Showing papers on "Dissipative system published in 1988"

Asymptotic Behavior of Dissipative Systems

Abstract: Discrete dynamical systems: Limit sets Stability of invariant sets and asymptotically smooth maps Examples of asymptotically smooth maps Dissipativeness and global attractors Dependence on parameters Fixed point theorems Stability relative to the global attractor and Morse-Smale maps Dimension of the global attractor Dissipativeness in two spaces Continuous dynamical systems: Limit sets Asymptotically smooth and $\alpha$-contracting semigroups Stability of invariant sets Dissipativeness and global attractors Dependence on parameters Periodic processes Skew product flows Gradient flows Dissipativeness in two spaces Properties of the flow on the attractor Applications: Retarded functional differential equations Sectorial evolutionary equations A scalar parabolic equation The Navier-Stokes equation Neutral functional differential equations Some abstract evolutionary equations A one-dimensional damped wave equation A three-dimensional damped wave equation Remarks on other applications Dependence on parameters and approximation of the attractor.

Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations

Abstract: Contents: Introduction.- Presentation of the Approach and of the Main Results.- The Transport of Finite Dimensional Contact Elements.- Spectral Blocking Property.- Strong Squeezing Property.- Cone Invariance Properties.- Consequences Regarding the Global Attractor.- Local Exponential Decay Toward Blocked Integral Surfaces.- Exponential Decay of Volume Elements and the Dimension of the Global Attractor.- Choice of the Initial Manifold.- Construction of the Inertial Mainfold.- Lower Bound for the Exponential Rate of Convergence to the Attractor.- Asymptotic Completeness: Preparation.- Asymptotic Completeness: Proof of Theorem 12.1.- Stability with Respect to Perturbations.- Application: The Kuramoto-Sivashinsky Equation.- Application: A Nonlocal Burgers Equation.- Application: The Cahn-Hilliard Equation.- Application: A parabolic Equation in Two Space Variables.- Application: The Chaffee-Infante Reaction Diffusion Equation.- References.- Index.

Stability of dynamical systems

Abstract: A new definition of the stability of ordinary differential equations is proposed as an alternative to structural stability. It is particularly aimed at dissipative nonlinear systems, including those with chaos or strange attractors. The definition is as follows. Given a vector field nu on an oriented manifold X, and given epsilon <0, let u be the steady state of the Fokker-Planck equation for nu with epsilon -diffusion. The existence, uniqueness and global attraction of u is proved in the case when X is compact (in the non-compact case a suitable boundary condition on nu is required for the existence of u). Vector fields are defined to be equivalent, or stable, according to whether their steady states are. A similar theory is developed for diffeomorphisms. The new definition has a number of advantages over structural stability. Stable systems are dense, and therefore most strange attractors are stable, including non-hyperbolic ones. The equivalence extends the Thom classification of gradient systems to non-gradient systems. The theory is closely related to applications, because the steady state u is an epsilon -smoothing of the measure on the attractors of the flow of nu , and therefore in numerical and physical experiments u can be used to model the data with epsilon -error.

Applications of Theoretical Contact Mechanics to Solid Particle System Simulation

Abstract: Summary The paper considers the application of theoretical contact mechanics to computer simulation of dissipative solid particle systems. Theoretical and computational requirements are discussed, and results obtained from simulated impact experiments are presented.

Chaotic dynamics of the cutting process

Abstract: In the article a manufacturing system is treated as a simple two dimensional elastic structure in which the cutting force is generated by material flow against the tool. Generalized empirical relations corresponding to orthogonal cutting are applied to describe the non-linear dependence of a cutting force on chip velocity and thickness. The formulated mechanical model of a two dimensional coupled oscillator thus represents a typical example of a non-linear dissipative system, the dynamics of which require description in a four dimensional phase space. The numerical solutions of the governing dynamical equations reveal chaotic oscillations if the characteristic cutting parameter is selected in a region corresponding to intensive cutting. The properties of chaotic oscillations are illustrated by the time dependence of tool displacement, acceleration, cutting force, and power dissipated for material deformation or exchanged with an oscillating tool. The manufactured surface profile is found to resemble a wavy water surface. The phase portraits, Lissajoux figures and spectral densities of calculated signals indicate a similarity with quasiperiodic movement. The information dimension of a corresponding strange attractor, estimated by the correlation exponent, is approximately 3 for a typical example of cutting chaos. The tool displacement X in the direction of input velocity appears to be the most characteristic variable. Its return map is constructed by successive observations of maximal displacement values. The corresponding approximate analytical expression X n+1 =5 X n (1.1- X n )-0.70 is similar to the prototypical map which is frequently applied in the study of chaotic phenomena.

Quantum distinction of regular and chaotic dissipative motion.

Abstract: We generalize the concept of level spacings to dissipative quantum maps. For periodically kicked tops with damping, we find linear and cubic level repulsions under conditions of classically regular and chaotic motion, respectively. The numerically obtained spacing distribution for the chaotic top appears to be universal: It compares favorably with the spacing distribution of general complex matrices of large dimension, the analytical form of which we also present.

Model for Flood Propagation on Initially Dry Land

Abstract: A model is presented for the simulation of shallow water flow and, specifically, flood waves propagating on a dry bed. The governing equations are transformed to an equivalent system valid on a deforming coordinate system and are solved by a dissipative finite-element technique. A second-order difference scheme is employed for the integration in time. The implicit nonlinear equations resulting from the weak formulations are solved by the Newton-Raphson method, and the set of linear algebraic equations generated is solved by a frontal algorithm. A deforming grid generation scheme is introduced in the dissipative finite-element formulation to account for the effects of the propagating or receding wave fronts on dry land. The accuracy and stability of the model is examined by comparing the model results with observed data from an experimental field test. Results of trial runs for the simulation of overland flow are also presented.

Localization of gravity waves on a channel with a random bottom

Abstract: We present a theoretical study of the localisation phenomenon of gravity waves by a rough bottom in a one-dimensional channel. After recalling localisation theory and applying it to the shallow-water case, we give the first study of the localisation problem in the framework of the full potential theory; in particular we develop a renormalised-transfer-matrix approach to this problem. Our results also yield numerical estimates of the localisation length, which we compare with the viscous dissipation length. This allows the prediction of which cases localisation should be observable in and in which cases it could be hidden by dissipative mechanisms.

Pulse interactions in an unstable dissipative‐dispersive nonlinear system

Abstract: An attempt is made to understand several features of the wave evolutions in an unstable dissipative‐dispersive nonlinear system in terms of the interactions of localized solitonlike pulses. It is found that the wave evolutions can be qualitatively well described by weak interactions of pulses, each of which is the steady solution to the original evolution equation. The oscillatory structure of a tail of the pulse for weakly dispersive cases is responsible for the existence of bound states of pulses, which explains the numerical result that the interpulse distances in the initial value problem take certain fixed values or values in the definite regions. In cases of monotone tails for strongly dispersive cases, the effects of pulse interactions become repulsive, which explains the result that the pulses asymptotically tend to be arranged periodically, adjusting to the periodic boundary conditions in the numerical simulation.

Nonlinear Oscillations in Rectangular Tanks

Abstract: A nonlinear, dispersive, dissipative model is developed to describe the fluid motion in a rectangular tank which is moved by an oscillatory and transient translational motion. A linear model for this motion is also presented. Experiments conducted indicate that for a continuous excitation at or near a resonant mode of oscillation, the linear theory becomes inadequate and the nonlinear, dispersive, dissipative theory must be used. For a transient excitation, the validity of the linear theory is found to depend on the magnitude of the Stokes parameter.

On the design of the dissipative LQG-type controllers

Abstract: The design of dissipative linear-quadratic-Gaussian-type compensators for positive real plants is considered. It is shown that if the noise covariance matrices (used as weighting matrices) satisfy certain conditions, the compensator has a strictly positive real transfer function matrix. The stability of the resulting closed-loop system is guaranteed regardless of modeling errors as long as the plant remains positive real. In view of this property, the controller is expected to be useful for vibration suppression in large flexible space structures. >

Strict Global Lyapunov Functions for Mechanical Systems

Abstract: This paper presents a strict, global Lyapunov function for the class of dissipative mechanical systems defined on a configurtion space which admits a trivial tangent bundle.

Experimental and numerical investigation of a forced circular shear layer

Abstract: In a previous article we introduced a dissipative circular geometry in which stationary states of the shear flow instability were obtained. We show here that the dynamical behaviour of this flow depends strongly on the aspect ratio of the cell. In large cells, where the number of vortices is large, transitions from a mode with m vortices to a mode with (m−1) vortices occur through localized processes. In contrast to that situation, in small cells, transition takes place after a series of bifurcations which correspond to the successive breaking of all the symmetries of the flow.We show that, provided an adequate forcing term is introduced, a two-dimensional numerical simulation of this flow is sufficient to recover all the dynamical processes which characterize the experimental flow.

Minimization of energy in quasi-static manipulation

Abstract: Quasistatic mechanical systems, in which mass or acceleration is sufficiently small for the inertial term ma in F=ma to be negligible compared to dissipative forces, are discussed. It is pointed out that many instances of robotic manipulation can be well approximated as quasistatic systems, with the dissipative force being dry friction. Energetic formulations of Newton's laws have often been found useful in the solution of mechanics problems involving multiple constraints. An intuitive minimum power principle is outlined which states that a system chooses at every instant the lowest-energy, or 'easiest', motion in conformity with the constraints. Surprisingly, the principle is in general false; but it is proved that the principle is correct in the useful special case that Coulomb friction is the only dissipative or velocity-dependent force acting in the system. >

Numerical Modeling in Science and Engineering

Abstract: Basic Equations of Macroscopic Systems. Introduction to Numerical Methods. Steady-state Systems. Dissipative Systems. Nondissipative Systems. High-order, Nonlinear, and Coupled Systems. Appendix: Summary of Vector and Tensor Analysis. Index.

Unified approach to the quantum‐Kramers reaction rate

Abstract: The quantum analog of Kramers rate theory is derived from a unique many‐body rate approach (Miller formula), being valid at all temperatures. In contrast to the imaginary free energy method (‘‘bounce’’ method) for a dissipative system we do not have to invoke a different prescription of the rate formula for temperatures below the crossover temperature T0 to tunneling dominated escape. Miller’s many‐body quantum transition state theory is shown to produce the results of the imaginary free energy technique; in particular it also describes correctly the subtle regime near crossover T∼T0.

Dissipative systems: Implications for geomorphology

Abstract: A recent revolution in the study of nonlinear dynamical systems in the physical sciences has shown the worth of regarding systems as dissipative. The nature of dissipative systems in the physical sciences is briefly described by reference to process equations, bifurcations, and fluctuations. Some speculations are then made concerning the implications of dissipative system theory for geomorphology. It is suggested that geomorphological systems containing bifurcations will have both deterministic (universal and necessary) and probabilistic (historical happenstance) elements; they will have more than one solution (configuration) and this fact calls into question notions of process domains leading to the development of characteristic forms; they will possess varying degrees of susceptibility to change induced by fluctuations; and they will respond differently to local, regional, and global fluctuations. Kf,Y WORDS Bifurcation Catastrophe Cataclysm Dissipative system Geomorphological systems Thresholds

Practical performance of the θ 1 -method and comparison with other dissipative algorithms in structural dynamics

Abstract: A single-step time marching scheme, the θ 1 - method , is presented. The method leads to an unconditionally stable implicit algorithm with controllable numerical dissipation. A comparison with other known dissipative algorithms is made. The accuracy, the spectral properties, and the overshooting behaviour are investigated. Numerical results for linear single and multidegree of freedom systems are presented. Among the class of unconditionally stable implicit algorithms with numerical dissipation the θ 1 - method shows some advantages over other known methods, especially in accuracy and overshooting behaviour. The computational effort for nonlinear problems is comparable to Newmark's trapezoidal rule.

Vertex Dynamics of Two-Dimensional Cellular Patterns

Abstract: Time evolution of a two-dimensional cellular structure is studied by computer for a deterministic dissipative model system of interacting vertices. Despite its completely deterministic nature, our model gives rise to fully-developed random cellular structures which abound in nature. Both the growth law and the shape distribution for cells are obtained and are compared with other existing studies.

A Quantitative Analysis of the Dissipation Inherent in Semi-Lagrangian Advection

Abstract: Semi-Lagrangian advection schemes are known to be dissipative because of the interpolation required to estimate the values of the flow fields at each parcel's departure point. In this study, the amplification factors of first, second, third, and fourth-order Largrangian interpolation schemes are used to calculate the dissipative decay time scale and the resulting effective eddy viscocity as functions of wavelength and residual Courant number. The dissipation inherent in the semi-Lagrangian advection can then be compared to more traditional forms of dissipation, such as Laplacian of biharmonic eddy viscosity. The correspondence between semi-Lagrangian advection and more traditional Eulerian techniques is emphasized. The dependence of the dissipation on the time step and grid spacing is also discussed, with a view to selecting the discretionary parameters to meet conservation criteria.

Third-order nonlinear dissipative oscillator with an initial squeezed state

Abstract: The statistical properties of radiation passing through a nonlinear medium modelled as a third-order nonlinear dissipative oscillator interacting with squeezed light have been investigated. Whereas the photon statistics, being insensitive to the nonlinearity, are determined exactly, an approximative approach has been adopted in the description of squeezing in order to involve fluctuations. As a consequence of the general treatment, new results for a damped linear oscillator with initial squeezed light are provided.

Dynamical Systems Approaches to Nonlinear Problems in Systems and Circuits

Abstract: Applications of dynamical-systems analysis to nonlinear circuits and physical systems are discussed in reviews and reports. Topics addressed include general analytical methods, general simulation methods, nonlinear circuits and systems in electrical engineering, control systems, solids and vibrations, and mechanical systems. Consideration is given to the applicability of the Mel'nikov method to highly dissipative systems, damping in nonlinear solid mechanics, a three-dimensional rotation instrument for displaying strange attractors, a chaotic saddle catastrophe in forced oscillators, soliton experiments in annular Josephson junctions, local bifurcation control, periodic and chaotic motions of a buckled beam experiencing parametric and external excitation, and robust nonlinear computed torque control for robot manipulators.

Waves in liquids with vapour bubbles

Abstract: An investigation of wave processes in liquids with vapour bubbles with interphase heat and mass transfer is presented. A single-velocity two-pressure model is used which takes into account both the liquid radial inertia due to medium volume changes, and the temperature distribution around the bubbles. An analysis of the microscopic fields of physical parameters is aimed at closing the system of equations for averaged characteristics. The original system of differential equations of the model is modified to a form suitable for numerical integration. An elliptic equation is obtained to determine the field of the mixture average pressure at an arbitrary time through the known fields of the remaining quantities. The existence of the steady structure of shock waves, either monotonic or oscillatory, is proved. The effect of the initial conditions, shock strength, volume fraction, and dispersity of the vapour phase and of the thermophysical properties of the phases on shock-wave structure and relaxation time is studied. The influence of nonlinear, dispersion and dissipative effects on the wave evolution is also investigated. The shock adiabat for reflected waves is analysed. The results obtained have proved that the interphase heat and mass transfer determined by the thermal diffusivity of the liquid greatly influences the wave structure. The possible enhancement of disturbances in the region of their initiation is shown. The model has been tested for suitability and the results of calculations have been compared with experimental data.

The evolution of cross helicity in driven/dissipative two‐dimensional magnetohydrodynamics

Abstract: A numerical study of the evolution of cross helicity in driven/dissipative magnetohydrodynamics (MHD) is presented. The magnetofluid is incompressible and a two‐dimensional (2‐D) periodic geometry is considered. Cross helicity, a measure of the correlation between fluctuations in the magnetic field and the velocity field, is injected by use of correlated Gaussian forcing over a finite bandwidth in wavenumber. Numerical experiments include the driving of initially uncorrelated spectra with highly correlated forcing and the driving of correlated spectra with anticorrelated forcing. A recurring and persistent feature of the simulations is the appearance of oppositely signed cross helicity at small scales relative to large scales. A simple argument based on the Elsasser variables and used previously in the context of decaying turbulence explains many of the observed features. The effect of a uniform external magnetic field is considered and the relation to purely decaying 2‐D MHD turbulence is discussed.

The whole as a result of self-organization

Abstract: On the basis of the experimental discovery of characteristic eigenfrequencies of the human body, the living organism is considered as a quantum system and a dissipative structure, the long-range coherence of which is provided by electromagnetic interaction. Among dissipative structures a special class is discerned for stable intact systems. Included along with living objects in this class are other fundamental structural material units with discrete characteristic frequencies of single-particle type (the nucleus, atom, molecule).