Showing papers on "Dissipative system published in 1984"
Book•
01 Jan 1984
TL;DR: In this article, the authors present a model for the detection of deterministic chaos in the Lorenz model, which is based on the idea of the Bernoulli shift and the Kicked Quantum Rotator.
Abstract: Preface.Color Plates.1 Introduction.2 Experiments and Simple Models.2.1 Experimental Detection of Deterministic Chaos.2.2 The Periodically Kicked Rotator.3 Piecewise Linear Maps and Deterministic Chaos.3.1 The Bernoulli Shift.3.2 Characterization of Chaotic Motion.3.3 Deterministic Diffusion.4 Universal Behavior of Quadratic Maps.4.1 Parameter Dependence of the Iterates.4.2 Pitchfork Bifurcation and the Doubling Transformation.4.3 Self-Similarity, Universal Power Spectrum, and the Influence of External Noise.4.4 Behavior of the Logistic Map for r ≤ r.4.5 Parallels between Period Doubling and Phase Transitions.4.6 Experimental Support for the Bifurcation Route.5 The Intermittency Route to Chaos.5.1 Mechanisms for Intermittency.5.2 Renormalization-Group Treatment of Intermittency.5.3 Intermittency and 1/f-Noise.5.4 Experimental Observation of the Intermittency Route.6 Strange Attractors in Dissipative Dynamical Systems.6.1 Introduction and Definition of Strange Attractors.6.2 The Kolmogorov Entropy.6.3 Characterization of the Attractor by a Measured Signal.6.4 Pictures of Strange Attractors and Fractal Boundaries.7 The Transition from Quasiperiodicity to Chaos.7.1 Strange Attractors and the Onset of Turbulence.7.2 Universal Properties of the Transition from Quasiperiodicity to Chaos.7.3 Experiments and Circle Maps.7.4 Routes to Chaos.8 Regular and Irregular Motion in Conservative Systems.8.1 Coexistence of Regular and Irregular Motion.8.2 Strongly Irregular Motion and Ergodicity.9 Chaos in Quantum Systems?9.1 The Quantum Cat Map.9.2 A Quantum Particle in a Stadium.9.3 The Kicked Quantum Rotator.10 Controlling Chaos.10.1 Stabilization of Unstable Orbits.10.2 The OGY Method.10.3 Time-Delayed Feedback Control.10.4 Parametric Resonance from Unstable Periodic Orbits.11 Synchronization of Chaotic Systems.11.1 Identical Systems with Symmetric Coupling.11.2 Master-Slave Configurations.11.3 Generalized Synchronization.11.4 Phase Synchronization of Chaotic Systems.12 Spatiotemporal Chaos.12.1 Models for Space-Time Chaos.12.2 Characterization of Space-Time Chaos.12.3 Nonlinear Nonequilibrium Space-Time Dynamics.Outlook.Appendix.A Derivation of the Lorenz Model.B Stability Analysis and the Onset of Convection and Turbulence in the Lorenz Model.C The Schwarzian Derivative.D Renormalization of the One-Dimensional Ising Model.E Decimation and Path Integrals for External Noise.F Shannon's Measure of Information.F.1 Information Capacity of a Store.F.2 Information Gain.G Period Doubling for the Conservative H-enon Map.H Unstable Periodic Orbits.Remarks and References.Index.
1,693 citations
Book•
01 Jan 1984
TL;DR: In this article, the Fourier Transform Poincare is used to describe a dynamical system to chaos, which is a type of dynamical systems to chaos in dissipative systems.
Abstract: Free Oscillator - Damped Oscillator Forced Oscillator - Parametric Oscillator The Fourier Transform Poincare Sections Three Examples of Dynamical Systems To Chaos: Temporal Chaos in Dissipative System Strange Attractors Quasiperiodicity The Subharmonic Cascade Intermittency Debate Appendixes Index.
367 citations
TL;DR: In this paper, the authors investigated the transition to chaos in dissipative dynamical systems with two competing frequencies and showed that the transition is caused by interaction and overlap of mode-locked resonances and occurs at a critical line where the map loses invertibility.
Abstract: Dissipative dynamical systems with two competing frequencies exhibit transitions to chaos. We have investigated the transition through a study of discrete maps of the circle onto itself. The transition is caused by interaction and overlap of mode-locked resonances and occurs at a critical line where the map loses invertibility. At this line the mode-locked intervals trace up a complete devil's staircase whose complementary set is a Cantor set with fractal dimension $D\ensuremath{\sim}0.87$. Numerical results indicate that the dimension is universal for maps with cubic inflection points. Below criticality the staircase is incomplete, leaving room for quasiperiodic behavior. The Lebesgue measure of the quasiperiodic orbits seems to be given by an exponent $\ensuremath{\beta}\ensuremath{\sim}0.35$ which can be related to $D$ through the scaling relation $D=1\ensuremath{-}\frac{\ensuremath{\beta}}{\ensuremath{
u}}$. The exponent $\ensuremath{
u}$ characterizes the cutoff of narrow plateaus near the transition. A variety of other exponents describing the transition to chaos is defined and estimated numerically.
293 citations
TL;DR: In this article, the concept of hamiltonian sysem was generalized to include a wide class of dissipative processes, such as particle collisions, three-wave interactions, and wave-particle resonances.
236 citations
TL;DR: In this article, it was shown that the existence of a potential is not a generic property of a continuous nonlinear dissipative system but requires the complete integrability of an associated Hamiltonian system.
Abstract: It is shown that the existence of a potential is not a generic property of a continuous nonlinear dissipative system but requires the complete integrability of an associated Hamiltonian system.
195 citations
TL;DR: In this article, the authors studied the transition to chaos caused by interaction and overlap of resonances in some condensed-matter systems by constructing and analyzing appropriate return maps, and they concluded that recent theoretical results on universal behavior can readily be checked experimentally by studying systems in condensed matter physics.
Abstract: We have studied the transition to chaos caused by interaction and overlap of resonances in some condensed-matter systems by constructing and analyzing appropriate return maps. In particular, the resistively shunted Josephson junction in microwave fields and charge-density waves in rf electric fields may be described by the differential equation of the damped driven pendulum in a periodic force. The two-dimensional return map for this equation is shown to collapse to a one-dimensional circle map in a parameter regime including the transition to chaos. Phase locking, noise, and hysteresis in these systems can thus be understood in a simple and coherent way by taking over theoretical results for the circle map, some of which were derived in the preceding paper. In order to understand the contraction to one dimensionality we have studied the two-dimensional Chirikov standard map with dissipation. A well-defined transition line along which the system exhibits circle-map critical behavior was found. At this line the system is always phase locked. We conclude that recent theoretical results on universal behavior can readily be checked experimentally by studying systems in condensed-matter physics. The relation between theory and experiment is simple and direct.
191 citations
TL;DR: In this article, the bracket formulation of the Euler fluid mechanics equations is extended to the fluid mechanics equation corresponding to the Navier-Stokes-Fourier and the Edelen constitutive relations.
156 citations
TL;DR: In this paper, a weak shock theory is developed to treat inviscid motions, and the method of multiple scales is used to derive the nonlinear parabolic equation governing the evolution of weakly dissipative waves.
Abstract: One-dimensional small-amplitude waves in which the local value of the fundamental derivative changes sign are examined. The undisturbed medium is taken to be a Navier–Stokes fluid which is at rest and uniform with a pressure and density such that the fundamental derivative is small. A weak shock theory is developed to treat inviscid motions, and the method of multiple scales is used to derive the nonlinear parabolic equation governing the evolution of weakly dissipative waves. The latter is used to compute the viscous shock structure. New phenomena of interest include shock waves having an entropy jump of the fourth order in the shock strength, shock waves having sonic conditions either upstream or downstream of the shock, and collisions between expansion and compression shocks. When the fundamental derivative of the undisturbed media is identically zero it is shown that the ultimate decay of a one-signed pulse is proportional to the negative 1/3-power of the propagation time.
138 citations
TL;DR: In this article, it was shown that the chaotic behavior in the Belousov-Zhabotinsky reaction can arise from the interaction of propagating waves and stationary dissipative structures.
Abstract: Order–disorder transitions are among the most important aspects of the problem of self-organization1. The Belousov–Zhabotinsky (B–Z) reaction is a convenient model for studying these transitions because it displays not only a variety of regular wave patterns but also chemical turbulence. Chaos in this reaction is usually thought to be a local property2, corresponding to a strange attractor in ordinary differential equations. In an active system, however, a different type of chaos (autowave chaos) may occur, which is not based on local mechanisms3,4. We have now found that this type of chaos develops from the interaction of propagating waves and stationary dissipative structures. This interaction initiates a chain reaction of spiral wave production resulting in a reduction of the characteristic scales of the wave pattern and in the occurrence of chaos. If the physical conditions are inappropriate for the formation of dissipative structures, then no chemical turbulence occurs and the wave pattern remains regular. Therefore, chaos appears in the reaction as a result of structure formation.
118 citations
TL;DR: In this article, a general scheme for the theoretical treatment of self-synchronization of many-body oscillators with variable amplitudes, close to harmonic ones with small nonlinearity and dissipative interactions, in the presence of external noises and a native frequency distribution is presented.
112 citations
TL;DR: In this article, a variance of the Galerkinetic finite element method is proposed that exhibits highly selective damping characteristics, which produces a clean, sharp jump structure that agrees favorably with the exact solution of some test problems.
Abstract: The finite element method based on the classical Galerkin formulation produces very poor results when applied to discontinuous channel flow. A variance of the Galerkin method is proposed that exhibits highly selective damping characteristics. The dissipation affects only the numerically-generated high-frequency parasitic waves, while maintaining remarkable accuracy in the approximation to the true solution of the problem. In fact, it is shown that the phase error of the finite element simulation is improved by the introduction of dissipation. The resulting model is second-order accurate with respect to the time step and produces a clean, sharp jump structure that agrees favorably with the exact solution of some test problems. The method is based on discontinuous weighting functions that produce “upwind” effects but at the same time maintain the accuracy of a central difference scheme. The dissipation level is selected by analytical investigations, so that the numerical error is minimized. No second-order pseudo viscosity terms are required, which relaxes the inter-element continuity conditions and results in a very simple and inexpensive scheme.
TL;DR: Experimental evidence and theoretical concepts for the equilibration of the neutron-toproton ratio, N Z, of the fragments from binary dissipative heavy-ion collisions are reviewed in this paper.
TL;DR: For jet-cooled anthracene with E_(vib) = 1792 cm^(−1), the unrelaxed fluorescence displays a fast decay of ≈75 ps, while the relaxed fluorescence shows a corresponding rise as mentioned in this paper.
TL;DR: In this paper, a new idea that the spatial structures in various galaxies are regarded as ''dissipative structures'' which are found in nonequilibrium and open systems, is presented.
Abstract: A new idea that the spatial structures in various galaxies are regarded as ''dissipative structures,'' which are found in nonequilibrium and open systems, is presented. Fundamental physical processes in galaxies are mutual interchange processes among three components of the interstellar medium (ISM). First, the temporal behaviors ( = time structures) of the ISM are investigated by a set of model equations. Second, adding the diffusion term to each component, we explore the physical mechanisms fo the formation of spatial structures. We prove three different types of ''dissipative structures'' may ppear in our ISM system. Third, the pattern formation is studied in a differentially rotating disk. It is shown that our idea is successful in the formation of spiral structures in galaxies. Finally, this resultg is compared with that of the stochastic self-propagating star formation model.
TL;DR: In this paper, a theory of macroscopic systems which takes as independent variables the slow (conserved) ones plus the fast dissipative fluxes is carefully analyzed at three levels of description.
Abstract: A theory of macroscopic systems which takes as independent variables the slow (conserved) ones plus the fast dissipative fluxes is carefully analyzed at three levels of description: macroscopic (thermodynamic), microscopic (projection operators) and mesoscopic (fluctuation theory). Such a description is compared with the memory function approach based only on the conserved variables. We find that the first theory is richer and wider than the second one, and some misunderstandings in this connection are clarified and discussed.
TL;DR: Using renormalized one-point turbulence theory for the nonlinear gyrokinetic equation in the ballooning representation, it was shown that ion Compton scattering is an effective saturation mechanism as discussed by the authors.
Abstract: Drift modes in toroidal geometry are destabilized by trapped electron inverse dissipation and evolve to a nonlinearly saturated state. Using renormalized one‐point turbulence theory for the nonlinear gyrokinetic equation in the ballooning representation, it is shown that ion Compton scattering is an effective saturation mechanism. Ion Compton scattering transfers wave energy from short to long perpendicular wavelength, where it is absorbed by ion resonance with extended, linearly stable, long‐wavelength modes. The fluctuation spectrum and fluctuation levels are calculated using the condition of nonlinear saturation. Transport coefficients and energy confinement time scalings are determined for several regimes. Specifically, the predicted confinement time density scaling for an Ohmically heated discharge increases from n3/8 in the collisionless regime to n9/8 in the dissipative trapped electron regime.
TL;DR: In this paper, the wall-and-window formula was generalized to include the dissipation associated with a time rate of change of the mass asymmetry degree of freedom, which is crucial for understanding the existence of deep-inelastic nuclear reactions.
TL;DR: In this article, the authors present an accurate numerical calculation of the tunneling rate of a system from a metastable well, at zero temperature, in the presence of dissipative coupling to the environment.
Abstract: In view of recent interest in the problem of macroscopic quantum tunneling in systems involving the Josephson effect, we present an accurate numerical calculation of the tunneling rate of a system from a metastable well, at zero temperature, in the presence of dissipative coupling to the environment. Although we concentrate on a specific form of dissipation, as discussed by Caldeira and Leggett, we believe that such a numerical method can be extended to other forms of dissipation as well. Our method is based on the framework recently described by Caldeira and Leggett, and requires (a) a novel treatment of a nonlinear integro-differential equation and (b) an extension of the usual Fredholm scattering theory so as to be applicable to the present dissipative problem. We present explicit results for wide ranges of dissipation and estimate our error in the calculation of the exponent to be no larger than 0.1% and of the prefactor to be no larger than 2%.
TL;DR: In this article, a new theory for the description of dissipative systems by nonlinear Schrodinger-type field equations (NLSEs) with logarithmic nonlinearity, which has been recently developed by the authors, is applied to investigate the frictionally damped free motion and similar spatially unrestricted aperiodic problems.
Abstract: A new theory for the description of dissipative systems by nonlinear Schrodinger‐type field equations (NLSE’s) with logarithmic nonlinearity, which has been recently developed by the authors, is applied to investigate the frictionally damped free motion and similar spatially unrestricted aperiodic problems. Wave‐packet solutions as well as time‐dependent wave‐function solutions are derived and discussed. In the limit of vanishing friction (friction constant γ→0) these solutions turn into the well‐known solutions of the respective linear Schrodinger field equation. The same applies to the mean values of position, momentum, and energy, as well as to the uncertainty product of position and momentum. For γ≠0, however, interesting new effects appear. In contrast to the linear theory the uncertainty product of position and momentum does not diverge any more for infinitely long times, t→∞, but asymptotically approaches a definite constant value which depends on characteristic parameters of the system like its ma...
TL;DR: In this paper, a single soliton and a bound state of two solitons are shown to approximate simple or strange attractors of a forced dissipative nonlinear Schrodinger equation when forcing and dissipative terms are small.
TL;DR: In this paper, a method is presented to obtain stochastic equations of motion for topological defects from the underlying TDGL-like field equations by making use of virtual displacements of the Goldstone coordinates of topological defect lines.
TL;DR: Using the generalized time-dependent Ginzburg-Landau equations for dirty superconductors near T c ≥ 2, the authors studied the oscillatory phase-slip solutions in quasione-dimensional (QD)-superconductors at arbitrary pair-breaking parameter and calculate their properties.
Abstract: Using the generalized time-dependent Ginzburg-Landau equations for dirty superconductors nearT
c
we study the oscillatory phase-slip solutions in quasione-dimensional superconductors at arbitrary pair-breaking parameter and calculate their properties. For small pair breaking (“low temperature”), a static approximation can be used, which allows analytic evaluation of most quantities and connects the theory to simple models. The influence of two types of short weak regions that pin the phase slips is considered. They tend to decrease the differential resistance and influence its temperature dependence.
TL;DR: In this paper, the authors provide an understanding of what high order viscosity terms smooth the physical discontinuities, and determine a class of degenerate second order viscoity terms of physical type which are admissible.
Abstract: : Many equations of mathematical physics take the form of nonlinear hyperbolic systems of conservation laws. With small dissipative effects neglected, typically smooth solutions must develop discontinuities (shocks) in finite time. Re-incorporating dissipation helps select those discontinuities which are physically relevant. For this purpose, many different sorts of dissipation will do; in particular, the physical viscosity is typically degenerate and not convenient. In this paper the author provide an understanding of what high order viscosity terms smooth the physical discontinuities. A natural class of admissible viscosity terms is determined based on a simple linearized stability criterion. In addition, they determine a class of degenerate second order viscosity terms of physical type which are admissible. These results are applied to the equations of compressible fluid dynamics, to determine what conditions ensure the existence of the shock layer with viscosity and heat conduction. This should be of interest to others interested in general equations of state for compressible fluids, such as those investigating phase transitions.
TL;DR: In this article, a general formula for the decay rate at finite temperatures is obtained by a method which is based on the framework recently described by Caldeira and Leggett, where the form of the potential enters only through the frequency of small oscillations about the metastable minimum and the "length" of the zero temperature bounce trajectory.
Abstract: The quantum decay of a metastable system which interacts with an environment at temperatureT is considered. A general formula for the decay rate at finite temperatures is obtained by a method which is based on the framework recently described by Caldeira and Leggett. The thermal enhancement of the tunnelling rate at low temperatures is discussed for arbitrary metastable potentials, and it is found that the exponent of the rate obeys a power law in a dissipative system. The power law exponent is shown to be a characteristic feature of the dissipative mechanism. Finally, a universally valid formula for the thermal enhancement factor is given, where the form of the potential enters only through the frequency of small oscillations about the metastable minimum and the “length” of the zero temperature bounce trajectory.
TL;DR: In this paper, the authors report experimental measurements and calculations using a model on a driven, dissipative, dynamical system which shows chaotic behavior and verify the scaling law both experimentally and with model calculations.
Abstract: We report experimental measurements and calculations using a model on a driven, dissipative, dynamical system which shows chaotic behavior. The system is the diode resonator composed of the series combination of a generator, inductor, and a $p\ensuremath{-}n$-junction diode. It is studied where there are sudden transient changes in the strange attractor, phenomena called crises by Grebogi, Ott, and Yorke, for which a universal scaling law exists. We verify the scaling law both experimentally and with model calculations. Furthermore, the Lyapunov exponent, a measure of sensitivity to initial conditions, is shown by both methods to increase rapidly but continuously through the crisis region.
TL;DR: In this article, the authors consider a class of simple materials characterized by a history-dependent viscosity tensor which they call purely dissipative, and show how a dissipative model of the fluid-like state associated with plastic yielding in solids can be employed to formulate the usual plastic yield condition and flow rule in a single equation.
Abstract: We consider a class of simple materials characterized by a history-dependent viscosity tensor which we call purely dissipative. These materials encompass several existing models of thixotropy and viscoplasticity, including a generalized Oldroyd-Bingham material. We lay down a theoretical framework for classifying these and other simple materials which extends and modifies the well-known scheme of Noll. Finally, we show how a dissipative model of the fluid-like state associated with plastic yielding in solids can be employed to formulate the usual plastic yield condition and flow rule in a single equation.
01 Aug 1984
TL;DR: In this paper, the authors provide an overview of the use of strange attractors in fluid dynamics, including Rayleigh-Benard convection and Couette flow, and provide a mathematical route to turbulence.
Abstract: Publisher Summary This chapter provides an overview of strange attractors in fluid dynamics. Strange attractors are relevant to the transition from regular to chaotic flow in some parametric regimes, whether or not the chaotic flow is called turbulent, weakly turbulent, or simply aperiodic. The term strange attractor for chaotic solutions of dissipative systems with regular forcing was introduced by Ruelle and Takens (1971), who discovered the phenomenon and suggested its relevance to turbulence. Ruelle and Takens conjectured that a strange attractor would appear at the third bifurcation in the Landau sequence so that at most two incommensurate frequencies could appear prior to the transition to chaotic flow. This scenario, with some variation in the number of incommensurate frequencies that appear prior to chaos, has been observed in both Rayleigh-Benard convection and circular Couette flow. The chapter describes concepts related to Lorenz's convection model. It provides details about the Howard–Malkus–Welander convection model. Mathematical routes to turbulence are also elaborated in the chapter.
TL;DR: In this paper, the authors provide an overview of the use of strange attractors in fluid dynamics, including Rayleigh-Benard convection and Couette flow, and provide a mathematical route to turbulence.
Abstract: Publisher Summary This chapter provides an overview of strange attractors in fluid dynamics. Strange attractors are relevant to the transition from regular to chaotic flow in some parametric regimes, whether or not the chaotic flow is called turbulent, weakly turbulent, or simply aperiodic. The term strange attractor for chaotic solutions of dissipative systems with regular forcing was introduced by Ruelle and Takens (1971), who discovered the phenomenon and suggested its relevance to turbulence. Ruelle and Takens conjectured that a strange attractor would appear at the third bifurcation in the Landau sequence so that at most two incommensurate frequencies could appear prior to the transition to chaotic flow. This scenario, with some variation in the number of incommensurate frequencies that appear prior to chaos, has been observed in both Rayleigh-Benard convection and circular Couette flow. The chapter describes concepts related to Lorenz's convection model. It provides details about the Howard–Malkus–Welander convection model. Mathematical routes to turbulence are also elaborated in the chapter.
TL;DR: In this paper, the cubic nonlinear Schroedinger equation in the presence of driving and Landau damping is studied numerically, and the system exhibits a transition from intermittency to a two-torus to chaos.
Abstract: The cubic nonlinear Schroedinger equation, in the presence of driving and Landau damping, is studied numerically. As the pump intensity is increased, the system exhibits a transition from intermittency to a two-torus to chaos. The laminar phase of the intermittency is also a two-torus motion which corresponds in physical space to two identical solitons of amplitude determined by a power-balance equation.
TL;DR: In this article, a simple model of an elastic-viscoplastic material with internal imperfections is proposed, justified by physical mechanisms of polycrystalline matter flow in some regions of temperature and strain rate changes.
Abstract: In the paper the description of the postcritical behaviour of dissipative solids is presented. A simple model of an elastic-viscoplastic material with internal imperfections is proposed. This model is justified by physical mechanisms of polycrystalline matter flow in some regions of temperature and strain rate changes. A model proposed satisfies the requirement that during the deformation process in which the effective strain rate is equal to the assumed static value the response of a material becomes elastic-plastic. The identification procedure for all material functions and constants are based on available experimental data. Both the mechanical test data and physical, metallurgical observations are used. As an example of a quasi-static, isothermal flow process the boundary-initial-value problem describing the necking phenomenon is considered. The problem is formulated in such a way that enables discussion of the influence of strain rate effects, as well as of imperfection and diffusion effects on the onset of localization. Comparison of theoretical predictions with available experimental results is given.