• Volume 64, Issue 3

March 2005,   pages  305-466

• Foreword

• A perspective on nonlinear dynamics

We present a brief report on the conference, a summary of the proceedings, and a discussion on the field of nonlinear science studies and its current frontiers.

• Logarithmic scaling in the near-dissipation range of turbulence

A logarithmic scaling for structure functions, in the formSp [ln(r/ η)]ζp, where η is the Kolmogorov dissipation scale and ζp are the scaling exponents, is suggested for the statistical description of the near-dissipation range for which classical power-law scaling does not apply. From experimental data at moderate Reynolds numbers, it is shown that the logarithmic scaling, deduced from general considerations for the near-dissipation range, covers almost the entire range of scales (about two decades) of structure functions, for both velocity and passive scalar fields. This new scaling requires two empirical constants, just as the classical scaling does, and can be considered the basis for extended self-similarity.

• Instabilities and transition in boundary layers

Some recent developments in boundary layer instabilities and transition are reviewed. Background disturbance levels determine the instability mechanism that ultimately leads to turbulence. At low noise levels, the traditional Tollmien-Schlichting route is followed, while at high levels, a ‘by-pass’ route is more likely. Our recent work shows that spot birth is related to the pattern of secondary instability in either route.

• Incompressible turbulence as non-local field theory

It is well-known that incompressible turbulence is non-local in real space because sound speed is infinite in incompressible fluids. The equation in Fourier space indicates that it is non-local in Fourier space as well. However, the shell-to-shell energy transfer is local. Contrast this with Burgers equation which is local in real space. Note that the sound speed in Burgers equation is zero. In our presentation we will contrast these two equations using non-local field theory. Energy spectrum and renormalized parameters will be discussed.

• On the dynamical mechanism of cross-over from chaotic to turbulent states

The Portevin-Le Chatelier effect is one of the few examples of organization of defects. Here the spatio-temporal dynamics emerges from the cooperative behavior of the constituent defects, namely dislocations and point defects. Recent dynamical approach to the study of experimental time series reports an intriguing cross-over phenomenon from a low dimensional chaotic to an infinite dimensional scale invariant power-law regime of stress drops in experiments on CuAl single crystals and AlMg polycrystals, as a function of strain rate. We show that an extension of a dynamical model due to Ananthakrishna and coworkers for the Portevin-Le Chatelier effect reproduces this cross-over. At low and medium strain rates, the model shows chaos with the structure of the attractor resembling the reconstructed experimental attractor. At high strain rates, the model exhibits a power-law statistics for the magnitudes and durations of the stress drops as in experiments. Concomitantly, the largest Lyapunov exponent is zero. In this regime, there is a finite density of null exponents which itself follows a power law. This feature is similar to the Lyapunov spectrum of a shell model for turbulence. The marginal nature of this state is visualized through slow manifold approach.

• Statistical methods in nonlinear dynamics

Sensitivity to initial conditions in nonlinear dynamical systems leads to exponential divergence of trajectories that are initially arbitrarily close, and hence to unpredictability. Statistical methods have been found to be helpful in extracting useful information about such systems. In this paper, we review briefly some statistical methods employed in the study of deterministic and stochastic dynamical systems. These include power spectral analysis and aliasing, extreme value statistics and order statistics, recurrence time statistics, the characterization of intermittency in the Sinai disorder problem, random walk analysis of diffusion in the chaotic pendulum, and long-range correlations in stochastic sequences of symbols.

• Instantaneous frequencies of a chaotic system

The structure and geometry of high-dimensional, complex dynamical systems is usually hidden under a profusion of numerical data. We show that time-frequency analysis allows one to analyze these data regardless of the number of degrees of freedom. Our method takes snapshots of the system in terms of its instantaneous frequencies defined as ridges of the time-frequency landscape. Using the wavelet transform of a single trajectory, it can characterize key dynamical properties like the extent of chaos, resonance transitions and trappings.

• Local dimension and finite time prediction in coupled map lattices

Forecasting, for obvious reasons, often become the most important goal to be achieved. For spatially extended systems (e.g. atmospheric system) where the local nonlinearities lead to the most unpredictable chaotic evolution, it is highly desirable to have a simple diagnostic tool to identify regions of predictable behaviour. In this paper, we discuss the use of the bred vector (BV) dimension, a recently introduced statistics, to identify the regimes where a finite time forecast is feasible. Using the tools from dynamical systems theory and Bayesian modelling, we show the finite time predictability in two-dimensional coupled map lattices in the regions of low BV dimension.

• Fractal differential equations and fractal-time dynamical systems

Differential equations and maps are the most frequently studied examples of dynamical systems and may be considered as continuous and discrete time-evolution processes respectively. The processes in which time evolution takes place on Cantor-like fractal subsets of the real line may be termed as fractal-time dynamical systems. Formulation of these systems requires an appropriate framework. A new calculus calledFα-calculus, is a natural calculus on subsetsF⊂ R of dimension α,0 &lt; α ≤ 1. It involves integral and derivative of order α, calledFα-integral andFα-derivative respectively. TheFα-integral is suitable for integrating functions with fractal support of dimension α, while theFα-derivative enables us to differentiate functions like the Cantor staircase. The functions like the Cantor staircase function occur naturally as solutions ofFα-differential equations. Hence the latter can be used to model fractal-time processes or sublinear dynamical systems.

We discuss construction and solutions of some fractal differential equations of the form$$D_{F,t}^\alpha x = h(x,t),$$ whereh is a vector field andDF,tα is a fractal differential operator of order α in timet. We also consider some equations of the form$$D_{F,t}^\alpha W(x,t) = L[W(x,t)],$$ whereL is an ordinary differential operator in the real variablex, and(t,x)F × Rn whereF is a Cantor-like set of dimension α.

Further, we discuss a method of finding solutions toFα-differential equations: They can be mapped to ordinary differential equations, and the solutions of the latter can be transformed back to get those of the former. This is illustrated with a couple of examples.

• q-Deformed nonlinear maps

Motivated by studies onq-deformed physical systems related to quantum group structures, and by the elements of Tsallis statistical mechanics, the concept ofq-deformed nonlinear maps is introduced. As a specific example, aq-deformation procedure is applied to the logistic map. Compared to the canonical logistic map, the resulting family ofq-logistic maps is shown to have a wider spectrum of interesting behaviours, including the co-existence of attractors — a phenomenon rare in one-dimensional maps.

• The quasi-equilibrium phase of nonlinear chains

We show that time evolution initiated via kinetic energy perturbations in conservative, discrete, spring-mass chains with purely nonlinear, non-integrable, algebraic potentials of the formV(xi− xi+1) ∼ (xi− xi+1)2n,n ≥ 2 and an integer, occurs via discrete solitary waves (DSWs) and discrete antisolitary waves (DASWs). Presence of reflecting and periodic boundaries in the system leads to collisions between the DSWs and DASWs. Such collisions lead to the breakage and subsequent reformation of (different) DSWs and DASWs. Our calculations show that the system eventually reaches a stable ‘quasi-equilibrium’ phase that appears to be independent of initial conditions, possesses Gaussian velocity distribution, and has a higher mean kinetic energy and larger range of kinetic energy fluctuations as compared to the pure harmonic system withn = 1; the latter indicates possible violation of equipartition.

• Construction of a reconfigurable dynamic logic cell

We report the first experimental realization of all the fundamental logic gates, flexibly, using a chaotic circuit. In our scheme a simple threshold mechanism allows the chaotic unit to switch easily between behaviours emulating the different gates. We also demonstrate the combination of gates through a half-adder implementation.

• Homoclinic bifurcation in Chua’s circuit

We report our experimental observations of the Shil’nikov-type homoclinic chaos in asymmetry-induced Chua’s oscillator. The asymmetry plays a crucial role in the related homoclinic bifurcations. The asymmetry is introduced in the circuit by forcing a DC voltage. For a selected asymmetry, when a system parameter is controlled, we observed transition from large amplitude limit cycle to homoclinic chaos via a sequence of periodic mixed-mode oscillations interspersed by chaotic states. Moreover, we observed two intermediate bursting regimes. Experimental evidences of homoclinic chaos are verified with PSPICE simulations.

• Synchronization of coupled chaotic dynamics on networks

We review some recent work on the synchronization of coupled dynamical systems on a variety of networks. When nodes show synchronized behaviour, two interesting phenomena can be observed. First, there are some nodes of the floating type that show intermittent behaviour between getting attached to some clusters and evolving independently. Secondly, two different ways of cluster formation can be identified, namely self-organized clusters which have mostly intra-cluster couplings and driven clusters which have mostly inter-cluster couplings.

• List of participants

• # Pramana – Journal of Physics

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