5,436 research outputs found
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
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
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
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