Chairman’s introduction

1987 ◽  
Vol 413 (1844) ◽  
pp. 3-4
Keyword(s):  
Asymptotic Behaviour ◽  
Frequency Spectrum ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Strange Attractors ◽  
Qualitative Properties ◽  
Differential Topology ◽  
Deterministic Systems ◽  
Sensitive Dependence ◽  
Quantitative Properties

The understanding of chaos and strange attractors is one of the most exciting areas of mathematics today. It is the question of how the asymptotic behaviour of deterministic systems can exhibit unpredictability and apparent chaos, due to sensitive dependence upon initial conditions, and yet at the same time preserve a coherent global structure. The field represents a remarkable confluence of several different strands of thought. 1. Firstly came the influence of differential topology, giving global geometric insight and emphasis on qualitative properties. By qualitative properties I mean invariants under differentiable changes of coordinates, as opposed to quantitative properties which are invariant only under linear changes of coordinates. To give an example of this influence, I recall a year-long symposium at Warwick in 1979/80, which involved sustained interaction between pure mathematicians and experimentalists, and one of the most striking consequences of that interaction was a transformation in the way that experimentalists now present their data. It is generally in a much more translucent form: instead of merely plotting a frequency spectrum and calling the incomprehensible part ‘noise’, they began to draw computer pictures of underlying three-dimensional strange attractors.

Computational Characterization of Passive Fluid Mixing in Microfluidics

Volume 3 ◽  
2004 ◽  
Author(s):  
Erik D. Svensson
Keyword(s):  
Stokes Equations ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Fluid Mixing ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Fluid Systems ◽  
Sensitive Dependence ◽  
Microfluidic Mixers

In this work we computationally characterize fluid mixing in a number of passive microfluidic mixers. Generally, in order to systematically study and characterize mixing in realistic fluid systems we (1) compute the fluid flow in the systems by solving the stationary three-dimensional Navier-Stokes equations or Stokes equations with a finite element method, and (2) compute various measures indicating the degree of mixing based on concepts from dynamical systems theory, i.e., the sensitive dependence on initial conditions and mixing variance.

Strange Attractors and Chaos in Nonlinear Mechanics

1983 ◽  
Vol 50 (4b) ◽  
pp. 1021-1032 ◽  
Author(s):  
P. J. Holmes ◽  
F. C. Moon
Keyword(s):  
Mathematical Models ◽  
Initial Conditions ◽  
Piecewise Linear ◽  
Nonlinear Differential Equations ◽  
Homoclinic Orbits ◽  
Strange Attractors ◽  
Electrical Systems ◽  
Nonlinear Mechanical ◽  
Sensitive Dependence

We review several examples of nonlinear mechanical and electrical systems and related mathematical models that display chaotic dynamics or strange attractors. Some simple mathematical models — iterated piecewise linear mappings — are introduced to explain and illustrate the concepts of sensitive dependence on initial conditions and chaos. In particular, we describe the role of homoclinic orbits and the horseshoe map in the generation of chaos, and indicate how the existence of such features can be detected in specific nonlinear differential equations.

A SIMPLE GEOMETRICAL STRUCTURE UNDERLYING SPEECH SIGNALS OF THE JAPANESE VOWEL /a/

1996 ◽  
Vol 06 (01) ◽  
pp. 149-160 ◽  
Author(s):  
ISAO TOKUDA ◽  
RYUJI TOKUNAGA ◽  
KAZUYUKI AIHARA
Keyword(s):  
Cross Sections ◽  
Initial Conditions ◽  
Nonlinear Modeling ◽  
Three Dimensional ◽  
Principal Component ◽  
Coordinate Space ◽  
Speech Signals ◽  
Vowel Sound ◽  
Modeling Analysis ◽  
Sensitive Dependence

We provide several pieces of evidence for possible chaotic dynamics in the irregular behavior of normal speech signals of the Japanese vowel /a/. First, principal component analysis demonstrates that a simple geometric structure underlying the complex speech signal is well reconstructed in a three-dimensional delay-coordinate space. Observations of the reconstructed speech trajectory at multiple cross sections also display speech dynamics with stretching, folding and compressing. Second, Lyapunov spectrum analysis indicates sensitive dependence on initial conditions with a positive Lyapunov exponent for the speech signals of several different speakers. Third, nonlinear modeling analysis with an artificial neural network shows that the nonlinear dynamics of the vowel sound is well reproduced by a deterministic dynamical model.

scholarly journals On some kinds of dense orbits in general nonautonomous dynamical systems

2020 ◽  
Vol 7 (1) ◽  
pp. 163-175
Author(s):  
Mehdi Pourbarat
Keyword(s):  
Dynamical Systems ◽  
Initial Conditions ◽  
Point Of View ◽  
Topological Conjugacy ◽  
Topological Transitivity ◽  
Nonautonomous Systems ◽  
Nonautonomous Dynamical Systems ◽  
Sensitive Dependence ◽  
Dense Orbits

AbstractWe study the theory of universality for the nonautonomous dynamical systems from topological point of view related to hypercyclicity. The conditions are provided in a way that Birkhoff transitivity theorem can be extended. In the context of generalized linear nonautonomous systems, we show that either one of the topological transitivity or hypercyclicity give sensitive dependence on initial conditions. Meanwhile, some examples are presented for topological transitivity, hypercyclicity and topological conjugacy.

IS SENSITIVE DEPENDENCE ON INITIAL CONDITIONS NATURE’S SENSORY DEVICE?

1992 ◽  
Vol 02 (01) ◽  
pp. 193-199 ◽  
Author(s):  
RAY BROWN ◽  
LEON CHUA ◽  
BECKY POPP
Keyword(s):  
Initial Conditions ◽  
Sensory Perception ◽  
Sensory Features ◽  
Sensitive Dependence ◽  
Nonlinear Devices

In this letter we illustrate three methods of using nonlinear devices as sensors. We show that the sensory features of these devices is a result of sensitive dependence on parameters which we show is equivalent to sensitive dependence on initial conditions. As a result, we conjecture that sensitive dependence on initial conditions is nature’s sensory device in cases where remarkable feats of sensory perception are seen.

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

A Quasi-three-dimensional Finite-volume Shallow Water Model for Green Water on Deck

2013 ◽  
Vol 57 (03) ◽  
pp. 125-140
Author(s):  
Daniel A. Liut ◽  
Kenneth M. Weems ◽  
Tin-Guen Yen
Keyword(s):  
Shallow Water ◽  
Finite Volume ◽  
Time Domain ◽  
Degrees Of Freedom ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Green Water ◽  
Water Model ◽  
Ship Motions ◽  
Shallow Water Model

A quasi-three-dimensional hydrodynamic model is presented to simulate shallow water phenomena. The method is based on a finite-volume approach designed to solve shallow water equations in the time domain. The nonlinearities of the governing equations are considered. The methodology can be used to compute green water effects on a variety of platforms with six-degrees-of-freedom motions. Different boundary and initial conditions can be applied for multiple types of moving platforms, like a ship's deck, tanks, etc. Comparisons with experimental data are discussed. The shallow water model has been integrated with the Large Amplitude Motions Program to compute the effects of green water flow over decks within a time-domain simulation of ship motions in waves. Results associated to this implementation are presented.

A Combined Eulerian and Lagrangian Method for Prediction of Evaporating Sprays

2002 ◽  
Vol 124 (3) ◽  
pp. 481-488 ◽  
Author(s):  
M. Burger ◽  
G. Klose ◽  
G. Rottenkolber ◽  
R. Schmehl ◽  
D. Giebert ◽  
...  
Keyword(s):  
Two Phase Flow ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Lagrangian Method ◽  
Phase Flow ◽  
Two Phase Flows ◽  
Two Phase ◽  
Lagrangian Methods ◽  
Eulerian Method ◽  
Lagrangian Modeling

Polydisperse sprays in complex three-dimensional flow systems are important in many technical applications. Numerical descriptions of sprays are used to achieve a fast and accurate prediction of complex two-phase flows. The Eulerian and Lagrangian methods are two essentially different approaches for the modeling of disperse two-phase flows. Both methods have been implemented into the same computational fluid dynamics package which is based on a three-dimensional body-fitted finite volume method. Considering sprays represented by a small number of droplet starting conditions, the Eulerian method is clearly superior in terms of computational efficiency. However, with respect to complex polydisperse sprays, the Lagrangian technique gives a higher accuracy. In addition, Lagrangian modeling of secondary effects such as spray-wall interaction enhances the physical description of the two-phase flow. Therefore, in the present approach the Eulerian and the Lagrangian methods have been combined in a hybrid method. The Eulerian method is used to determine a preliminary solution of the two-phase flow field. Subsequently, the Lagrangian method is employed to improve the accuracy of the first solution using detailed sets of initial conditions. Consequently, this combined approach improves the overall convergence behavior of the simulation. In the final section, the advantages of each method are discussed when predicting an evaporating spray in an intake manifold of an internal combustion engine.

Effect of Initial Velocity Field on Smoke Diffusion Characteristic in Subway Tunnel

2009 ◽  
Author(s):  
Hui Yang ◽  
Li Jia ◽  
Lixin Yang
Keyword(s):  
Velocity Field ◽  
Field Condition ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Wind Effect ◽  
Diffusion Characteristic ◽  
Subway Tunnel ◽  
Air Velocity ◽  
Initial Field ◽  
Fire Disaster

In this paper, piston wind effect on smoke diffusion characteristic in subway tunnel is studied by using three-dimensional transient computational fluid dynamics (CFD) method. In the first simulation case, fire disaster is simulated with homogeneous resting initial field condition. In the second simulation case, the train’s decelerating process till stopping in the tunnel is simulated for getting three-dimensional tunnel air velocity field distribution. Then the final heterogeneous air velocity field when the train stops in the tunnel is taken as initial field condition and the same fire scenario as the first case is simulated again. The data obtained under both initial conditions are compared by detecting people evacuation safety and the influence of initial air velocity field is analyzed. The results show that the inertial air velocity field caused by train’s movement has significant influence on smoke diffusion at the first few minutes of fire disaster, which is the key time for people’s evacuation. The adopted method in this paper and the simulation result could be used in establishing more effective subway fire evacuation plan.

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