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Dhurjati Prasad Datta

Publications and source records attributed to Dhurjati Prasad Datta.

At least 19 recordsLinked to original sources
arXiv2021

Novel Excitation of local fractional dynamics

The question of a possible excitation and emergence of fractional type dynamics, as a more realistic framework for understanding emergence of complex systems, directly from a conventional integral order dynamics, in the form a continuous transition or deformation, is of significant interest. Although there have been a lot of activities in nonlinear, fractional or not, dynamical systems, the above question appears yet to be addressed systematically in the current literature. The present work may be considered to be a step forward in this direction. Based on a novel concept of asymptotic duality structure, we present here an extended analytical framework that would provide a scenario for realizing the above stated continuous deformation of integral order dynamics to a local fractional order dynamics on a fractal and fractional space. The related concepts of self dual and strictly dual asymptotics are introduced and there relevance in connection with smooth and nonsmooth deformation of the real line are pointed out. The relationship of the duality structure and renormalization group is examined. The ordinary derivation operator is shown to be invariant under this duality enabled renormalization group transformation, leading thereby to a {\em natural} realization of local fractional type derivative in a fractal space. As an application we discuss linear wave equation in one and two dimensions and show how the underlying integral order wave equation could be deformed and renormalized suitably to yield meaningful results for vibration of a fractal string or wave propagation in a region with fractal boundary.

math.GM↗
arXiv2021

Efficient Computation of Periodic Orbits of Forced Rayleigh Equation in the Framework of Novel Asymptotic Structures

Higher precision efficient computation of period 1 relaxation oscillations of strongly nonlinear and singularly perturbed Rayleigh equations with external periodic forcing is presented. The computations are performed in the context of conventional renormalization group method (RGM). We demonstrate that although a slight homotopically modified RGM could generate approximate periodic orbits that agree qualitatively with the exact orbits, the method, nevertheless, fails miserably to reduce the large quantitative disagreement between the theoretically computed results with that of exact numerical orbits. In the second part of the work we present a novel asymptotic analysis incorporating SL(2,R) invariant nonlinear deformation of slower time scales, $t_{n} =\varepsilon^{n}t, \ n\rightarrow\infty, \ \varepsilon<1$, for asymptotic late time $t$, to a nonlinear time $T_{n}=t_{n}σ(t_{n})$, where the deformation factor $σ(t_{n})>0$ respects some well defined SL(2,R) constraints. Motivations and detailed applications of such nonlinear asymptotic structures are explained in performing very high accuracy ($> 98\%$) computations of relaxation orbits. Existence of an interesting condensation and rarefaction phenomenon in connection with dynamically adjustable scales in the context of a slow-fast dynamical system is explained and verified numerically.

nlin.CD↗
arXiv2019

Duality Structure, Asymptotic analysis and Emergent Fractal sets

A new, extended nonlinear framework of the ordinary real analysis incorporating a novel concept of {\em duality structure} and its applications into various nonlinear dynamical problems is presented. The duality structure is an asymptotic property that should affect the late time asymptotic behaviour of a nonlinear dynamical system in a nontrivial way leading naturally to signatures generic to a complex system. We argue that the present formalism would offer a natural framework to understand the abundance of complex systems in natural, biological, financial and related problems. We show that the power law attenuation of a dispersive, lossy wave equation, conventionally deduced from fractional calculus techniques, could actually arise from the present asymptotic duality structure. Differentiability on a Cantor type fractal set is also formulated.

math.CA↗
arXiv2015

Comparative Study of Homotopy Analysis and Renormalization Group Methods on Rayleigh and Van der Pol Equations

A comparative study of the Homotopy Analysis method and an improved Renormalization Group method is presented in the context of the Rayleigh and the Van der Pol equations. Efficient approximate formulae as functions of the nonlinearity parameter $\varepsilon$ for the amplitudes $a(\varepsilon)$ of the limit cycles for both these oscillators are derived. The improvement in the Renormalization group analysis is achieved by invoking the idea of nonlinear time that should have significance in a nonlinear system. Good approximate plots of limit cycles of the concerned oscillators are also presented within this framework.

math.CA↗
arXiv2014

Excitation of Flow Instabilities due to Nonlinear Scale Invariance

A novel route to instabilities and turbulence in fluid and plasma flows is presented in kinetic Vlasov-Maxwell model. New kind of flow instabilities is shown to arise due to the availability of new kinetic energy sources which are absent in conventional treatments. The present approach is based on a scale invariant nonlinear analytic formalism developed to address irregular motions on a chaotic attractor or in turbulence in a more coherent manner. We have studied two specific applications of this turbulence generating mechanism. The warm plasma Langmuir wave dispersion relation is shown to become unstable in the presence of these multifractal measures. In the second application, these multifractal measures are shown to induce naturally non-Gaussian i.e. a stretched -Gaussian distribution and anomalous transport for tracer particles from the turbulent advection-diffusion transport equation in a Vlasov plasma flow.

physics.plasm-ph↗
arXiv2011

Turbulence in Fluid and Plasma: Search for a New Paradigm

A first principle explanation of the origin of intermittency and nonlinear structure formation in the Lagrangian velocity increments of a turbulent flow is presented in the context of a scale invariant analytical formalism that is being developed recently. The copious generation of power laws and nonlinear exponents in the structure functions are shown to follow quite naturally in the present formalism.

math.CA↗
arXiv2011

On a new proof of the Prime Number Theorem

A new elementary proof of the prime number theorem presented recently in the framework of a scale invariant extension of the ordinary analysis is re-examined and clarified further. Both the formalism and proof are presented in a much more simplified manner. Basic properties of some key concepts such as infinitesimals, the associated nonarchimedean absolute values, invariance of measure and cardinality of a compact subset of the real line under an IFS are discussed more thoroughly. Some interesting applications of the formalism in analytic number theory are also presented. The error term as dictated by the Riemann hypothesis also follows naturally thus leading to an indirect proof of the hypothesis.

math.GM↗
arXiv2010

Scale Free Analysis and the Prime Number Theorem

We present an elementary proof of the prime number theorem. The relative error follows a golden ratio scaling law and respects the bound obtained from the Riemann's hypothesis. The proof is derived in the framework of a scale free nonarchimedean extension of the real number system exploiting the concept of relative infinitesimals introduced recently in connection with ultrametric models of Cantor sets. The extended real number system is realized as a completion of the field of rational numbers $Q$ under a {\em new} nonarchimedean absolute value, which treats arbitrarily small and large numbers separately from a finite real number.

math.GM↗
arXiv2010

Ultrametric Cantor sets and Origin of Anomalous Diffusion

The anomalous mean square fluctuations are shown to arise naturally from the ordinary diffusion equation interpreted scale invariantly in a formalism endowing real numbers with a nonarchimedean multiplicative structure. A variable $t$ approaching 0 linearly in the ordinary analysis is shown to enjoy instead a sublinear $t\log t^{-1}$ flow in the presence of this scale invariant structure. Diffusion on an ultrametric Cantor set is also generically subdiffusive with the above sublinear mean square deviation. The present study seems to offer a new interpretation of a possible emergence of complex patterns from an apparently simple system.

math.CA↗
arXiv2010

On the determination of exact number of limit cycles in Lienard Systems

We present a simpler proof of the existence of an exact number of one or more limit cycles to the Lienard system $\dot{x}=y-F(x) $, $\dot {y}=-g(xt)$, under weaker conditions on the odd functions $F(x) $ and $g(x) $ as compared to those available in literature. We also give improved estimates of amplitudes of the limit cycle of the Van Der Pol equation for various values of the nonlinearity parameter. Moreover, the amplitude is shown to be independent of the asymptotic nature of $F$ as $|x| \to\infty$.

math.CA↗
arXiv2010

Dynamical real numbers and living systems

Recently uncovered second derivative discontinuous solutions of the simplest linear ordinary differential equation define not only an nonstandard extension of the framework of the ordinary calculus, but also provide a dynamical representation of the ordinary real number system. Every real number can be visualized as a living cell -like structure, endowed with a definite evolutionary arrow. We discuss the relevance of this extended calculus in the study of living systems. We also present an intelligent version of the Newton's first law of motion.

math.GM↗
arXiv2010

Scale free SL(2,R) analysis and the Picard's existence and uniqueness theorem

The existence of higher derivative discontinuous solutions to a first order ordinary differential equation is shown to reveal a nonlinear SL(2,R) structure of analysis in the sense that a real variable $t$ can now accomplish changes not only by linear translations $t \to t + h$ but also by inversions $t \to 1/t$. We show that the real number set has the structure of a positive Lebesgue measure Cantor set. We also present an extension of the Picard's theorem in this new light.

math.CA↗
arXiv2010

Analysis on a Fractal Set

The formulation of a new analysis on a zero measure Cantor set $C (\subset I=[0,1])$ is presented. A non-archimedean absolute value is introduced in $C$ exploiting the concept of {\em relative} infinitesimals and a scale invariant ultrametric valuation of the form $\log_{\varepsilon^{-1}} (\varepsilon/x) $ for a given scale $\varepsilon>0$ and infinitesimals $0<x<\varepsilon, x\in I\backslash C$. Using this new absolute value, a valued (metric) measure is defined on $C $ and is shown to be equal to the finite Hausdorff measure of the set, if it exists. The formulation of a scale invariant real analysis is also outlined, when the singleton $\{0\}$ of the real line $R$ is replaced by a zero measure Cantor set. The Cantor function is realised as a locally constant function in this setting. The ordinary derivative $dx/dt$ in $R$ is replaced by the scale invariant logarithmic derivative $d\log x/d\log t$ on the set of valued infinitesimals. As a result, the ordinary real valued functions are expected to enjoy some novel asymptotic properties, which might have important applications in number theory and in other areas of mathematics.

math.GM↗
arXiv2010

Non-Archimedean Scale Invariance and Cantor Sets

The framework of a new scale invariant analysis on a Cantor set $C\subset $ $% I=[0,1] $, presented originally in {\it S. Raut and D. P. Datta, Fractals, 17, 45-52, (2009)}, is clarified and extended further. For an arbitrarily small $\varepsilon >0$, elements $\tilde{x}$ in $I\backslash C$ satisfying $0<\tilde{x}<\varepsilon <x, x\in C $ together with an inversion rule are called relative infinitesimals relative to the scale $\varepsilon$. A non-archimedean absolute value $v(% \tilde{x})=\log_{\varepsilon ^{-1}}\frac{\varepsilon}{\tilde{x}}, \varepsilon \to 0$ is assigned to each such infinitesimal which is then shown to induce a non-archimedean structure in the full Cantor set $C$. A valued measure constructed using the new absolute value is shown to give rise to the finite Hausdorff measure of the set. The definition of differentiability on $% C$ in the non-archimedean sense is introduced. The associated Cantor function is shown to relate to the valuation on $C$ which is then reinterpreated as a locally constant function in the extended non-archimedean space. The definitions and the constructions are verified explicitly on a Cantor set which is defined recursively from $I$ deleting $q$ number of open intervals each of length $\frac{1}{r}$ leaving out $p$ numbers of closed intervals so that $p+q=r.$

math.GM↗
arXiv2010

The arrow of time, complexity and the scale free analysis

The origin of complex structures, randomness, and irreversibility are analyzed in the scale free SL(2,R) analysis, which is an extension of the ordinary analysis based on the recently uncovered scale free $C^{2^n-1}$ solutions to linear ordinary differential equations. The role of an intelligent decision making is discussed. We offer an explanation of the recently observed universal renormalization group dynamics at the edge of chaos in logistic maps. The present formalism is also applied to give a first principle explanation of 1/$f$ noise in electrical circuits and solid state devices. Its relevance to heavy tailed (hyperbolic) distributions is pointed out.

math.GM↗
arXiv2010

Higher derivative discontinuous solutions to linear ordinary differential equations: A new route to complexity?

We present a new one parameter family of second derivative discontinuous solutions to the simplest scale invariant linear ordinary differential equation. We also point out how the construction could be extended to generate families of higher derivative discontinuous solutions as well. The discontinuity can occur only for a subset of even order derivatives, viz.,2nd, 4th, 8th, 16th, ....The solutions are shown to break the discrete parity (reflection) symmetry of the underlying equation. These results are expected to gain significance in the contemporary search of a new {\em dynamical principle} for understanding complex phenomena in Nature.

math.GM↗
arXiv2002

A new class of scale free solutions to linear ordinary differential equations and the universality of the Golden Mean ${\frac{\sqrt{5}-1}{2}}$}

A new class of finitely differentiable scale free solutions to the simplest class of ordinary differential equations is presented. Consequently, the real number set gets replaced by an extended physical set, each element of which is endowed with an equivalence class of infinitesimally separated neighbours in the form of random fluctuations. We show how a sense of time and evolution is intrinsically defined by the infinite continued fraction of the golden mean irrational number ${\frac{\sqrt{5}-1}{2}}$, which plays a key role in this extended SL(2,R) formalism of Calculus. Time may thereby undergo random inversions generating well defined random scales, thus allowing a dynamical system to evolve self similarly over the set of multiple scales. The late time stochastic fluctuations of a dynamical system enjoys the generic $1/f$ spectrum. A universal form of the related probability density is also derived. We prove that the golden mean number is intrinsically random, letting all measurements in the physical universe fundamentally uncertain. The present analysis offers an explanation of the universal occurrence of the golden mean in diverse natural and biological processes.

nlin.CD↗