Dissipative Oscillator Research Articles

Comparison of bifurcation structures of driven dissipative nonlinear oscillators.

The bifurcation sets of driven strictly dissipative nonlinear oscillators are compared in terms of phase diagrams and fixed-point diagrams. The comparison reveals distinctive bifurcation patterns that occur for all models. In particular, two subpatterns of bifurcation curves within the period-doubling hierarchy can be identified. The results suggest that there exists a universal bifurcation structure for oscillators of the type investigated.

ON MODELING DRIVEN OSCILLATORS BY MAPS

A family of two-dimensional maps showing essential features of the bifurcation set of some periodically driven, strictly dissipative oscillators is constructed.

Quantum-mechanical dissipation: From the weak- to the strong-coupling limit.

We study a spin-1/2 entity interacting with a set of quantum-mechanical oscillators using the Heisenberg picture. In the weak-coupling limit, the Heisenberg equation is given an analytical solution, suggesting that upon increase of the coupling the system should exhibit a critical transition to an overdamped regime, like that of a heavily damped oscillator. Thus the lifetime of the trapped state increases with increasing temperature. Under the assumption of very small tunneling frequency, another analytical solution is found, which predicts the occurrence of trapped states, whose lifetime decreases with an increase of temperature. As a result of the competition between the two trapping processes, the lifetime of the localized state is shown to be a nonmonotonic function of temperature with a minimum lifetime at a certain critical value of temperature. The predictions of this theoretical treatment are carefully checked by numerical solution of the interaction between a 1/2-spin entity and a dissipative oscillator, and a very satisfactory agreement between theory and numerical solution is found over the whole range of coupling strength.

Ruelle's rotation frequency for a symplectic chain of dissipative oscillators.

Ruelle's rotation frequency for a symplectic chain of dissipative oscillators.

Third-order Nonlinear Dissipative Oscillator with Initial Gaussian Pure and Mixed States

Abstract A two-fold generalization of results concerning the third-order nonlinear dissipative oscillator is presented. The exact solution to the master equation for an arbitrary initial state is given, and closed formulae for the squeeze variances in the case of initial Gaussian pure and mixed states are derived.

Exact quantum statistics of a nonlinear dissipative oscillator evolving from an arbitrary state.

The statistical properties of the third-order nonlinear dissipative oscillator, which evolves from any state, are derived on the basis of the exact solution to the master equation. Some important features of the nonlinear oscillator model, such as the recurrences of the initial state, are related to the properties of quasidistributions connected with the phase of the complex field amplitude.

Multiparameter direct variational solutions for linear dissipative oscillators

The validity of Lagrangian functional for linear dissipative oscillator is checked by using it as a tool for approximate solutions via direct methods of variational calculus. The results of a single-parameter approximation, presented in an earlier work, are considerably improved by increasing the number of parameters in test functions. The same method is applied to the more convential Lagrangian functions, formulated by the multiplier method and the image (adjoint) method. These results allow a comparative estimate of the accuracy obtained by three different methods.

DC injection alters spontaneous otoacoustic emission frequency in the frog

A permanent wire electrode was implanted in the inner ear of the green frog ( Rana esculenta). Through this electrode direct current was delivered to the inner ear fluid of the frog, during recording of a spontaneous otoacoustic emission (SOAE). Both SOAE frequency and amplitude turned out to depend on DC level and polarity. Possible explanations for the observed phenomena are given, based on a dissipative limit-cycle oscillator model for SOAE generation, and the assumption that SOAE frequency depends on hair cell ion channel inductance.

Quantum correlations of coupled nonlinear dissipative oscillators

We present a quantum solution for two-mode Kerr interaction, including losses and noise. Moments up to fourth order are calculated and discussed. A connection between revivals of a coherent part of a field and squeezing is found. The influence of losses and noise on quantum effects is examined as well.

Intrinsic Fluctuation and Its Critical Scaling in a Class of Populations of Oscillators with Distributed Frequencies

Large populations of coupled dissipative oscillators exhibit macroscopic mutual entrainment when coupling is strong enough to compensate for the disordering effect due to the distribution of natural frequencies of oscillators. This phenomenon has much relevance to a variety of fields in science (e.g., see Ref. 1», and quite a few investigations have been devoted to elucidating it. Of these, studies with emphasis on the aspect of a ph,ase transition may be particularly interesting and important since they open a new area in the field of critical phenomena to clarify how the onset of mutual entrainment in oscillator assemblies compares to conventional phase transi­ tions in diverse equilibrium systems. 2 ) In this paper we attempt a study along such a line using the following model :3)

Quantum statistics of nonclassical radiation in dissipative forward four-wave mixing. II. Coupled third-order nonlinear dissipative oscillators.

The statistical properties of nonlinear oscillators modeling dissipative forward four-wave mixing are investigated, adopting the Fokker-Planck equation approach. The photon statistics are shown to be insensitive to the nonlinearities and are formulated in terms of the superposition of coherent and chaotic fields. The only approximation in describing the squeezing properties consists in the way of including thermal noise. The effect of the initial coherent amplitudes, that of the interplay of the coupling constants, and that of the reservoir on the squeezing properties are studied. An interesting connection between the revivals of coherent amplitudes and squeezing is pointed out.

Finite-temperature field theory and quantum noise in an electrical network.

Finite-temperature (0\ensuremath{\le}T\ensuremath{\infty}) field (FTF) theory with an effective spectral Lagrangian density formulation is used to study quantum noise in an electrical network. Solutions for the finite second moments that satisfy the uncertainty principle bound are given for a dissipative quantum oscillator. A regularization method, based on the analysis of a semi-infinite low-pass filter, is employed, and it leads to results which differ from those of the Drude model. To illustrate the FTF method, an example is given using an ideal finite-temperature coherent state.

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.

Periodic and aperiodic regimes in coupled dissipative chemical oscillators

Dynamic behavior of two identical reaction cells with linear symmetric coupling is studied in detail. The standard model reaction scheme “Brusselator” is used as the description of the kinetics. The uncoupled cells can exhibit either a stable stationary state or stable periodic oscillations. A number of stationary and periodic oscillatory patterns arise as a result of the coupling. A non-homogeneous spatio-temporal organization includes homoclinic and heteroclinic oscillations as well as chaotic regimes. Numerical continuation algorithms are used to determine the dependence of stationary and periodic solutions on parameters. Stable stationary nonhomogeneous regimes exist typically at intermediate levels of coupling intensity. The nonhomogeneous periodic solutions arise either via Hopf bifurcatios from stationary solutions or via period-doubling bifurcations from the homogeneous periodic solutions. The results obtained may serve as a standard for the study of the behavior of other coupled systems in which either a stable stationary state or stable oscillations exist in the single cell.

Resonances and Torsion Numbers of Driven Dissipative Nonlinear Oscillators

The torsion of the local flow around closed orbits and its relation to the superstructure in the bifurcation set of strictly dissipative nonlinear oscillators is investigated. The torsion number describing the twisting behaviour of the flow turns out to be a suitable invariant for the classification of local bifurcations and resonances in those systems. Furthermore, the notions of winding number and resonance are generalized to arbitrary one-dimensional dissipative oscillators.

Impurity-Induced Pinning, Damping and Metastability of Interchain-Coupled Charge Density Waves

The impurity pinning of the charge density waves that exhibit three-dimensional order due to interchain coupling is investigated in the model of Fukuyama and Lee by taking account of phase distortions in the transverse directions as well as in the chain direction. Calculated adiabatic potentials elucidate the presence of metastable states, which is responsible for strong pinning and depinning. The linear response function calculated for the pinned state is in the form used for a dissipative harmonic oscillator and explains the overall feature of observed frequency dependent conductivities. Memory effects recently found in experiments are also qualitatively explained in the present model.

Dynamics of nonlinear dissipative oscillators

In this paper we review some of the important concepts and theorems dealing with nonlinear dissipative oscillators in the presence of random and deterministic periodic forces, with particular reference to the Van der Pol oscillator. We display simple techniques such as the Krylov–Bogoliubov averaging method and generalize this method to random oscillator systems exhibiting new noise-induced phenomena

Connection between dissipative and resonant conservative nonlinear oscillators

It is shown how to add dissipation to the “resonant” nonlinear oscillators studied by Ford and Lunsford in such a way that the system remains on the energy surface. In the dissipative system, the energy surface is stable in some directions and neutrally stable in other directions. The dissipative oscillators are special cases of the general type investigated by Sherman and McLaughlin. The connection between resonant conservative nonlinear oscillators and dissipative oscillators may make it easier to extend the theorem of Arnol'd to dissipative systems.

Quantum theory of a dissipative oscillator

The nonlinear equation of dissipative quantum mechanics is considered in the relaxation-time approximation. It is shown that the steady current-free state does not change when dissipation is taken into account; in particular, there is no ground-state damping, and the zero energy is conserved. A solution is obtained to the problem of the excitation of a harmonic oscillator, serving as a model of single-mode radiation in an open resonator; the solution obtained describes the evolution of the oscillator from an arbitrary steady state under the action of a constraining force. Transition probabilities between the oscillator steady states are calculated. The results are found to be in agreement with the classical theory of damping oscillations.