Class Of Nonlinear Oscillators Research Articles

Wave analysis by Slepian models

Ocean waves exhibit more or less a Gaussian distribution for the instantaneous water surface height, and there is a need to develop simple models for generation of the characteristic non-Gaussian statistics, namely the asymmetric distributions of water surface height and wave slope. We argue that a simple class of non-linear oscillators can reproduce some of the characteristic features of random water wave processes and linear or non-linear response to ocean waves. We describe the Slepian model for the Gaussian case, and explain the use of the regression approximation for level crossing distances and associated variables, such as wave period and amplitude. Finally we speculate about a generalization of the regression technique to the non-linear Markov process case.

Approximate Solution of a Class of Nonlinear Oscillators in Resonance with a Periodic Excitation

This paper deals with the most important characteristics of a generalized Van der Pol–Duffing oscillator in resonance with a periodic excitation. We use an asymptotic perturbation method based on Fourier expansion and time rescaling and demonstrate through a second order perturbation analysis the existence of one or two limit cycles. Moreover, we identify a sufficient condition to obtain a doubly periodic motion, when a second low frequency appears, in addition to the forcing frequency. Comparison with the solution obtained by the numerical integration confirms the validity of our analysis.

Response of a non-linear system with strong damping to multifrequency excitations

In this paper, a new asymptotic calculation method is presented for a class of nonlinear nonautonomous vibration systems with multiple external periodic interferences. Simple calculation formulae of an asymptotic solution for resonance and off-resonance vibration are derived. The paper is concerned with a vibration system representing a class of nonlinear oscillators. Consequently, the calculation of a class of nonlinear oscillators is routinized. Two different Duffing equations are verified which shows that the results are completely in accordance with the solutions of references [3, 4, 6]. The derivation of solutions of Duffing equations becomes easier and simpler. In addition, some errors in reference [6] are pointed out.

Application of high-order difference methods for the study of period doubling bifurcations in nonlinear oscillators

For the study of period doubling bifurcations in nonlinear T-periodic forced oscillators depending on one parameter, it is necessary to use a quick and accurate method to obtain periodic solutions of a second-order nonlinear differential equation. Using a periodic approximation, obtained earlier for another parametric value, a Newton method allows its iterative refinement, by the successive solution of linear periodic boundary value problems. In these problems, the derivatives of the unknown periodic solution are approximated by high-order difference formulae, yielding accurate results in the N abscissae x i = iT N , i = 0, …, N − 1. To determine the parametric values where bifurcation takes place, the solution of the variational equations was calculated using a Runge-Kutta-Hǔta method. Therefore the values of the corresponding nT-periodic solution ( n = 2 k , k ∈ N ) were determined in some points inside the discretization intervals using trigonometric interpolation. From the resulting fundamental matrix Φ( nT), the multipliers are readily determined, allowing the detection of the period doubling bifurcations when the largest multiplier passes through −1. The method has been applied successfully to study bifurcations leading to chaos in several classical nonlinear oscillators.

Exact Solutions and the Numeration Program for Free Oscillation in a Nonlinear System with a Quadratic Spring Function and a Single Degree of Freedom.

A classical nonlinear oscillator possesses a spring function expressed in terms of the 2nd-or the 3rd-order polynomial, viz., the so-called Helmholtz or Duffing systems. The latter, a symmetrical system, has been studied extensively since Duffing proposed the mathematical models ; however, the former, an asymmetrical system, is less well known than the latter. The exact solution of their free oscillation seems to be at the same stage of development, in that an expression in terms of the Jacobian elliptic functions, i. e., sn, cn or dn for the Duffing system, exists already, whereas sn2 or cn2 have only recently been found for the Helmholtz system. This paper summarizes existing numeration algorithms and proposes a program for exact solutions of the free oscillation in the Helmholtz system. This solution successufully verifies the accuracy of a numerical or an approximate scheme applicable to solve a nonlinear problem which may essentially have no exact solution.

Instabilities and chaos in a bistable thin layer of two-level atoms

A theoretical analysis is made of the interaction between a bistable thin layer and an electromagnetic wave of modulated amplitude. It is shown that complex (including random) regimes appear in the reflection and transmission of light by a thin layer. The model equations are presented in a form correpsonding to the motion of a classical nonlinear oscillator with parametric and external force excitation. The behavior of the system is analyzed in terms of the dynamics of this oscillator.

Nonlinear parametric oscillations in certain stochastic systems: A random van der Pol oscillator

Astochastic model for some class ofnonlinear oscillators, which includes a van der Pol-type oscillator with random parameters, is analyzed in thediffusion limit. That is, small random fluctuations and long time are considered, while the nonlinearity is also assumed to be small. We show that there existstationary distributions, independent of the phase of the oscillator, a result proved earlier by R. L. Stratonovich assuming the random perturbations of the frequency to be delta correlated. The time behavior of the moments of the displacement of the oscillator from its rest position is also investigated and the results are compared with the corresponding ones for the linear random oscillator. A numerical study is also performed for the first two moments and plots are given.

Analysis of chaotic systems using the cell mapping approach

Chaotic motions in deterministic nonlinear systems are an important topic both from a theoretical and a practical point of view. In particular, there have been many studies of systems which yield bounded nonperiodic trajectories converging to attractors of a rather complicated nature, so-called strange attractors. Their existence was demonstrated in a class of nonlinear oscillators with periodic forcing which occur in electric circuit theory and mechanics. The determination of the domain of attraction of such attractors, depending on the parameters, is an interesting problem. It is shown, that the cell mapping approach, i.e., a discrete version of a Poincare map, represents a very efficient method for analyzing this problem.

New Tamm-Dancoff treatment of the classical nonlinear oscillator

Heisenberg’s new Tamm-Dancoff method is applied to the 1-dimensional classical nonlinear oscillator. The procedure is seen formally to lead to an explicit solution to the initial-value problem, which is expressed in terms of infinite-dimensional matrices. The matrix transformation leading from the initial to the final data is defined by the eigenvectors and eigenvalues of the linear-eigenvalue problem. The truncation of the characteristic matrix is seen numerically to lead to a convergent approximation procedure in which the classical secular problem does not arise.

Cooling of internal degrees of freedom of atoms and molecules using resonance radiation

An analysis is made of the statistics of an ensemble of one-dimensional (classical) nonlinear oscillators in a thermostat excited by an external harmonic force. It is found that, under certain conditions, the action of a harmonic force may suppress the thermal excitation of oscillator vibrations. On the basis of the proposed model, it is concluded that internal degrees of freedom of gas atoms and molecules may be cooled by resonance radiation.

On statistics of the ensemble of oscillators under excitation. I. Classical nonlinear oscillators excited by a weak external force

The distribution function is searched out of the Fokker—Planck kinetic equation with the small parameter at higher derivatives. The analytic solution is produced by means of a double successive application of the averaging method. It is shown that an external harmonic force leads to the appearance of excessive population in some regions of phase space, where the vibrational frequency of oscillator is close to the frequency of the external force. The present model shows that laser radiation can stimulate chemical reactions with high activation energy, whereas reactions with low activation energy will not be stimulated.

Transformation of Nonlinear Oscillators to Neural Equations

Two types of classical nonlinear oscillators are shown to be transformed into neurodynamical equations of Cowan. Symmetric properties of these transformations are also discussed.

Computational Aspects of Random Vibration Analysis

Computational aspects of the eigenfunction associated with the envelope of the nonstationary random response of a lightly damped linear oscillator are examined. Analytic formulas are given which may be used for an easily mechanized numerical computation of the probability density function and the moments of the response envelope. Formulas are also given facilitating the numerical determination, based on a perturbation type approach, of the statistics of the response envelope of a broad class of nonlinear oscillators. Specific computational problems are examined in the context of a Duffing oscillator.

The excitation of anharmonic molecules for steady-state and pulsed radiation

Excitation levels by multiple-photon absorption are calculated using a quantized model for a classical nonlinear oscillator. Substantial red shifting is necessary in order to reach high vibrational excitation. Increased damping has a linearizing effect and reduces red shifting as well as the excitation. Cross sections for the various excited states are calculated. The decay of the excited molecule to the ground state is discussed. In principle the model can be used to calculate photodissociation.

The classical nonlinear oscillator and the coherent state

The coherent state constructed out of quantum oscillator states is employed to develop a method for solving the classical nonlinear oscillator problem. The perturbation solution of the Duffing oscillator is used to illustrate the method and to obtain the result of the classical procedure.

A Note on Dynamic Susceptibility of Classical Non-Linear Oscillator

A Note on Dynamic Susceptibility of Classical Non-Linear Oscillator

Adiabatic invariance to all orders

The purpose of this article is to extend adiabatic invariance theorems to all orders in the slowness parameter. The concept of adiabatic invariance to all orders is formulated precisely in Section I. In Section II, a classical one-dimensional nonlinear oscillator is treated. It is proved that the action integral extended over a period of the instantaneous time independent problem is an adiabatic invariant to all orders. Section III is devoted to the extension of the Born-Fock quantum mechanical adiabatic theorem to all orders. The systems to which the proof applies in all rigor are those which have a finite number of nondegenerate quantum states which do not cross in the process of adiabatic change. The proofs are based on a systematic construction of the asymptotic expansion in the slowness parameter of the statistical distribution function which describes an initially stationary ensemble of systems.