STATIONARY PATTERNS IN CNN-LIKE ENSEMBLES WITH MODIFIED CELL OUTPUT FUNCTIONS

2006 ◽  
Vol 16 (08) ◽  
pp. 2207-2220 ◽  
Author(s):  
OLEG I. KANAKOV ◽  
VLADIMIR D. SHALFEEV ◽  
GIAN LUIGI FORTI
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Basins Of Attraction ◽  
Point Of View ◽  
Lattice Systems ◽  
Nonlinear Networks ◽  
Halftone Image ◽  
Contour Lines ◽  
Edge Detecting ◽  
Negative Coupling

Stationary pattern formation in ensembles of coupled bistable elements is investigated both analytically and by means of numerical simulation. The models considered are similar to cellular nonlinear networks (CNNs) — a well-known class of collective dynamical systems intended mainly for image processing — but differ from them in the type of nonlinear functions contained in their equations of motion — cell output functions. The main subject of interest is the transformation of initial conditions, treated as a representation of a half-tone image, into a steady-state pattern. In the analytical part the location of attractors and their basins of attraction in the phase space are estimated for two types of CNN-like systems. In the simulation part the equations of two lattice systems — a CNN and a CNN-like system with local negative coupling — are integrated numerically with initial conditions taken in the form of a sample halftone image. The dependence of the patterns established in both systems upon parameters is studied and compared. The present paper proves, that CNN-like systems with modified cell output functions may be studied analytically, and results available for conventional CNNs may be adapted to this wider class of lattice systems. From the application point of view, it is shown, that the modification of cell output functions under certain conditions does not lead to a breakdown of the system functionality. Moreover, an example is presented, where such a modification allows to introduce new functionality into a system (namely, controlling contour lines width in an edge-detecting system).

BASIN CONFIGURATION OF A SIX-DIMENSIONAL MODEL OF AN ELECTRIC POWER SYSTEM

2002 ◽  
Vol 12 (06) ◽  
pp. 1333-1356 ◽  
Author(s):  
YOSHISUKE UEDA ◽  
HIROYUKI AMANO ◽  
RALPH H. ABRAHAM ◽  
H. BRUCE STEWART
Keyword(s):  
Power Systems ◽  
Power System ◽  
Initial Conditions ◽  
Electrical Power ◽  
Practical Importance ◽  
Intervention Strategy ◽  
Basins Of Attraction ◽  
Electric Power System ◽  
Point Of View ◽  
Electrical Power System

As part of an ongoing project on the stability of massively complex electrical power systems, we discuss the global geometric structure of contacts among the basins of attraction of a six-dimensional dynamical system. This system represents a simple model of an electrical power system involving three machines and an infinite bus. Apart from the possible occurrence of attractors representing pathological states, the contacts between the basins have a practical importance, from the point of view of the operation of a real electrical power system. With the aid of a global map of basins, one could hope to design an intervention strategy to boot the power system back into its normal state. Our method involves taking two-dimensional sections of the six-dimensional state space, and then determining the basins directly by numerical simulation from a dense grid of initial conditions. The relations among all the basins are given for a specific numerical example, that is, choosing particular values for the parameters in our model.

Potential Well Escape of an Eccentric Disk

2012 ◽  
Author(s):  
Genevieve M. Lipp ◽  
Brian P. Mann
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Basins Of Attraction ◽  
Potential Well ◽  
Mass Approximation ◽  
Amplitude Parameter ◽  
Related Point ◽  
Eccentric Disk ◽  
Disk Rolling ◽  
Cubic Function

This paper investigates the dynamic behavior of an eccentric disk rolling on a curve of arbitrary shape and then on a curve defined as a cubic function. Comparisons are made to a disk with no eccentricity and the related point mass approximation. The curve is subject to base excitation, and the system is considered from the perspective of a potential well problem where escape is possible on one side. The equations of motion are derived using a roll-without-slip constraint, and the behavior is investigated by means of simulated frequency and amplitude parameter sweeps and by considering the basins of attraction when initial conditions or forcing parameters are varied.

Sensitivity of Final Field Position to the Punt Initial Conditions in American Football

2016 ◽  
Author(s):  
James D. Turner ◽  
Brian P. Mann
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Nonlinear Behavior ◽  
Basins Of Attraction ◽  
Three Dimensions ◽  
American Football ◽  
Impact Forces ◽  
Highly Nonlinear ◽  
Angular Velocities ◽  
Field Position

The starting field position is often a deciding factor in an American football game. In the case of a defensive stop, a kick, known as a punt, is used to give the receiving team a field position that is more advantageous to the kicking team when possession changes. The goal of the punter is to kick the ball along a desired flight path, where a delicate balance between the distance traveled before impact, hang time in the air, and the distance traveled after bouncing is favorable for the kicking team. However, the punter has only imprecise control over the initial conditions, such as the angular velocity, linear velocity, and orientation of the football. Due to the highly nonlinear behavior of the football, from aerodynamic and impact forces, even small changes in initial conditions can produce large changes in the final position of the football, but there may be regions of initial conditions with relatively consistent results. If punters could target such large contiguous regions of initial conditions with desirable football paths, they could improve their chances of successful kicks. For nonlinear systems, basins of attraction diagrams are often used to graphically display the initial conditions that lead to different final attractors. In this case, the regions of initial conditions that lead to a desirable final field position can be grouped and shown graphically. A numerical simulation program was developed including models for aerodynamic flight and bouncing of the irregularly shaped football. The flight model used fourth order Runge-Kutta integration of the equations of motion of the football, including gravitational and aerodynamic forces and moments with empirical lift, drag, and yaw coefficients in three dimensions. The bounce model was based on an empirical two-dimensional coefficient of restitution model that was published in the literature. The behavior of a football in flight and during bouncing was simulated for a range of initial angular velocities and launch angles, and the characteristics of the flight paths were analyzed. The characteristics of some regions of initial conditions were relatively sensitive to small changes, while other regions were relatively uniform. This shows that this approach, with a quantitatively accurate bounce model, could be practically applied to develop a guide for punters to optimize their kicks. With such a guide and sufficient practice, punters could select and target the larger regions of initial conditions that produced desirable behavior, which would improve their chances of successful punts.

scholarly journals On Kaufmann's theory of the impact of the pianoforte hammer

1920 ◽  
Vol 97 (682) ◽  
pp. 99-110 ◽  
Keyword(s):  
Initial Conditions ◽  
Practical Importance ◽  
Prima Facie ◽  
Point Of View ◽  
Infinite Length ◽  
Modes Of Vibration ◽  
Rigorous Treatment ◽  
The Subject ◽  
Set Up ◽  
The Impact

The theory of the vibrations of the pianoforte string put forward by Kaufmann in a well-known paper has figured prominently in recent discussions on the acoustics of this instrument. It proceeds on lines radically different from those adopted by Helmholtz in his classical treatment of the subject. While recognising that the elasticity of the pianoforte hammer is not a negligible factor, Kaufmann set out to simplify the mathematical analysis by ignoring its effect altogether, and treating the hammer as a particle possessing only inertia without spring. The motion of the string following the impact of the hammer is found from the initial conditions and from the functional solutions of the equation of wave-propagation on the string. On this basis he gave a rigorous treatment of two cases: (1) a particle impinging on a stretched string of infinite length, and (2) a particle impinging on the centre of a finite string, neither of which cases is of much interest from an acoustical point of view. The case of practical importance treated by him is that in which a particle impinges on the string near one end. For this case, he gave only an approximate theory from which the duration of contact, the motion of the point struck, and the form of the vibration-curves for various points of the string could be found. There can be no doubt of the importance of Kaufmann’s work, and it naturally becomes necessary to extend and revise his theory in various directions. In several respects, the theory awaits fuller development, especially as regards the harmonic analysis of the modes of vibration set up by impact, and the detailed discussion of the influence of the elasticity of the hammer and of varying velocities of impact. Apart from these points, the question arises whether the approximate method used by Kaufmann is sufficiently accurate for practical purposes, and whether it may be regarded as applicable when, as in the pianoforte, the point struck is distant one-eighth or one-ninth of the length of the string from one end. Kaufmann’s treatment is practically based on the assumption that the part of the string between the end and the point struck remains straight as long as the hammer and string remain in contact. Primâ facie , it is clear that this assumption would introduce error when the part of the string under reference is an appreciable fraction of the whole. For the effect of the impact would obviously be to excite the vibrations of this portion of the string, which continue so long as the hammer is in contact, and would also influence the mode of vibration of the string as a whole when the hammer loses contact. A mathematical theory which is not subject to this error, and which is applicable for any position of the striking point, thus seems called for.

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.

Artificial Planning Experience by Means of a Heuristic Cell-Space Model: Simulating International Migration in the Urban Process

1995 ◽  
Vol 27 (10) ◽  
pp. 1647-1665 ◽  
Author(s):  
J Portugali ◽  
I Benenson
Keyword(s):  
Urban Policy ◽  
Initial Conditions ◽  
Point Of View ◽  
Town Planning ◽  
Space Model ◽  
Self Organized ◽  
A Cell ◽  
Policy And Planning ◽  
The City ◽  
Theory And Model

We suggest considering the city as a complex, open, and thus self-organized system, and describing it by means of a cell-space model. A central property of self-organizing systems is that they are not controllable—not by individuals, nor by economic, political, and planning institutions. The city, in this respect, is complex and untamable. Inability to recognize and accept this property is one of the reasons for the difficulties and problems of modernist town planning. The theory and model we present are built to describe the urban process as a historical one in which, given identical initial conditions, each simulation run is unique and never fully repeats itself. From the point of view of urban policy and planning, our heuristic model can guide decisionmakers by answering the following question: ‘given the initial conditions of an inflow of new immigrants (that is, from the ex-USSR), what possible urban scenarios can result, and what are their global structural properties?’.

3D Printing — The Basins of Tristability in the Lorenz System

2017 ◽  
Vol 27 (08) ◽  
pp. 1750128 ◽  
Author(s):  
Anda Xiong ◽  
Julien C. Sprott ◽  
Jingxuan Lyu ◽  
Xilu Wang
Keyword(s):  
Lorenz System ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Basins Of Attraction ◽  
Symmetric Pair ◽  
State Space Approach ◽  
3D Printers ◽  
The Lorenz System ◽  
Almost All ◽  
Space Approach

The famous Lorenz system is studied and analyzed for a particular set of parameters originally proposed by Lorenz. With those parameters, the system has a single globally attracting strange attractor, meaning that almost all initial conditions in its 3D state space approach the attractor as time advances. However, with a slight change in one of the parameters, the chaotic attractor coexists with a symmetric pair of stable equilibrium points, and the resulting tri-stable system has three intertwined basins of attraction. The advent of 3D printers now makes it possible to visualize the topology of such basins of attraction as the results presented here illustrate.

Fractal Boundary Explorations for a Nonlinear Two Degree-of-Freedom System

2012 ◽  
Author(s):  
Benjamin A. M. Owens ◽  
Brian P. Mann
Keyword(s):  
Initial Conditions ◽  
Coupled System ◽  
Basins Of Attraction ◽  
Degree Of Freedom ◽  
Fractal Boundary ◽  
Final State ◽  
Potential Wells ◽  
Mechanical Coupling ◽  
Mass Spring ◽  
Two Degree Of Freedom

This paper explores a two degree-of-freedom nonlinearly coupled system with two distinct potential wells. The system consists of a pair of linear mass-spring-dampers with a non-linear, mechanical coupling between them. This nonlinearity creates fractal boundaries for basins of attraction and forced well-escape response. The inherent uncertainty of these fractal boundaries is quantified for errors in the initial conditions and parameter space. This uncertainty relationship provides a measure of the final state and transient sensitivity of the system.

scholarly journals Dynamics of Space Particles and Spacecrafts Passing by the Atmosphere of the Earth

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Vivian Martins Gomes ◽  
Antonio Fernando Bertachini de Almeida Prado ◽  
Justyna Golebiewska
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
The Sun ◽  
Three Body Problem ◽  
The Earth ◽  
Before And After ◽  
Three Body ◽  
Body Energy ◽  
Spacecraft Position

The present research studies the motion of a particle or a spacecraft that comes from an orbit around the Sun, which can be elliptic or hyperbolic, and that makes a passage close enough to the Earth such that it crosses its atmosphere. The idea is to measure the Sun-particle two-body energy before and after this passage in order to verify its variation as a function of the periapsis distance, angle of approach, and velocity at the periapsis of the particle. The full system is formed by the Sun, the Earth, and the particle or the spacecraft. The Sun and the Earth are in circular orbits around their center of mass and the motion is planar for all the bodies involved. The equations of motion consider the restricted circular planar three-body problem with the addition of the atmospheric drag. The initial conditions of the particle or spacecraft (position and velocity) are given at the periapsis of its trajectory around the Earth.

scholarly journals Numerical solution for consistent initial conditions of constrained mechanical systems

2003 ◽  
Vol 25 (3) ◽  
pp. 170-185
Author(s):  
Dinh Van Phong
Keyword(s):  
Numerical Solution ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Flexible Approach ◽  
Constrained Mechanical Systems ◽  
Consistent Initial Values

The article deals with the problem of consistent initial values of the system of equations of motion which has the form of the system of differential-algebraic equations. Direct treating the equations of mechanical systems with particular properties enables to study the system of DAE in a more flexible approach. Algorithms and examples are shown in order to illustrate the considered technique.

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