The adiabatic approximation for coupled oscillators

The adiabatic approximation is applied to determine the quantum states of coupled oscillators described by a generalized Hénon-Heiles hamiltonian. Comparison with exact quantum and other results show that numerically calculated adiabatic energy levels are accurate even for excited states.     

New unitary transformation for the three coupled oscillators

A new unitary operator U, which can transform the Fock space of a three-dimensional isotropic harmonic oscillator into the space in which the Hamiltion of three coupled oscillators is diagonized, is found. The coordinate representation of U is presented and is used to directly derive the wave function of the energy eigenstate of the coupled oscillators.     

Symmetry broken self-consistent field approximation for coupled oscillators

A self-consistent field approximation allowing for a change of coordinates is applied to a model of two coupled oscillators considered by Davis and Heller [J. Chem. Phys. 75 , 246 (1981)]. The model accounts well for the unsymmetrical nonstationary states involved in the quantum dynamical tunneling phenomenon.     

Quasi-bound states of coupled Morse oscillators

Quasi-bound (resonance) states are present in the continuous spectrum of the Hamiltonian of two coupled Morse oscillators. Two different methods for approximating these as localized states are compared. The algebraic approach is shown to be in very good accord with the other method which is formulated in coordinate space and hence is differential in character. For these highly excited states an intermultiplet mixing term is included in the algebraic Hamiltonian.     

Complex behavior in coupled bromate oscillators

In this study, coupled bromate-oscillators constructed by adding 1,4-cyclohexanedione (1,4-CHD) to the ferroin-catalyzed Belousov-Zhabotinsky reaction are investigated in a batch reactor under anaerobic conditions. Various complex behaviors such as sequential oscillations and bursting phenomena are observed. At low concentrations of ferroin or malonic acid (MA), the development of sequential oscillations is found to depend on the ratio of [1,4-CHD]/[ferroin] and [1,4-CHD]/[MA] rather than their absolute concentrations. As the concentration of MA or ferroin was increased gradually, however, the minimum 1,4-CHD concentration required to induce complex oscillations reaches a plateau. Perturbations by light illustrate that the first oscillatory window is governed by the ferroin-MA-BZ mechanism, whereas the 1,4-CHD-bromate oscillator plays a prominent role during the non-oscillatory evolution and the second oscillatory window. Our conclusion is further supported by numerical simulations in which sequential oscillations observed in experiments are qualitatively reproduced by a modified FKN mechanism.     

Relation of algebraic models of coupled anharmonic oscillators

It is shown that the authors's SU(n) invariance approach to coupled anharmonic oscillators and the coupled SU(2) non-invariance approach of Levine and co-workers are complementary methods related by a transformation between the raising and lowering operators used in each.     

Experimental and theoretical studies of coupled chemical oscillators

Chemical oscillators may be coupled together in a variety of ways. Two of the most important forms of coupling are physical (via transport) and chemical (via common species). Such coupling can result in new phenomena. Here we focus on rhythmogenesis, the onset of oscillations when two steady state systems are coupled, and oscillator death, the cessation of oscillations when two oscillatory systems are coupled. We also discuss briefly a biological example, the crustacean stomatogastric ganglion, and the important role of delay, which may be brought on by coupling, in chemical oscillation.     

Internal stochastic resonance in two coupled chemical oscillators

The dynamics of two coupled chemical oscillators was investigated numerically when the first subsystem was subjected to external parametric noise. The signal-to-noise ratio (SNR) of the response of each subsystem to external noise shows internal stochastic resonance (SR). In addition, the SNR also shows resonance behavior with the variation of coupling strength.     

Collective behavior of a population of chemically coupled oscillators

Experiments are performed in which a large number (approximately 10(4)) of relaxation oscillators are globally coupled through the concentration of chemicals in the surrounding solution. Each oscillator consists of a microscopic catalyst-loaded particle that displays oscillations in the concentrations of chemical species when suspended in catalyst-free Belousov-Zhabotinsky (BZ) reaction solution. In the absence of stirring, the uncoupled particles display a range of oscillatory frequencies. In the well-stirred system, oscillations appear in the surrounding solution for greater than a critical number density of particles (n(crit)). There is a growth in the amplitude of oscillations with increasing n, accompanied by a slight increase or no change in frequency. A model is proposed to account for the behavior, in which the transfer of activator and inhibitor to and from the bulk medium is considered for each particle. We demonstrate that the appearance and subsequent growth in the amplitude of oscillations may be associated with partial synchronization of the oscillators.     

The anharmonic effect study of coupled Morse oscillators for the unimolecular reaction

The importance of anharmonic effect in dissociation of molecular systems especially clusters has been noted. In this paper, we shall study the effect of coupled anharmonic oscillator of the standard bilinear form (SBF) Morse oscillator (MO) potential on unimolecular reaction. We shall use the systematic theoretical approach, YL method, proposed by Yao and Lin (YAO L, et. al. J Phys Chem A, 2007, 111(29): 6722-6729), which can evaluate anharmonic effects on the rate constants based on the transition state theory. In treating the anharmonic effect with the Morse oscillator potential on unimolecular reactions under collision-free conditions by using the RRKM (Rice-Ramsperger-Kassel-Marcus) theory, the in-verse Laplace transformation of the partition functions was used to obtain the total amount of state and density of state by using the first-order and the second-order approximations of the saddle-point method. To demonstrate the anharmonic effect of the SBF Morse model, we choose some model systems and a real reaction as examples.     

The connected moments expansion for the zero-point energy of coupled anharmonic oscillators

The connected moment expansion (CMX) technique is used to calculate the zero-point energy of an arbitrary system of coupled anharmonic oscillators. When the anharmonic term has the form of a polynomial with respect to the normal coordinates, it is possible to calculate the zero-point energy in a completely automated way. A numerical example is presented, demonstrating the power of the new method.     

Hypervirial-perturbative approximation to the SCF eigenvalues of coupled oscillators

The hypervirial relations and perturbation theory are used in order to obtain analytical non-numerical expressions for the SCF eigenvalues of coupled oscillators.     

Energy level distributions in spectra of coupled harmonic oscillators

The spectral density distribution of a hamiltonian which represents a system of N coupled harmonic oscillators, and hence may approximately describe molecular vibrations in the local mode picture, is analyzed. The spectral density moments are expressed as linear combination of products of coefficients which depend on the molecular structure and of one-particle moments describing individual bonds and interactions between them. Detailed expressions for linear and tetrahedral molecules are analyzed. Moreover, general formulae for matrix elements of powers of momentum in the harmonic oscillator basis are given.     

Loss of coherence in a population of diffusively coupled oscillators

The authors investigate the relationship between the natural frequency distribution of diffusively coupled chemical oscillators and their entrainment by pacemakers. The system consists of micrometer-sized catalyst beads which are coupled to their neighbors by diffusion of the activator/inhibitor species through the catalyst-free Belousov-Zhabotinsky (BZ) reaction solution. The frequency distribution is measured as a function of the beads' number of neighbors. With the maximum number of neighbors, either target waves or disordered patterns are observed in the reaction domain and there is a shift to higher frequencies than those observed in the natural frequency distribution. The loss of coherence between neighbor oscillators is quantified by a decrease in the phase synchronization index. The experimental results are reproduced in simulations which demonstrate that the decrease in the degree of synchronization is correlated with the appearance of a small fraction of permanently excited beads in BZ populations of high mean frequency and/or large width.     

Roles of noise in single and coupled multiple genetic oscillators

The noisy fluctuation of chemical reactions should profoundly affect the oscillatory dynamics of gene circuit. In this paper a prototypical genetic oscillator, repressilator, is numerically simulated to analyze effects of noise on oscillatory dynamics. The oscillation is coherent when the protein number and the rate of the DNA state alteration are within appropriate ranges, showing the phenomenon of coherence resonance. Stochastic fluctuation not only disturbs the coherent oscillation in a chaotic way but also destabilizes the stationary state to make the oscillation relatively stable. Bursting in translation, which is a source of intense stochastic fluctuation in protein numbers, suppresses the destructive effects of the finite leakage rate of protein production and thus plays a constructive role for the persistent oscillation. When multiple repressilators are coupled to each other, the cooperative interactions among repressilators enhance the coherence in oscillation but the dephasing fluctuation among multiple repressilators induces the amplitude fluctuation in the collective oscillation.     

Approximate and exact quantum mechanical energies and eigenfunctions for a system of coupled oscillators

A non-separable model hamiltonian, which describes the normal vibrations of a pair of coupled oscillators, has been solved by a number of approximation procedures, and approximate solutions compared with the “exact” solutions. The results of a simple perturbation theory calculation are seen to be rather more reliable than those of self-consistent-field model.     

Squeezing transformation caused by frequency jumps for solving the problem of two coupled oscillators

For the Hamiltonian a unitary operator that is closely related to the squeezing transformation caused by frequency jumps is found and its coordinate representation provides an alternative approach for calculating the wave function of the eigenstate of ??, whereas its normally ordered form is convenient for deriving the resolvent kernel in a coherent-state basis.     

Adiabatic multi-step separation method and its application to coupled oscillators

Extension of the adiabatic approach to a multi-step separation method is presented. This method step by step reduces the multi-dimensional Schrödinger equation to the effective equations of lower dimensions. The reduction procedure allows to take advantage of multi-level hierarchy of various physical systems. The multi-step separation method is applied in the calculation of vibrational energies of coupled oscillators. The new method is found to be very effective and accurate.