Theory Of Dissipative Systems Research Articles

A new Lyapunov function for interconnected power systems

A Lyapunov function is constructed for a general n-machine power system with nonzero transfer conductance. The theory of dissipative dynamic systems and its modifications are used. The Lyapunov function construction procedure calls for checking positive definiteness of a sparse matrix. Results are reported only for the case of nonuniform damping, which is more complex than the uniformly damped system. The procedure is iterative and is guaranteed to converge for uniform damping but may fail to converge for a very lightly nonuniformly damped system with strong interconnections. The Lyapunov function is a quadratic term plus a weighted sum of integrals, a form not reported before. This is an extension of a previously reported quadratic Lyapunov function.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Comparison of rate theories for generalized Langevin dynamics

Rate constants evaluated from (1) the energy-loss turnover theory of Pollak, Grabert, and Hänggi (PGH), (2) the Grote–Hynes extension of Kramers theory (GH), and (3) the microcanonical variational transition state theory for dissipative systems of Tucker and Pollak (μVTST) are compared with rate constants determined from direct computer simulations of generalized Langevin dynamics. The comparisons are made for a cubic oscillator under the influence of a slow bath characterized by a Gaussian friction kernel. In the μVTST calculations, which are based on an effective two degree of freedom Hamiltonian, barrier crossing due to energy transfer from the bath to the effective Hamiltonian is neglected. This neglect is significant only at very strong coupling, where it causes the μVTST results to drop below the simulation results. Both GH and μVTST theories fail (as expected) in the energy diffusion regime, while PGH theory is only moderately successful. The μVTST results agree extremely well with the simulation results in the spatial diffusion regime, providing a significant improvement over the GH results at intermediate coupling strengths and over the PGH results at strong coupling strengths. This improvement is a result of nonlinear effects which are included in the μVTST approach but neglected in the PGH and GH theories.

Global existence and asymptotic behavior for a quasilinear reaction-diffusion system from climate modeling

This paper is concerned with a weakly coupled system of quasilinear autonomous strongly parabolic equations on a compact two-dimensional manifold without boundary; the system arises from an energy balance climate model. We establish L ∞, Hoelder, and Sobolev estimates, and apply general results on quasilinear evolution equations in order to guarantee the existence of classical nonnegative solutions. More precisely, it is shown that the system generates a global solution semiflow in the positive cone of some fractional order Sobolev space. Employing elements of the theory of infinite-dimensional dissipative systems, we prove the existence of a connected global attractor. Finally, we present some results about stationary solutions and forced periodic oscillations. The present paper extends earlier work of the authors on a semilinear problem.

The limits of synthesis: Some comments on Habermas' recent sociological writings

There is a growing schism in the ranks of critical theory. The division has become serious enough that we now find it necessary to dis? tinguish between the "classical critical theory" of the Frankfurt School and those revisions of critical thought which presently abound in the so? cial sciences. This revisionism is characterized by its rejection of the clas? sical paradigm's commitment to the premises of Western Marxism. It replaces the materialism and dialectical method of classical critical theory with a variety of theoretical alternatives such as language theory, an upgraded form of hermeneutic interpretation, or a recrudescent neo functionalism. In turning to these idealist conceptions this revisionism seems bent on constructing a sociological parody of that "critical criticism" which Marx and Engels dispatched in The German Ideology (1976; 23-31) more than a century ago. This paper discusses this revisionist trend as it has developed in the works of J?rgen Habermas. It begins by first tracing Habermas' theoreti? cal Odyssey from its origins in Western Marxism, to its current terminus in a neo-functionalist systems theory. Once having charted this trek, we will then critically evaluate the adequacy of the political economy that grounds Habermas' mature sociological theory. Finally, on the basis of that critique, we will take up the central theme of this paper: a con? sideration of the synthetic limits of Habermas' systems theory and the possibilities which recent advances in the theory of dissipative systems offer critical theory.

Phonon-mediated excitonic optical bistability in polymers.

For a polymer inside an optical cavity, optical multistability mediated by phonon modes and virtual exciton coupling is investigated. The cavity field, excitons, and phonon modes are treated as damped oscillators, and the equations of motion for the dynamical variables are obtained by applying Dekker's theory for dissipative systems. Steady-state results are obtained numerically for polydiacetylene [bis(p-toluene sulfonate)] with the input field frequency on or off resonance with the cavity field. The variation of the cavity-field intensity follows a reversed hysteresis loop on resonance, while it may follow either a normal or a reversed hysteresis loop off resonance depending on the course of change of the driving-field intensity at the unstable turning point. The variation is inverted when the detuning changes its sign. It is also found that the virtual excitons, phonon modes, and feedback of the cavity are all indispensable to such tristability.

Dissipative systems: Implications for geomorphology

AbstractA 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 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.

Renormalized theory of dissipative dispersive turbulent systems.

Renormalized theory of dissipative dispersive turbulent systems.

Squeezing of intracavity and traveling-wave light fields produced in parametric amplification

A general input-output theory for quantum dissipative systems is developed in which it is possible to relate output to input via the internal dynamics of a system. This is applied to the problem of computing the squeezing produced by a degenerate parametric amplifier located inside a cavity. The results for the internal modes are identical with those of Milburn and Walls [Opt. Commun. 39, 401 (1981)]. The output field is also found to have only 50% of maximal squeezing. However, by taking the output for a degenerate parametric amplifier inside a single-ended cavity and feeding this into an empty single-ended cavity, one can produce a maximally squeezed state inside this second cavity.

Irreversibility, time, and fluctuations in quantum statistical theory of dissipative molecular systems. II. Application to fluorescence spectroscopy

AbstractSome basic concepts of the Prigogine quantum statistical theory of irreversible processes are discussed in connection with a previous paper by Chatzidimitriou‐Dreismann and Lippert. The removal of the “dual role” that the Hamiltonian plays within conventional quantum theory corresponds to the introduction of a star‐unitary operator 𝒯 called “intrinsic time.” The formalism of the previous paper is extended to systems that interact with external fields. A formal specification of 𝒯, furthermore, follows from the Lt invariance of the basic equations of motion and its connection with the intrinsic time. As an application to fluorescence spectroscopy, a photon counting experiment is presented. It allows us to detect certain predicted “dynamical” fluctuations in emission spectra. Some current experimental results are reported concerning the resolution of the broad fluorescence band of fluoranthene (dissolved in ethanol) through detection of the underlying finer vibrational structure.

General Instability Results for Interconnected Systems

General results giving conditions for an interconnected system to be input-output and/or Lyapunov unstable are considered. These results are derived in terms of the theory of dissipative systems. This enables a very simple formulation of the requirements for instability. In particular, the restrictions of linearity and unstable subsystems, that appear in previous results, are seen to be unnecessary. Consequently, the relationship between instability and stability conditions is made clearer. A wide variety of useful instability criteria can be easily obtained as special cases.

Dissipative Dynamical Systems: Basic Input-Output and State Properties

A complete account is given of the theory of so-called dissipative dynamical systems. The concept of dissipativeness is defined as a general input-output property which includes, as notable special cases, passivity and other properties related to finite-gain. The aim is to treat input-output and state properties side-by-side with emphasis on exploring connections between them. The key connection is that a dissipative system in general possesses a set of energy-like functions of the state. The properties of these functions are studied in some detail. It is demonstrated that this connection represents a direct generalization of the well-known Kalman-Yakubovich lemma to arbitrary dynamical systems. Applications to stability theory and passive system synthesis are briefly discussed for non-linear systems.

Use of the Hamilton-Ostrogradskii principle in the vibration theory of dissipative systems

Use of the Hamilton-Ostrogradskii principle in the vibration theory of dissipative systems

A dynamical field theory for dissipative systems: The hierarchical structure of field thermodynamics

A dynamical field theory for dissipative systems: The hierarchical structure of field thermodynamics

A dynamical field theory for dissipative systems: The hierarchical structure of field thermodynamics

The analytical exploration of hierarchical structures is a central problem in modern biology. In many ways, the most basic tenet of any theory of any biological system is that its behavior lies within the laws of thermodynamics. Thus an investigation into the hierarchical structure of field thermodynamics is in effect an investigation into the underlying structures common to all biological systems. In field thermodynamics, the dynamics is prescribed at the hierarchical level of description concerned with the motions of classes of the various molecular species flowing in the system. Of crucial importance are scale factors which convert potentials into potential energies per mole of the various molecular species. On the other hand, the overall behavior of the system is presented at a higher hierarchical level of description in terms of systemic properties such as charge density or entropy density. Three questions are investigated: (1) how is the transition from the specific to the systemic level of description effected, (2) given the scale factors at the specific level how does one find those at the systemic level, and (3) what relationship can one find between the scale factors at the two levels of description? Answering these questions necessitates an examination of the nature of the forces at the two levels of the hierarchy of descriptions.

The time-invariant linear-quadratic optimal control problem

This paper provides a review of one of the basic problems of systems theory—the general time-invariant optimal control problem involving linear systems and quadratic costs. The problem includes on one hand the regulator problem of optimal control and on the other, the theory of linear dissipative systems, itself central to network theory and to the stability theory of feedback systems. The theory is developed using simple properties of dynamical systems and involves a minimum of ‘hard’ analysis or algebra. It includes a full existence theory of the matrix quadratic equation, of interest in its own right.

Stability results for nonlinear feedback systems

This paper presents an approach towards deriving sufficient conditions for the stability of nonlinear feedback systems. The central features of the approach are twofold. Firstly, useful stability tests are obtained for the case when the subsystems have nonlinear dynamics; secondly, a unifying set of general stability criteria are given, from which known situations can be treated as special cases and new ones are handled with equal ease. The results are obtained by use of a recently developed theory of dissipative systems.

A dynamical field theory for dissipative thermal systems. The first and second laws of irreversible thermodynamics

A dynamical field theory for dissipative thermal systems. The first and second laws of irreversible thermodynamics

A dynamical field theory for dissipative thermal systems. The first and second laws of irreversible thermodynamics

The thermodynamics of irreversible processes is derived from the principles of dynamical field theory independently of all elements of thermostatics, in particular the assumption of local equilibrium. Field thermodynamics proceeds from the premise that all driving forces experienced by the molecules in a continuum are conservative and arise from scalar potential functions. Dynamically the temperature potential T is no different from the pressure potential p. A field is converted to a force upon multiplication by a scale factor. A potential is converted to potential energy by the same scale factor. To scale the field −▿ p to the force per mole of molecular species k, the partial molar volume V¯ k is the scale factor. Similarly the partial molar entropy, S¯ k , scales the temperature field. The transition from the scale factors (which are physical parameters) to the systemic variables, for example S¯ k → s(x, y, z; t) , is not trivial. From the dynamics and the structure of the derived potential energy function are inducted the conjugate variables such as ( p, V I ) and ( T, s). The meta-mechanical properties of the thermal variables ( T, s) are discovered via the local First Law of Thermodynamics, which relates internal energy, thermal flux, and work, and from the local Second Law, which prescribes the possible partitions of internal energy between kinetic, potential, and thermal energies. From the form of the potential energy come Maxwell's relationships. From the energy partition comes the equation of continuity for entropy, with its important source term. In contrast to earlier theories of irreversible thermodynamics, the dissipation function does not include the stress tensor, a constitutive parameter.

Conformal invariance and Hamilton Jacobi theory for dissipative systems

For certain dissipative systems, a comparison can be made between the Hamilton–Jacobi theory and the conformal invariance of action theory. The two concepts are not identical, but the conformal action theory covers the Hamilton–Jacobi theory.

Scattering theory for dissipative hyperbolic systems

An abstract theory of scattering is developed for dissipative hyperbolic systems, a typical example is the wave equation: u tt = Δu in an exterior domain with lossy boundary conditions: u n + αu t = 0, α ⩾ 0. In this theory, as in an earlier theory developed by the authors for conservative systems, a central role is played by two distinguished subspaces of data common to both the perturbed and unperturbed problems. Associated with each subspace is a translation representation of the unperturbed system. When these representations coincide they provide a convenient tool for extending the data so as to include a large class of generalized eigenfunctions for both the perturbed and unperturbed generators. The scattering matrix is characterized in terms of these generalized eigenfunctions; it is shown to be meromorphic in the whole complex plane and holomorphic in the lower half plane. The zeroes and poles of the scattering matrix correspond, respectively, to incoming and outgoing generalized eigenfunctions of the perturbed generator. In the second part of the paper the assumptions introduced in the abstract theory are verified for the wave equation problem cited above.