Fractional Dynamics with Non-Local Scaling

Abstract In this work, synchronization of fractional dynamics of chaotic system is presented. The suggested dynamics is governed by a system of fractional differential equations, where the fractional derivative operator is modeled by the novel Caputo operator. The nature of fractional dynamical system is non-local which often rules out a closed-form solution. As a result, an efficient numerical method based on shifted Chebychev spectral collocation method is proposed. The error and convergence analysis of this scheme is also given. Numerical results are given for different values of fractional order and other parameters when applied to solve chaotic system, to address any points or queries that may occur naturally.

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Non-local network dynamics via fractional graph Laplacians

Journal of Complex Networks ◽

10.1093/comnet/cnaa017 ◽

2020 ◽

Vol 8 (3) ◽

Cited By ~ 2

Author(s):

Michele Benzi ◽

Daniele Bertaccini ◽

Fabio Durastante ◽

Igor Simunec

Keyword(s):

Directed Graphs ◽

Local Network ◽

Fractional Dynamics ◽

Multi Agent Systems ◽

Directed Networks ◽

Arbitrary Length ◽

Agent Systems ◽

Graph Laplacians ◽

Multi Agent ◽

Non Local

Abstract We introduce non-local dynamics on directed networks through the construction of a fractional version of a non-symmetric Laplacian for weighted directed graphs. Furthermore, we provide an analytic treatment of fractional dynamics for both directed and undirected graphs, showing the possibility of exploring the network employing random walks with jumps of arbitrary length. We also provide some examples of the applicability of the proposed dynamics, including consensus over multi-agent systems described by directed networks.

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Renormalization group analysis of a turbulent compressible fluid neard= 4: Crossover between local and non-local scaling regimes

EPJ Web of Conferences ◽

10.1051/epjconf/201612505006 ◽

2016 ◽

Vol 125 ◽

pp. 05006 ◽

Cited By ~ 7

Author(s):

A.V. Antonov ◽

N.M. Gulitskiy ◽

M.M. Kostenko ◽

T. Lučivjanský

Keyword(s):

Renormalization Group ◽

Compressible Fluid ◽

Group Analysis ◽

Renormalization Group Analysis ◽

Local Scaling ◽

Non Local

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Toward High-Resolution Low-Voltage SEM

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100103152 ◽

1988 ◽

Vol 46 ◽

pp. 218-219

Author(s):

Zhifeng Shao

Keyword(s):

Low Voltage ◽

Spherical Aberration ◽

Chromatic Aberration ◽

Limiting Factor ◽

Operating Voltage ◽

Probe Radius ◽

Non Local ◽

Electron Microscopes ◽

Image Position ◽

Scanning Electron Microscopes

Recently, low voltage (≤5kV) scanning electron microscopes have become popular because of their unprecedented advantages, such as minimized charging effects and smaller specimen damage, etc. Perhaps the most important advantage of LVSEM is that they may be able to provide ultrahigh resolution since the interaction volume decreases when electron energy is reduced. It is obvious that no matter how low the operating voltage is, the resolution is always poorer than the probe radius. To achieve 10Å resolution at 5kV (including non-local effects), we would require a probe radius of 5∽6 Å. At low voltages, we can no longer ignore the effects of chromatic aberration because of the increased ratio δV/V. The 3rd order spherical aberration is another major limiting factor. The optimized aperture should be calculated as

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Chromatic aberration and low-voltage SEM

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100127293 ◽

1987 ◽

Vol 45 ◽

pp. 546-547

Author(s):

Zhifeng Shao ◽

A.V. Crewe

Keyword(s):

Electron Energy ◽

Low Voltage ◽

Spherical Aberration ◽

Chromatic Aberration ◽

Primary Electron ◽

Probe Size ◽

Non Local ◽

Optical Treatment ◽

Electron Microscopes ◽

Scanning Electron Microscopes

For scanning electron microscopes, it is plausible that by lowering the primary electron energy, one can decrease the volume of interaction and improve resolution. As shown by Crewe /1/, at V0 =5kV a 10Å resolution (including non-local effects) is possible. To achieve this, we would need a probe size about 5Å. However, at low voltages, the chromatic aberration becomes the major concern even for field emission sources. In this case, δV/V = 0.1 V/5kV = 2x10-5. As a rough estimate, it has been shown that /2/ the chromatic aberration δC should be less than ⅓ of δ0 the probe size determined by diffraction and spherical aberration in order to neglect its effect. But this did not take into account the distribution of electron energy. We will show that by using a wave optical treatment, the tolerance on the chromatic aberration is much larger than we expected.

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