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Showing papers on "Dissipative system published in 1972"


Journal ArticleDOI
TL;DR: In this paper, a general theory of dissipative dynamical systems is presented, where dissipativeness is defined in terms of an inequality involving the storage function and the supply function, which is bounded from below by the available storage and from above by the required supply.
Abstract: The first part of this two-part paper presents a general theory of dissipative dynamical systems. The mathematical model used is a state space model and dissipativeness is defined in terms of an inequality involving the storage function and the supply function. It is shown that the storage function satisfies an a priori inequality: it is bounded from below by the available storage and from above by the required supply. The available storage is the amount of internal storage which may be recovered from the system and the required supply is the amount of supply which has to be delivered to the system in order to transfer it from the state of minimum storage to a given state. These functions are themselves possible storage functions, i.e., they satisfy the dissipation inequality. Moreover, since the class of possible storage functions forms a convex set, there is thus a continuum of possible storage functions ranging from its lower bound, the available storage, to its upper bound, the required supply. The paper then considers interconnected systems. It is shown that dissipative systems which are interconnected via a neutral interconnection constraint define a new dissipative dynamical system and that the sum of the storage functions of the individual subsystems is a storage function for the interconnected system. The stability of dissipative systems is then investigated and it is shown that a point in the state space where the storage function attains a local minimum defines a stable equilibrium and that the storage function is a Lyapunov function for this equilibrium. These results are then applied to several examples. These concepts and results will be applied to linear dynamical systems with quadratic supply rates in the second part of this paper.

3,124 citations


Journal ArticleDOI
TL;DR: The theory of dissipative systems in the context of finite dimensional stationary linear systems with quadratic supply rates has been studied in this paper, where a necessary and sufficient frequency domain condition for dissipativeness is derived.
Abstract: This paper presents the theory of dissipative systems in the context of finite dimensional stationary linear systems with quadratic supply rates. A necessary and sufficient frequency domain condition for dissipativeness is derived. This is followed by the evaluation of the available storage and the required supply and of a time-domain criterion for dissipativeness involving certain matrix inequalities. The quadratic storage functions and the dissipation functions are then examined. The discussion then turns to reciprocal systems and it is shown that external reciprocity and dissipativeness imply the existence of a state space realization which is also internally reciprocal and dissipative. The paper proceeds with an examination of reversible systems and of relaxation systems. In particular, it is shown how a unique internal storage function may be defined for relaxation systems. These results are applied to the synthesis of electrical networks and the theory of linear viscoelastic materials.

1,061 citations


Journal ArticleDOI
TL;DR: In this paper, the simple microfluid theory of Eringen is extended to include the heat conduction and heat dissipation effects, and the exact nonlinear theory is presented and restricted by the axioms of constitution and the second law of thermodynamics.

683 citations


Journal ArticleDOI
TL;DR: In this paper, the Burgers equation and Korteweg-de Vries equation are applied to wave propagation processes where some balance occurs in the competition between a nonlinear effect and a higher order derivative effect which might be of a dispersive or a dissipative nature.
Abstract: This article is concerned with wave propagation processes where some balance occurs in the competition between a nonlinear effect and a higher order derivative effect which might be of a dispersive or a dissipative nature. The Burgers equation and the Korteweg–de Vries equation, which are prototype scalar nonlinear dissipative and dispersive equations, are shown to be fundamental to this study, even when quite general systems of equations are involved. The role of the solitary wave solution is shown to be central to the study which is applied to gravity waves, plasma waves and to waves in lattices. Both steady state solutions and initial value problems are reviewed together with questions of stability, existence and uniqueness.

343 citations


Journal ArticleDOI
TL;DR: In this article, the problem of chemical instabilities and the subsequent emergence of dissipative structures is studied in the case of systems which are maintained spatially nonuniform, and a simple model is analyzed numerically and shown to exhibit, for different ranges of values of the parameters, two types of solution of the kinetic equations: a localized steady state dissipative structure and a time-dependent solution describing a nonlinear propagating concentration wave.
Abstract: The problem of chemical instabilities and the subsequent emergence of dissipative structures is studied in the case of systems which are maintained spatially nonuniform. A simple model is analyzed numerically and shown to exhibit, for different ranges of values of the parameters, two types of solution of the kinetic equations: a localized steady state dissipative structure and a time‐dependent solution describing a non‐linear propagating concentration wave. The biological implications of such solutions are briefly discussed.

115 citations


Journal ArticleDOI
A. Davey1
TL;DR: In this article, the authors considered the propagation of a weak nonlinear wave whose energy is concentrated in a narrow band of wavenumbers in a fluid which is both dispersive and dissipative, and they used the small amplitude equations of Whitham's theory of slowly varying wave trains, modified slightly to include dissipation, to show that the modulation of the wave may be described by a nonlinear Schrodinger equation.
Abstract: We consider the propagation of a weak nonlinear wave whose energy is concentrated in a narrow band of wavenumbers in a fluid which is both dispersive and dissipative. We use the small amplitude equations of Whitham's theory of slowly varying wave trains, modified slightly to include dissipation, to show that the modulation of the wave may be described by a nonlinear Schrodinger equation. For long waves which are purely dispersive we obtain the Kortewegde Vries equation, and for long waves which are dissipative we obtain Burgers’ equation by suitable transformations of the nonlinear Schrodinger equation. We mention the problem of Stokes waves in deep water and comment briefly upon invariant far-field theory.

80 citations


Journal ArticleDOI
TL;DR: In the presence of heterogeneous catalysis, time independent local, undulatory inhomogeneities may occur as mentioned in this paper in chemical systems which are stable and may never have symmetry-breaking transitions to (global) dissipative structures, and then the local structure appears under far less restrictive conditions.
Abstract: In the presence of heterogeneous catalysis, time independent local, undulatory inhomogeneities may occur (1) in chemical systems which are stable and may never have symmetry‐breaking transitions to (global) dissipative structures, and (2) in chemical systems which may have such transitions, but then the local structure appears under far less restrictive conditions. The results of the theory are demonstrated with two reaction mechanisms, one due to Prigogine and Lefever, the other to Lotka and Volterra. Local structures may serve better in a role in morphogenesis than global ones. Experiments on spatial patterns must distinguish between global and local structure.

75 citations


Journal ArticleDOI
TL;DR: In this article, a general theory of dissipative periodic systems with a wide range of applications is presented, including systems defined by partial differential equations (distributed parameter systems) and functional differential equations of the retarded and neutral type (hereditary systems).

60 citations


Journal ArticleDOI
TL;DR: The theory of multiplicative stochastic processes is contrasted with the theory of additive processes as discussed by the authors, and it is suggested that multiplicative processes lead to a conceptual foundation for none-quilibrium thermodynamics and nonequilibrium statistical mechanics of marked generality.
Abstract: The theory of multiplicative stochastic processes is contrasted with the theory of additive stochastic processes. The case of multiplicative factors which are purely random, Gaussian, stochastic processes is treated in detail. In a spirit originally introduced by theoretical work in nuclear magnetic resonance and greatly extended by Kubo, dissipative behavior is demonstrated, on the average, for dynamical equations which do not show dissipative behavior without averaging. It is suggested that multiplicative stochastic processes lead to a conceptual foundation for nonequilibrium thermodynamics and nonequilibrium statistical mechanics, of marked generality.

59 citations




Journal ArticleDOI
TL;DR: For well defined ranges of values of certain parameters such systems may exhibit transitions between several stable steady states, and the thermodynamic aspects of such transitions are discussed within the framework of the recent theory of dissipative instabilities.

Journal ArticleDOI
TL;DR: In this paper, a unified formulation of the hydrodynamics suitable for normal liquids, liquid crystals, and solids is presented and applied to the nematic phase, where the phase and the symmetry of the system are characterized by the static stresses which it can sustain, i.e., by static response functions as the elastic constants.

Journal ArticleDOI
TL;DR: In this article, a three-dimensional model of the thermosphere dynamics is developed in terms of the eigenfunctions of the atmospheric system, and a three dimensional model for the external heat inputs like solar XUV-radiation and corpuscular heating during geomagnetic storms is derived.

Journal ArticleDOI
01 Nov 1972
TL;DR: In this paper, a nonstationary flow of an electrically conducting fluid in a transverse magnetic field is analyzed and the solution to this highly non-linear problem is obtained by means of a new variational principle developed by the authors.
Abstract: An investigation is made of the magnetic Rayleigh problem where a semi-infinite plate is given an impulsive motion and thereafter moves with constant velocity in a non-Newtonian power law fluid of infinite extent. The nonstationary flow of this electrically conducting fluid in a transverse magnetic field is then analyzed. The solution to this highly non-linear problem is obtained by means of a new variational principle developed by the authors. This new principle allows one to obtain the solution in a straightforward manner. Unlike other variational techniques in dissipative physics the authors' possesses a pure Hamiltonian structure and obeys all the laws of the classical variational calculus. It is shown that the influence of the magnetic field is greater on the coefficient of friction of dilatant fluids than pseudo-plastic fluids. In the absence of a magnetic field the thickness of the boundary layer increases with increasing powers of the fluid. Finally, the shape of the velocity profile is more strongly dependent on the magnetic field strength for pseudo-plastic fluids than for dilatant fluids.

Journal ArticleDOI
G. Fix1, Nabil Nassif1
TL;DR: Best possible error estimates are proved for spline semi-discrete approximations to dissipative initial value problems and error bounds are established for suitable difference quotients.
Abstract: Best possible error estimates are proved for spline semi-discrete approximations to dissipative initial value problems. Error bounds are also established for suitable difference quotients.

Journal ArticleDOI
TL;DR: In this article, a detailed multicomponent model is developed for dissipative processes in Euclidean homogeneous cosmological models, which involve neutrinos which might have long mean free times in interaction with other constituents which are thermalized by electromagnetic interactions.
Abstract: Consideration of dissipative processes in anisotropic homogeneous world models, showing that dissipation reduces the anisotropy. The viscosity approximation and its range of applicability is discussed. Examples are presented which have been calculated by the use of a simple approximation to the collision-time method, using the cross section appropriate to weak interaction neutrino scattering. It is found that such dissipation is quite effective except for one particular cosmological model which is axisymmetric and in which the entire expansion of the model is taken up by expansion along the axis. A detailed multicomponent model is developed for dissipative processes in Euclidean homogeneous cosmological models. These processes involve neutrinos which might have long mean free times in interaction with other constituents which are thermalized by electromagnetic interactions, and whose weak interactions produce thermal neutrinos.

Journal ArticleDOI
TL;DR: In this article, the evolution in time of explosively unstable systems under the influence of time-dependent dissipative and nonlinear coupling effects is studied, and generalized criteria for instability as well as new mode solutions are constructed by introducing a suitable transformation in time.
Abstract: The present study concerns the evolution in time of explosively unstable systems under the influence of time-dependent dissipative and nonlinear coupling effects. Generalized criteria for instability as well as new mode solutions are constructed by introducing a suitable transformation in time.

Journal ArticleDOI
TL;DR: The special role played by the invariants that are functions of the Hamiltonion is shown to be a direct consequence of the existence of a nonvanishing collision operator, and this reformulation of the ergodic problem may be used in statistical mechanics to study the er godicity of large quantum systems containing a small physical parameter such as the coupling constant or the concentration.
Abstract: We consider the dissipative properties of large quantum systems from the point of view of kinetic theory. The existence of a nontrivial collision operator imposes restrictions on the possible collisional invariants of the system. We consider a model in which a discrete level is coupled to a set of quantum states and which, in the limit of a large “volume,” becomes the Friedrichs model. Because of its simplicity this model allows a direct calculation of the collision operator as well as of related operators and the constants of the motion. For a degenerate spectrum the calculations become more involved but the conclusions remain simple. The special role played by the invariants that are functions of the Hamiltonion is shown to be a direct consequence of the existence of a nonvanishing collision operator. For a class of observables we obtain ergodic behavior, and this reformulation of the ergodic problem may be used in statistical mechanics to study the ergodicity of large quantum systems containing a small physical parameter such as the coupling constant or the concentration.

Journal ArticleDOI
TL;DR: In this article, an eddy description of the Navier-Stokes equation is used to study the homogeneous, isotropic turbulence of incompressible fluids and an equation for the energy spectrum which holds in both the inertial and dissipative regions is derived by combining the contributions from small and large eddies.

Journal ArticleDOI
TL;DR: In this paper, a matching scheme consistent to an equivalent initial boundary value problem was proposed to verify the algebraic conditions for stability of dissipative difference schemes across a coordinate line, and it was shown that some unstable perturbations do not upset the stability of the Lax-Wendroff scheme.
Abstract: Approximations that result from the natural matching of two stable dissipative difference schemes across a coordinate line are shown to be stable. The basic idea is to reformulate the matching scheme consistent to an equivalent initial boundary value problem and to verify the algebraic conditions for stability of such systems. An interesting comparison to the above result is the case of redefinition of a scheme at a single point. In particular, we show that some unstable perturbations do not upset the stability of the Lax–Wendroff scheme.

Journal ArticleDOI
TL;DR: In this paper, the authors examined dissipative processes resulting from magnetic spin-crystal lattice and spin-spin interactions from the point of view of modern continuum mechanics in the theory of micromagnetics of deformable media and derived a formula which generalizes that of Gilbert and Kelley (1953) for deformable solids.
Abstract: Dissipative processes resulting from magnetic spin-crystal lattice and spin-spin interactions are examined from the point of view of modern continuum mechanics in the theory of micromagnetics of deformable media A formula which generalizes that of Gilbert and Kelley (1953) is derived for deformable solids The corresponding heat conduction equation is obtained


Journal ArticleDOI
TL;DR: In this paper, the exact analytical solution for the steady-state motion of a two-degree-of-freedom dissipative nonautonomous system is derived and its asymptotically stable regions are determined.
Abstract: The exact analytical solution for the steady-state motion of a two-degree-of-freedom dissipative nonautonomous system is derived and its asymptotically stable regions are determined. The system consists of a spring-mass-dashpot combination that is coupled to another mass by means of a spring and dashpot that are piecewise linear. Both masses are excited sinusoidally by forces of different amplitudes. Simulated motion on a digital computer and experimental studies with an analog computer corroborate the predictions of the theory.

Journal ArticleDOI
TL;DR: In this article, a general variational principle concerning non-stationary purely dissipative phenomena obeying ONSAGER's reciprocal relations is presented, expressing that the variation of the sum of the total entropy production and the time derivative of the MASSIEU function is equal to zero.
Abstract: A general variational principle concerning non-stationary purely dissipative phenomena, obeying ONSAGER's reciprocal relations, is presented. The principle expresses that the variation of the sum of the total entropy production and the time derivative of the MASSIEU function is equal to zero. It is shown that the EULER-LAGRANGE equations corresponding to arbitrary variations of the state variables yield the conservation laws. Other general criteria for non-stationary processes have been given by GLANSDORFF-PRIGOGINE and GYARMATI. They are compared with our principle. Two illustrative examples are considered: coupled heat and electrical conduction in an isotropic medium and chemical reactions coupled with diffusion of matter and heat conduction in a multi-component fluid.


Journal ArticleDOI
TL;DR: In this article, the asymptotic stability of linear isotropic Kelvin-Voigt solids subjected to external damping and non-conservative surface tractions which are linear functions of displacements and displacement gradients is considered.
Abstract: The asymptotic stability of linear isotropic Kelvin-Voigt solids subjected to external damping and noncon-servative surface tractions which are linear functions of displacements and displacement gradients is considered. The boundary value problem that is adjoint to the original system is derived, and a variational principle from which these two boundary value problems may be generated is stated. The variational principle serves as a basis for an approximate method (similar to the Ritz method) for solving nonconservative stability problems in which the effects of internal and external damping are present. The method is applied to determine the value of the critical load intensity Qcr in a cantilever and a clamped-simply supported beam subjected to a linearly distributed tangential load as well as to internal and external damping forces. Plots of the variation of Qcr with the two damping parameters are given, and it is shown the nature of the boundary conditions can have a significant effect on the manner in which Qcr varies as a function of the damping parameters.

Journal ArticleDOI
TL;DR: In this article, the spectrum of light scattered from a system of coupled nonlinear chemical reactions is considered for steady states far from equilibrium, and new features appear in the spectrum when compared to the light scattering spectrum from reacting systems at equilibrium.
Abstract: The spectrum of light scattered from a system of coupled nonlinear chemical reactions is considered for steady states far from equilibrium. Fluctuations from a steady state may exhibit oscillatory decay, marginally stable chemical oscillations, or dissipative structures. Qualitatively new features appear in the spectrum when compared to the light scattering spectrum from reacting systems at equilibrium. The principal new features found are splittings in the chemical lines and dispersive (non‐Lorentzian) contributions. Two model reaction mechanisms, the Volterra‐Lotka model and the Prigogine‐Lefever model, are examined in detail.

Journal ArticleDOI
TL;DR: In this article, the Green's function of the linear part of the equation for the external boundary problem of flow past a finite symmetric body is obtained in explicit form, which makes it possible to investigate further the Lipmann, Ashkenase and Cowl equation in integral form.
Abstract: GREEN'S function of the linearized viscous transonic equation for the problem of flow past a symmetric plane finite body is obtained in explicit form. Quite severe changes in the flow parameters often occur in narrow regions adjacent to a shock wave. The flow parameter gradients in such regions can be so large that the influence of viscosity and thermal conductivity has to be taken into account, as well as the non-linear features of the motion. Such flows are termed short waves. With the theory of short waves is associated the theory of transonic flows. A general theory of short waves and its connection with transonic flows was outlined in [1]. In some concrete physical problems, examples have been described of short waves arising in stationary flows, where dissipative processes occur. The Mach reflection of weak shock waves from a wedge was discussed in [2]. In [3–5], the interaction of a weak shock wave with a boundary layer was examined. In [1], the asymptotic picture of sonic unidealized gas flow past a finite body was considered. In the papers cited, the conclusions were based on an equation first derived by Lipmann, Ashkenase and Cowl (see [6], Chapter V, Section 5) when describing the structure of a weak shock wave arising at a wing during the onset of a shock stall. The solutions of this equation which are supported by physical considerations are as a rule connected with the behaviour of its linear part. But no satisfactory strict mathematical proof of this fact has yet been offered. In the present paper we construct the Green's function of the linear part of the equation for the external boundary problem of flow past a finite symmetric body. It has quite a simple form, which makes it possible to investigate further the Lipmann, Ashkenase and Cowl equation in integral form. It should be mentioned that this equation is being increasingly referred to in the literature as the viscous-transonic, or simply VT, equation. The VT equation describing rotationally symmetric flows was first obtained and investigated in [7].

Journal ArticleDOI
TL;DR: In this article, a linear dynamical theory of elastic rods is formulated from a theory of one-dimensional oriented continua and sufficient conditions for stability, uniqueness and instability of a rod under applied dissipative forces and couples are given.
Abstract: A linear dynamical theory of elastic rods is formulated from a theory of one-dimensional oriented continua. Sufficient conditions for stability, uniqueness and instability of a rod under applied dissipative forces and couples are given.