Showing papers by "Ramakrishna Ramaswamy published in 2001"

Strange Nonchaotic Attractors

Abstract: Aperiodic dynamics which is nonchaotic is realized on Strange Nonchaotic Attractors (SNAs). Such attractors are generic in quasiperiodically driven nonlinear systems, and like strange attractors, are geometrically fractal. The largest Lyapunov exponent is zero or negative: trajectories do not show exponential sensitivity to initial conditions. In recent years, SNAs have been seen in a number of diverse experimental situations ranging from quasiperiodically driven mechanical or electronic systems to plasma discharges. An important connection is the equivalence between a quasiperiodically driven system and the Schrodinger equation for a particle in a related quasiperiodic potential, showing a correspondence between the localized states of the quantum problem with SNAs in the related dynamical system. In this review we discuss the main conceptual issues in the study of SNAs, including the different bifurcations or routes for the creation of such attractors, the methods of characterization, and the nature of dynamical transitions in quasiperiodically forced systems. The variation of the Lyapunov exponent, and the qualitative and quantitative aspects of its local fluctuation properties, have emerged as an important means of studying fractal attractors, and this analysis finds useful application here. The ubiquity of such attractors, in conjunction with their several unusual properties, suggests novel applications.

Bifurcations and transitions in the quasiperiodically driven logistic map

Abstract: We discuss several bifurcation phenomena that occur in the quasiperiodically driven logistic map. This system can have strange nonchaotic attractors (SNAs) in addition to chaotic and regular attractors; on SNAs the dynamics is aperiodic, but the largest Lyapunov exponent is nonpositive. There are a number of different transitions that occur here, from periodic attractors to SNAs, from SNAs to chaotic attractors, etc. We describe some of these transitions by examining the behavior of the largest Lyapunov exponent, distributions of finite time Lyapunov exponents and the invariant densities in the phase space.

A plethora of strange nonchaotic attractors

Abstract: We show that it is possible to devise a large class of skew-product dynamical systems which have strange nonchaotic attractors (SNAs): the dynamics is asymptotically on fractal attractors and the largest Lyapunov exponent is non-positive. Furthermore, we show that quasiperiodic forcing, which has been a hallmark of essentially all hitherto known examples of such dynamics is not necessary for the creation of SNAs.

A Plethora of Strange Nonchaotic Attractors

Abstract: We show that it is possible to devise a large class of skew--product dynamical systems which have strange nonchaotic attractors (SNAs): the dynamics is asymptotically on fractal attractors and the largest Lyapunov exponent is nonpositive. Furthermore, we show that quasiperiodic forcing, which has been a hallmark of essentially allhitherto known examples of such dynamics is {\it not} necessary for the creation of SNAs.

Critical states and fractal attractors in fractal tongues: localization in the Harper map.

Abstract: Localized states of Harper's equation correspond to strange nonchaotic attractors in the related Harper mapping. In parameter space, these fractal attractors with nonpositive Lyapunov exponents occur in fractally organized tongue-like regions which emanate from the Cantor set of eigenvalues on the critical line epsilon=1. A topological invariant characterizes wave functions corresponding to energies in the gaps in the spectrum. This permits a unique integer labeling of the gaps and also determines their scaling properties as a function of potential strength.

Critical Localization and Strange Nonchaotic Dynamics : The Fibonacci Chain

Abstract: The discrete Schr\"odinger equation with a quasiperiodic dichotomous potential specified by the Fibonacci sequence is known to have a singular continuous eigenvalue spectrum with all states being critically localized. This equation can be transformed into a quasiperiodic skew product dynamical system. In this iterative mapping which is entirely equivalent to the Schr\"odinger problem, critically localized states correspond to fractal attractors which have all Lyapunov exponents equal to zero. This provides an alternate means of studying the spectrum, as has been done earlier for the Harper equation. We study the spectrum of the Fibonacci system and describe the scaling of gap widths with potential strength.

Strange Nonchaotic Attractors

Abstract: Aperiodic dynamics which is nonchaotic is realized on Strange Nonchaotic attractors (SNAs). Such attractors are generic in quasiperiodically driven nonlinear systems, and like strange attractors, are geometrically fractal. The largest Lyapunov exponent is zero or negative: trajectories do not show exponential sensitivity to initial conditions. In recent years, SNAs have been seen in a number of diverse experimental situations ranging from quasiperiodically driven mechanical or electronic systems to plasma discharges. An important connection is the equivalence between a quasiperiodically driven system and the Schr\"odinger equation for a particle in a related quasiperiodic potential, giving a correspondence between the localized states of the quantum problem with SNAs in the related dynamical system. In this review we discuss the main conceptual issues in the study of SNAs, including the different bifurcations or routes for the creation of such attractors, the methods of characterization, and the nature of dynamical transitions in quasiperiodically forced systems. The variation of the Lyapunov exponent, and the qualitative and quantitative aspects of its local fluctuation properties, has emerged as an important means of studying fractal attractors, and this analysis finds useful application here. The ubiquity of such attractors, in conjunction with their several unusual properties, suggest novel applications.

Dynamics of the Harper map: Localized states, Cantor spectra and Strange nonchaotic attractors

Abstract: The Harper (or ``almost Mathieu'') equation plays an important role in studies of localization Through a simple transformation, this equation can be converted into an iterative two dimensional skew--product mapping of the cylinder to itself Localized states of the Harper system correspond to fractal attractors with nonpositive maximal Lyapunov exponent in the dynamics of the associated Harper map We study this map and these strange nonchaotic attractors (SNAs) in detail in this paper The spectral gaps of the Harper system have a unique labeling through a topological invariant of orbits of the Harper map This labeling associates an integer index with each gap, and the scaling properties of the width of the gaps as a function of potential strength, $\epsilon$ depends on the index SNAs occur in a large region in parameter space: these regions have a tongue--like shape and end on a Cantor set on the line $\epsilon = 1$ where the states are critically localized, and the spectrum is singular continuous The SNAs of the Harper map are described in terms of their fractal properties, and the scaling behaviour of their power--spectra These are created by unusual bifurcations and differ in many respects from SNAs that have hitherto been studied The technique of studying a quantum eigenvalue problem in terms of the dynamics of an associated mapping can be applied to a number of related problems in 1~dimension We discuss generalizations of the Harper potential as well as other quasiperiodic potentials in this context