COEXISTENCE OF LOW- AND HIGH-DIMENSIONAL SPATIOTEMPORAL CHAOS IN A CHAIN OF DISSIPATIVELY COUPLED CHUA’S CIRCUITS

1994 ◽  
Vol 04 (03) ◽  
pp. 639-674 ◽  
Author(s):  
A.L. ZHELEZNYAK ◽  
L.O. CHUA
Keyword(s):  
Phase Space ◽  
Initial Conditions ◽  
Coupling Parameter ◽  
Reaction Diffusion ◽  
Cellular Neural Network ◽  
Spatiotemporal Chaos ◽  
Matrix Phase ◽  
High Dimensional ◽  
The Matrix ◽  
A Chain

Spatiotemporal dynamics of a one-dimensional cellular neural network (CNN) made of Chua’s circuits which mimics a reaction-diffusion medium is considered. An approach is presented to analyse the properties of this reaction-diffusion CNN through the characteristics of the attractors of an associated infinite-dimensional dynamical system with a matrix phase space. Using this approach, the spatiotemporal correlation dimension of the CNN’s spatiotemporal patterns is computed over various ranges of the diffusion coupling parameter, length of the chain, and initial conditions. It is shown that in a finite-dimensional projection of the matrix phase space of the CNN, both low- and high-dimensional attractors corresponding to different initial conditions coexist.

scholarly journals The Process of Stellar Tidal Disruption by Supermassive Black Holes

2021 ◽  
Vol 217 (3) ◽  
Author(s):  
E. M. Rossi ◽  
N. C. Stone ◽  
J. A. P. Law-Smith ◽  
M. Macleod ◽  
G. Lodato ◽  
...  
Keyword(s):  
Black Holes ◽  
Initial Conditions ◽  
Numerical Models ◽  
Binary Star ◽  
Tidal Disruption ◽  
Massive Black Hole ◽  
Accretion Flows ◽  
Simple Order ◽  
Order Of Magnitude ◽  
A Chain

AbstractTidal disruption events (TDEs) are among the brightest transients in the optical, ultraviolet, and X-ray sky. These flares are set into motion when a star is torn apart by the tidal field of a massive black hole, triggering a chain of events which is – so far – incompletely understood. However, the disruption process has been studied extensively for almost half a century, and unlike the later stages of a TDE, our understanding of the disruption itself is reasonably well converged. In this Chapter, we review both analytical and numerical models for stellar tidal disruption. Starting with relatively simple, order-of-magnitude physics, we review models of increasing sophistication, the semi-analytic “affine formalism,” hydrodynamic simulations of the disruption of polytropic stars, and the most recent hydrodynamic results concerning the disruption of realistic stellar models. Our review surveys the immediate aftermath of disruption in both typical and more unusual TDEs, exploring how the fate of the tidal debris changes if one considers non-main sequence stars, deeply penetrating tidal encounters, binary star systems, and sub-parabolic orbits. The stellar tidal disruption process provides the initial conditions needed to model the formation of accretion flows around quiescent massive black holes, and in some cases may also lead to directly observable emission, for example via shock breakout, gravitational waves or runaway nuclear fusion in deeply plunging TDEs.

scholarly journals Theorems and Application of Local Activity of CNN with Five State Variables and One Port

2012 ◽  
Vol 2012 ◽  
pp. 1-13
Author(s):  
Gang Xiong ◽  
Xisong Dong ◽  
Li Xie ◽  
Thomas Yang
Keyword(s):  
Bifurcation Diagram ◽  
Complex Dynamics ◽  
Nonlinear Dynamical Systems ◽  
Reaction Diffusion ◽  
Cellular Neural Network ◽  
State Variables ◽  
Nonlinear Dynamical ◽  
Local Activity ◽  
Homogeneous Lattice ◽  
Coupled Cells

Coupled nonlinear dynamical systems have been widely studied recently. However, the dynamical properties of these systems are difficult to deal with. The local activity of cellular neural network (CNN) has provided a powerful tool for studying the emergence of complex patterns in a homogeneous lattice, which is composed of coupled cells. In this paper, the analytical criteria for the local activity in reaction-diffusion CNN with five state variables and one port are presented, which consists of four theorems, including a serial of inequalities involving CNN parameters. These theorems can be used for calculating the bifurcation diagram to determine or analyze the emergence of complex dynamic patterns, such as chaos. As a case study, a reaction-diffusion CNN of hepatitis B Virus (HBV) mutation-selection model is analyzed and simulated, the bifurcation diagram is calculated. Using the diagram, numerical simulations of this CNN model provide reasonable explanations of complex mutant phenomena during therapy. Therefore, it is demonstrated that the local activity of CNN provides a practical tool for the complex dynamics study of some coupled nonlinear systems.

scholarly journals Efficient approximation of solutions of parametric linear transport equations by ReLU DNNs

2021 ◽  
Vol 47 (1) ◽  
Author(s):  
Fabian Laakmann ◽  
Philipp Petersen
Keyword(s):  
Neural Networks ◽  
Deep Neural Networks ◽  
Initial Conditions ◽  
Activation Function ◽  
Transport Equations ◽  
High Dimensional ◽  
Linear Transport ◽  
Approximation Rates ◽  
Curse Of Dimension ◽  
Efficient Approximation

AbstractWe demonstrate that deep neural networks with the ReLU activation function can efficiently approximate the solutions of various types of parametric linear transport equations. For non-smooth initial conditions, the solutions of these PDEs are high-dimensional and non-smooth. Therefore, approximation of these functions suffers from a curse of dimension. We demonstrate that through their inherent compositionality deep neural networks can resolve the characteristic flow underlying the transport equations and thereby allow approximation rates independent of the parameter dimension.

Karhunen-Loève local characterization of spatiotemporal chaos in a reaction-diffusion system

2000 ◽  
Vol 61 (2) ◽  
pp. 1382-1385 ◽  
Author(s):  
Matthias Meixner ◽  
Scott M. Zoldi ◽  
Sumit Bose ◽  
Eckehard Schöll
Keyword(s):  
Reaction Diffusion ◽  
Spatiotemporal Chaos ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Local Characterization

scholarly journals Necessary conditions for local and global existence to a reaction-diffusion system with fractional derivatives

2006 ◽  
Vol 2006 ◽  
pp. 1-8 ◽  
Author(s):  
Kamel Haouam ◽  
Mourad Sfaxi
Keyword(s):  
Global Existence ◽  
Fractional Derivatives ◽  
Necessary Conditions ◽  
Initial Conditions ◽  
Reaction Diffusion ◽  
Single Equation ◽  
Reaction Diffusion System ◽  
Diffusion System ◽  
Existence Of A Solution

We give some necessary conditions for local and global existence of a solution to reaction-diffusion system of type (FDS) with temporal and spacial fractional derivatives. As in the case of single equation of type (STFE) studied by M. Kirane et al. (2005), we prove that these conditions depend on the behavior of initial conditions for large|x|.

scholarly journals THE EFFECTIVENESS OF ENSURING THE FINANCIAL STABILITY OF THE RESTAURANT BUSINESS IN THE CRISIS COVID-19

2020 ◽  
Author(s):  
Liudmyla Batchenko ◽  
◽  
Liliia Honchar ◽  
Andrii Beliak ◽  
◽  
...  
Keyword(s):  
Financial Management ◽  
Financial Stability ◽  
Hospitality Industry ◽  
Business Environment ◽  
Financial Indicators ◽  
Financial Condition ◽  
The Matrix ◽  
A Chain ◽  
Key Indicators ◽  
Enterprise Policy

The study identifies and systematizes key indicators and criteria for ensuring the financial stability of the restaurant business. The complex and thorough analysis of features of maintenance of financial stability of the enterprises of restaurant business on an example of one of restaurants of a chain of the Japanese kitchen of LLC «Sushiya» is carried out. After analyzing the key indicators of financial and economic activity of the restaurant, using the method of complex calculation of the rating of the financial condition of enterprises in the hospitality industry, the level of financial stability of the studied enterprise is determined. Based on the results of practice-oriented analysis, the ranking of financial management goals by the degree of impact on the financial stability of the enterprise. The mechanism of ensuring financial stability of restaurant business enterprises is modeled. The developed and substantiated mechanism is based on a unique methodology, which, unlike existing ones, is adapted to the field of hospitality, is carried out by specific tactical and strategic tools of financial management, based on the chosen type of enterprise policy; takes into account the dynamics of the main financial indicators of the enterprise, which is planned to implement the mechanism and the possible impact of factors of the external changing business environment. With the help of the matrix of financial strategies of J. Franchon and I. Romane, the position of the restaurant «Sushiya-Lavina» is determined and the methodological tools for improving the efficiency of its financial stability are substantiated.

One-Dimensional Harmonic Oscillator in Quantum Phase Space

2011 ◽  
Vol 110-116 ◽  
pp. 3750-3754
Author(s):  
Jun Lu ◽  
Xue Mei Wang ◽  
Ping Wu
Keyword(s):  
Phase Space ◽  
Harmonic Oscillator ◽  
Quantum Phase ◽  
Space Representation ◽  
Wave Mechanics ◽  
One Dimensional ◽  
Matrix Mechanics ◽  
The Matrix ◽  
Quantum Wave ◽  
The One

Within the framework of the quantum phase space representation established by Torres-Vega and Frederick, we solve the rigorous solutions of the stationary Schrödinger equations for the one-dimensional harmonic oscillator by means of the quantum wave-mechanics method. The result shows that the wave mechanics and the matrix mechanics are equivalent in phase space, just as in position or momentum space.

scholarly journals On the apparent power law in CDM halo pseudo-phase space density profiles

2017 ◽  
Vol 470 (1) ◽  
pp. 500-511 ◽  
Author(s):  
Ethan O. Nadler ◽  
S. Peng Oh ◽  
Suoqing Ji
Keyword(s):  
Phase Space ◽  
Power Law ◽  
Cold Dark Matter ◽  
Initial Conditions ◽  
Phase Space Density ◽  
Fluid Approximation ◽  
Density Profiles ◽  
Space Density ◽  
Scale Free ◽  
Hydrodynamic Calculations

Abstract We investigate the apparent power-law scaling of the pseudo-phase space density (PPSD) in cold dark matter (CDM) haloes. We study fluid collapse, using the close analogy between the gas entropy and the PPSD in the fluid approximation. Our hydrodynamic calculations allow for a precise evaluation of logarithmic derivatives. For scale-free initial conditions, entropy is a power law in Lagrangian (mass) coordinates, but not in Eulerian (radial) coordinates. The deviation from a radial power law arises from incomplete hydrostatic equilibrium (HSE), linked to bulk inflow and mass accretion, and the convergence to the asymptotic central power-law slope is very slow. For more realistic collapse, entropy is not a power law with either radius or mass due to deviations from HSE and scale-dependent initial conditions. Instead, it is a slowly rolling power law that appears approximately linear on a log–log plot. Our fluid calculations recover PPSD power-law slopes and residual amplitudes similar to N-body simulations, indicating that deviations from a power law are not numerical artefacts. In addition, we find that realistic collapse is not self-similar; scalelengths such as the shock radius and the turnaround radius are not power-law functions of time. We therefore argue that the apparent power-law PPSD cannot be used to make detailed dynamical inferences or extrapolate halo profiles inwards, and that it does not indicate any hidden integrals of motion. We also suggest that the apparent agreement between the PPSD and the asymptotic Bertschinger slope is purely coincidental.

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