Brownian motion of fractal particles: Levy flights from white noise

We generalise the Langevin equation with Gaussian white noise by replacing the velocity term by a local fractional derivative. The solution of this equation is a Levy process. We further consider the Brownian motion of a fractal particle, for example, a colloidal aggregate or a biological molecule and argue that it leads to a Levy flight. This effect can also be described using the local fractional Langevin equation. The implications of this development to other complex data series are discussed.Comment: 5 pages, two column

Recursive Local Fractional Derivative

The definition of the local fractional derivative has been generalised to the orders beyond the critical order. This makes it possible to retain more terms in the local fractional Taylor expansion leading to better approximation. This also extends the validity of the product rule

Local Fractional Calculus: a Review

The purpose of this article is to review the developments related to the notion of local fractional derivative introduced in 1996. We consider its definition, properties, implications and possible applications. This involves the local fractional Taylor expansion, Leibnitz rule, chain rule, etc. Among applications we consider the local fractional diffusion equation for fractal time processes and the relation between stress and strain for fractal media. Finally, we indicate a stochastic version of local fractional differential equation.Comment: to appear in the proceedings of 'National Workshop on Fractional Calculus: Theory and Applications', University of Pun

Disordered Totally Asymmetric Simple Exclusion Process: Exact Results

We study the effect of quenched spatial disorder on the current-carrying steady states of the totally asymmetric simple exclusion process with spatially disordered jump rates. The exact analytical expressions for the steady-state weights, and the current are found for this model in one dimension. We demonstrate how these solutions can be exploited to study analytically the exact symmetries of the system. In particular, we prove that the magnitude of the steady-state current is left invariant when the direction of all the allowed particle jumps are reversed. Or equivalently, we prove that for any given filling and disorder configuration, particle-hole transformation is an exact symmetry that leaves the steady-state current invariant. This non-trivial symmetry was recently demonstrated in numerical simulations by Tripathy & Burma (preprint cond-mat/9711302).Comment: 4 pages RevTe

Global Analysis of Synchronization in Coupled Maps

We introduce a new method for determining the global stability of synchronization in systems of coupled identical maps. The method is based on the study of invariant measures. Besides the simplest non-trivial example, namely two symmetrically coupled tent maps, we also treat the case of two asymmetrically coupled tent maps as well as a globally coupled network. Our main result is the identification of the precise value of the coupling parameter where the synchronizing and desynchronizing transitions take place.Comment: 7 pages, 5 figures, 2 sections adde

Multifractal invariant measures in expanding piecewise linear coupled maps

We analyze invariant measures of two coupled piecewise linear and everywhere expanding maps on the synchronization manifold. We observe that though the individual maps have simple and smooth functions as their stationary densities, they become multifractal as soon as two of them are coupled nonlinearly even with a small coupling. For some maps, the multifractal spectrum seems to be robust with the coupling or map parameters and for some other maps, there is a substantial variation. The origin of the multifractal spectrum here is intriguing as it does not seem to conform to the existing theory of multifractal functions

Local Fractional Derivatives and Fractal Functions of Several Variables

The notion of a local fractional derivative (LFD) was introduced recently for functions of a single variable. LFD was shown to be useful in studying fractional differentiability properties of fractal and multifractal functions. It was demonstrated that the local Holder exponent/ dimension was directly related to the maximum order for which LFD existed. We have extended this definition to directional-LFD for functions of many variables and demonstrated its utility with the help of simple examples.Comment: 4 pages, Revtex, appeared in proceedings of 'Fractals in Engineering' Arcachon, France (1997

A simple method to estimate fractal dimension of mountain surfaces

Fractal surfaces are ubiquitous in nature as well as in the sciences. The examples range from the cloud boundaries to the corroded surfaces. Fractal dimension gives a measure of the irregularity in the object under study. We present a simple method to estimate the fractal dimension of mountain surface. We propose to use easily available satellite images of lakes for this purpose. The fractal dimension of the boundary of a lake, which can be extracted using image analysis softwares, can be determined easily which gives the estimate of the fractal dimension of the mountain surface and hence a quantitative characterization of the irregularity of the topography of the mountain surface. This value will be useful in validating models of mountain formationComment: in proceedings of Humboldt Kolleg 'Surface Science and Engineered Surfaces' held in Lavasa, Indi

Definition of fractal measures arising from fractional calculus

It is wellknown that the ordinary calculus is inadequate to handle fractal structures and processes and another suitable calculus needs to be developed for this purpose. Recently it was realized that fractional calculus with suitable constructions does offer such a possibility. This makes it necessary to have a definition of fractal measures based on the fractional calculus so that the fractals can be naturally incorporated in the calculus. With this motivation a definition of fractal measure has been systematically developed using the concepts of fractional calculus. It has been demonstrated that such a definition naturally arises in the solution of an equation describing diffusion in fractal time.Comment: 3 pages, short versio

Chaotic Properties of Single Element Nonlinear Chimney Model: Effect of Directionality

We generalize the chimney model by introducing nonlinear restoring and gravitational forces for the purpose of modeling swaying of trees at high wind speeds. We have derived general equations governing the system using Lagrangian formulation. We have studied the simplest case of a single element in more detail. The governing equation we arrive at for this case has not been studied so far. We study the chaotic properties of this simple building block and also the effect of directionality in the wind on the chaotic properties. We also consider the special case of two elements.Comment: To appear in IJB