The Stability of Steady and Periodic Solutions of a Low-Dimensional Dynamical System for 2D Driven Cavity Flows

1996 ◽  
pp. 313-316
Author(s):  
W. Cazemier ◽  
R. W. C. P. Verstappen ◽  
A. E. P. Veldman
Keyword(s):  
Dynamical System ◽  
Periodic Solutions ◽  
Cavity Flows ◽  
Driven Cavity ◽  
Dimensional Dynamical System ◽  
The Stability ◽  
Low Dimensional

Non-Newtonian Rotational Flow With Viscous Dissipation

2012 ◽  
Author(s):  
N. Ashrafi ◽  
A. Hazbavi
Keyword(s):  
Dynamical System ◽  
Shear Rate ◽  
Flow Field ◽  
Nonlinear Flow ◽  
Exponential Dependence ◽  
Viscous Heating ◽  
Rotational Flow ◽  
Dimensional Dynamical System ◽  
Highly Correlated ◽  
The Stability

The rotational flow of a non-Newtonian fluid whose viscosity is simultaneously dependent upon temperature and shear rate is analyzed while viscous heating is taken into account. Exponential dependence of viscosity on temperature is modeled through Nahme law and the shear dependency is modeled according to the non-isothermal Carreau equation. The six-dimensional dynamical system, resulted from equations of the conservation of mass and momentum and energy, includes highly correlated terms a solution of which is numerically difficult to converge. Inclusion of the temperature dependence of viscosity as well as viscous heating result in arriving at more realistic simulation of the nonlinear flow. In presence of viscous heating, the effect of Nahme number on the stability of the flow is examined. Complete flow field are given for different scenarios of the flow.

A Posterior Model Reduction Method Based on Weighted Residue for Linear Partial Differential Equations

2014 ◽  
Vol 684 ◽  
pp. 34-40
Author(s):  
Jie Sha ◽  
Li Xiang Zhang ◽  
Chui Jie Wu
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Linear Partial Differential Equation ◽  
Galerkin Projection ◽  
Lagrangian Multiplier ◽  
Partial Differential ◽  
Projection Equation ◽  
Dimensional Dynamical System ◽  
Low Dimensional

This paper is concerned with a new model reduced method based on optimal large truncated low-dimensional dynamical system, by which the solution of linear partial differential equation (PDE) is able to be approximate with highly accuracy. The method proposed is based on the weighted residue of PDE under consideration, and the weighted residue is used as an alternative optimal control condition (POT-WR) while solving the PDE. A set of bases is constructed to describe a dynamical system required in case. The Lagrangian multiplier is introduced to eliminate the constraints of the Galerkin projection equation, and the penalty function is used to remove the orthogonal constraint. According to the extreme principle, a set of the ordinary differential equations is obtained by taking the variational operation on generalized optimal function. A conjugate gradients algorithm on FORTRAN code is developed to solve these ordinary differential equations with Fourier polynomials as the initial bases for iterations. The heat transfer equation under a potential initial condition is used to verify the method proposed. Good agreement between the simulations and the analytical solutions of example was obtained, indicating that the POT-WR method presented in this paper provides the most effective posterior way of capturing the dominant characteristics of an infinite-dimensional dynamical system with only finitely few bases.

PROJECTION OF A HETEROGENOUS AGENT-BASED PRODUCTION ECONOMY MODEL TO A CLOSED DYNAMICS OF AGGREGATE VARIABLES

2015 ◽  
Vol 18 (05n06) ◽  
pp. 1550012 ◽  
Author(s):  
KRISTIAN LINDGREN ◽  
EMMA JONSON ◽  
LIV LUNDBERG
Keyword(s):  
Agricultural Land ◽  
Model Framework ◽  
Production Economy ◽  
Agent Based ◽  
Rational Agents ◽  
Regional Markets ◽  
Dimensional Dynamical System ◽  
The Stability ◽  
Low Dimensional ◽  
Model Features

A model framework that describes a simple production economy is presented, in which the micro-dynamics can be projected to a closed dynamics of aggregate variables. The construction is based on an agent-based model with heterogeneity both regarding production characteristics and the strategies agents use to predict future prices as a basis for choosing what to produce. The world is divided into a number of regional markets that collect the locally produced goods and via inter-market trade supply regional demands with goods. We discuss the model features that make it possible to project the agent-based micro-dynamics to a closed form dynamics on the level of regionally aggregate quantities. One advantage of such a projection from a high dimensional agent-based dynamics to a low dimensional dynamical system is that the stability characteristics can be analytically approached, and this is illustrated by a derived condition for when a mix of naive and rational agents can stabilize the system. Some illustrations of the general framework in a model of global agricultural land-use are also given.

scholarly journals Shaping Dynamics With Multiple Populations in Low-Rank Recurrent Networks

2021 ◽  
pp. 1-44
Author(s):  
Manuel Beiran ◽  
Alexis Dubreuil ◽  
Adrian Valente ◽  
Francesca Mastrogiuseppe ◽  
Srdjan Ostojic
Keyword(s):  
Dynamical System ◽  
Latent Variables ◽  
Gaussian Mixture ◽  
Low Rank ◽  
Connectivity Matrix ◽  
Recurrent Networks ◽  
Collective Dynamics ◽  
Neural Computations ◽  
Dimensional Dynamical System ◽  
Low Dimensional

An emerging paradigm proposes that neural computations can be understood at the level of dynamic systems that govern low-dimensional trajectories of collective neural activity. How the connectivity structure of a network determines the emergent dynamical system, however, remains to be clarified. Here we consider a novel class of models, gaussian-mixture, low-rank recurrent networks in which the rank of the connectivity matrix and the number of statistically defined populations are independent hyperparameters. We show that the resulting collective dynamics form a dynamical system, where the rank sets the dimensionality and the population structure shapes the dynamics. In particular, the collective dynamics can be described in terms of a simplified effective circuit of interacting latent variables. While having a single global population strongly restricts the possible dynamics, we demonstrate that if the number of populations is large enough, a rank R network can approximate any R-dimensional dynamical system.

A Novel Low-Dimensional Method for Analytically Solving Partial Differential Equations

2015 ◽  
Vol 7 (6) ◽  
pp. 754-779 ◽  
Author(s):  
Jie Sha ◽  
Lixiang Zhang ◽  
Chuijie Wu
Keyword(s):  
Dynamical System ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Galerkin Projection ◽  
Reference Database ◽  
Lagrangian Multiplier ◽  
Partial Differential ◽  
Dimensional Dynamical System ◽  
Low Dimensional

AbstractThis paper is concerned with a low-dimensional dynamical system model for analytically solving partial differential equations (PDEs). The model proposed is based on a posterior optimal truncated weighted residue (POT-WR) method, by which an infinite dimensional PDE is optimally truncated and analytically solved in required condition of accuracy. To end that, a POT-WR condition for PDE under consideration is used as a dynamically optimal control criterion with the solving process. A set of bases needs to be constructed without any reference database in order to establish a space to describe low-dimensional dynamical system that is required. The Lagrangian multiplier is introduced to release the constraints due to the Galerkin projection, and a penalty function is also employed to remove the orthogonal constraints. According to the extreme principle, a set of ordinary differential equations is thus obtained by taking the variational operation of the generalized optimal function. A conjugate gradient algorithm by FORTRAN code is developed to solve the ordinary differential equations. The two examples of one-dimensional heat transfer equation and nonlinear Burgers’ equation show that the analytical results on the method proposed are good agreement with the numerical simulations and analytical solutions in references, and the dominant characteristics of the dynamics are well captured in case of few bases used only.

scholarly journals THE LOW DIMENSIONAL DYNAMICAL SYSTEM APPROACH IN GENERAL RELATIVITY: AN EXAMPLE

2007 ◽  
Vol 18 (12) ◽  
pp. 1853-1864 ◽  
Author(s):  
H. P. de OLIVEIRA ◽  
E. L. RODRIGUES ◽  
I. DAMIÃO SOARES ◽  
E. V. TONINI
Keyword(s):  
Dynamical System ◽  
General Relativity ◽  
Galerkin Method ◽  
System Approach ◽  
High Accuracy ◽  
Computational Effort ◽  
Dynamical System Approach ◽  
Dimensional Dynamical System ◽  
Low Dimensional ◽  
The Galerkin Method

In this paper we explore one of the most important features of the Galerkin method, which is to achieve high accuracy with a relatively modest computational effort, in the dynamics of Robinson–Trautman spacetimes.

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