are relatively rare [14]. Further, these also tend to have exponentially small basins of attraction and are difficult to detect. With quasiperiodic forcing we find that multistability persists depending on forcing strength, but the SNAs have a large basin of attraction. This paper is organized as follows. In the following section, we recall the properties of the weakly dissipative regime that are germane to the present study. In Sec. III we consider the dynamics of the H´ enon and Standard mappings with a quasiperiodic drive. This is followed in Sec. IV by a discussion and summary of our results. II. THE WEAK DISSIPATION LIMIT For nearly conservative dynamical systems, we confine our attention here to the following situation. Consider a conservative system for which the motion is on quasiperiodic tori, or is chaotic [12]. If a sufficiently small dissipative term is added to the dynamical equations, the resulting dissipative system possesses invariant sets; this is analogous to the Kolmogorov-Arnold-Moser (KAM) theorem that holds in conservative Hamiltonian systems [5]. Each initial condition in a conservative system leads to a different marginally stable orbit, namely a periodic orbit or a chaotic island [12]. With weak dissipation, some of the invariant structures that are present in the conservative limit become attractors or semiattractors [15]. The large number of periodic orbits in a conservative system evolve into a large but finite number of periodic attractors with the introduction of weak dissipation. The number of attractors depends on the family of mappings considered as well as the amount of dissipation introduced [16]. The chaotic sea in the conservative case evolves into the complex basin boundaries of the coexisting periodic attractors. Both the area occupied by the chaotic attractors in parameter space as well as their basin sizes reduce upon reducing the dissipation. These features of the transition from dissipative to conservative systems have been studied numerically in a number of such systems [16,17], most notably the H´ enon and Standard maps. These are respectively defined by the equations x n+1 = 1 − ax 2 n − (1 − ν )y n , y n+1 = x n (1) for the H´ enon mapping [16,18], and x n+1 = (1 − ν )x n + a sin(x n + y n ), (2) y n+1 = y n + x n mod 2π for the Standard mapping [19], where 2  ν  0 and a are the dissipation and nonlinearity parameters respectively, with ν = 0,2 being the conservative cases. III. QUASIPERIODIC DRIVING We now introduce an external quasiperiodic drive, x n+1 = G(x n ,y n ,α) +  cos 2πθ n . (3) θ n+1 = θ n + ω mod 1, 034901-1 1539-3755/2013/87(3)/034901(4) ©2013 American Physical Society