Nonlinear Oscillator Model Research Articles

Modeling auditory system nonlinearities through Volterra series

A computational method is reviewed, in which the solution of systems of nonlinear differential equations is written in terms of a Volterra functional series. Results of implementing the aforementioned technique in a computer program (exploiting new software for symbolic manipulation) and of applying it to a nonlinear oscillator model are presented. The relevance of this approach to Auditory System modeling is discussed. Suggestions are given, regarding possible applications to Speech Recognition problems.

Method of fuel tank design development for future long-term space flights

The problem of developing propellant tanks for spacecrafts is discussed. Typical situations are analysed which require knowledge of hydrodynamic processes in propellant tanks. Complex approaches to developing propellant tanks based on joint experimental and theoretical methods (including computation) is proved. Two typical experimental plants are described. One of the plants is a construction providing side overloads for a propellant tank model by its moving in a horizontal plane allowing its turn about a horizontal axis. The other plant allows to investigate liquid drain out of the tank model while oscillating in two mutually perpendicular directions in a horizontal plane. Some results of experimental investigation of hydrodynamic processes in propellant tanks are given. It is shown that an investigator should have a wide set of mathematical models of different complexity, adequacy and efficiency (time and resources spent for computation). Two mathematical models are given with a description of their possibilities and applicability limits. Non-linear oscillator model is a non-linear mechanical analogue hydrodynamic processes with small filling of propellant tanks and spread damper construction. A model of depth-averaged equations of hydrodynamics is true for the case of small filling of tanks and allows to reduce computation of three-dimensional hydrodynamic processes to solving two-dimension problems. Comparative analysis of experimental results and theoretical design is carried out. Carrying out experiments for obtaining emperical parameters of mathematical models, adequacy and applicability limits for development is emphasized.

Intrinsic optical bistability in a cavity.

The nonlinear oscillator model including propagation has an interesting optical bistable characteristic behavior, which has been studied by Goldstone and Garmire1 and, extensively, by Haus et al2. The authors show that this medium exhibits optical bistability without a cavity due to increased absorption. This phenomenon has been coined intrinsic optical bistability (IOB) because it is intrinsic to the material and does not require an external feedback. A study of transverse effects in IOB3 shows that bistability is not significantly altered by varying the Fresnel number down to order unity, but that an interesting redistribution of the electromagnetic energy takes place due to increased diffraction and high absorption at the center of the input intensity profile

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.

Gaussian domination: Quantum nonlinear oscillator

The Gaussian domination for the special model of the quantum nonlinear oscillator is proved. From this condition the upper bound for the thermal expectation follows.

Classification of Second-Order Ordinary Differential Equations Admitting Lie Groups of Fibre-Preserving Point Symmetries

We use Elie Cartan's method of equivalence to give a complete classification, in terms of differential invariants, of second-order ordinary differential equations admitting Lie groups of fibre-preserving point symmetries. We then apply our results to the determination of all second-order equations which are equivalent, under fibre-preserving transformations, to the free particle equation. In addition we present those equations of Painlevé type which admit a transitive symmetry group. Finally we determine the symmetry group of some equations of physical interest, such as the Duffing and Holmes-Rand equations, which arise as models of non-linear oscillators.

Spatial effects in intrinsic optical bistability.

Using the nonlinear oscillator model as a prototype medium exhibiting intrinsic optical bistability, we investigate the inhomogeneous absorption of the electromagnetic field. The forward- and backward-field amplitudes and diffraction effects are retained in the mathematical description. Analytic results are given in the limit of plane-wave propagation under steady-state conditions. The transmitted and reflected intensity exhibit a structure that is determined by the spatial inhomogeneity of the absorption in the longitudinal direction. The transmitted intensity has a structure that is dependent on the length of the medium. The reflected intensity has an interference structure from light reflected at the front surface and the internal boundary separating a high-polarization from a low-polarization branch. A degenerate-four-wave-mixing experiment is predicted to be a very sensitive probe of the internal boundary and the interference between the forward and backward field. The phase-conjugate signal develops large oscillations as the input field is varied. Numerical results for diffraction effects are also given, and we find that the plane-wave results for the center of the beam remain reliable down to Fresnel numbers of order unity and in media that are smaller than the linear absorption length.

On the temporal fluctuations of pulsating auroral luminosity

From a study of all‐sky TV records, it is shown that the luminosity fluctuations of pulsating auroras can be understood in terms of a series of pulses with rapid on‐off switchings in burstlike fashion and that the widths of successive pulses (pulsation “on” times) are fairly constant. This is common even when luminosity fluctuations consist of complex‐irregular variations, in contrast to the pulsation “off” time that is significantly variable. Complex‐irregular variations are found to be due to simultaneous appearance of more pulsating patches that exhibit movements eastward and westward over the site, and each of the patches shows primarily isolated luminosity pulses. Several examples are presented and described in detail. The pulsation “on” time is found to statistically concentrates in two small ranges (0.2–0.5 s for flash including 3‐Hz modulation and 2–6 s for pure‐expansion and streaming‐propagating modes), in spite of variations in appearance. On the other hand, the pulsation “off” time is widely scattered over a range from nearly zero to longer than 1 min. A natural consequence of these observations is that the classical concept of “period” does not mean much and the luminosity fluctuations should be treated as a series of individual isolated pulses where the pulsation “on” time is the most essential quantity. These characteristics are briefly discussed in relation to VLF/ELF wave‐particle interactions in the magnetosphere. Then a new interpretation of the nonlinear relaxation oscillation model is proposed, where the propagation effect of VLF/ELF waves in low energy plasma irregularities near the magnetospheric equatorial plane plays an essential role to produce rapid on‐off switchings of precipitating energetic electron fluxes. Both electromagnetic and electrostatic waves are possibly related to the precipitation pulsations.

Non-linear itinerant oscillator model. Inclusion of barrier crossing by the method of Praestgaard and van Kampen

A closed-form expression is deduced for the after-effect function of the non-linear itinerant oscillator (escape from a potential well is permitted and the oscillations need not be small). The dielectric loss is computed and compared with that of the linear itinerant oscillator, indicating that the escape mechanism shifts the microwave peak absorption to higher frequencies and yields additional spectral contributions between the microwave and far-infrared maxima.

The non-linear itinerant oscillator model: the effect on the polarisability spectrum of including barrier-crossing by dipoles

Analysis of the itinerant oscillator model of polar fluids has hitherto been concentrated on the small-oscillation approximation. This has neglected one important contribution to the dynamic polarization, namely relaxation due to crossing ofdipoles over the hills of the cosine potential for dipole-dipole coupling. In this Letter a numerical algorithm for solution of the Fokker-Planck equation due to Reid is used to calculate the Fourier transform of the angular velocity autocorrelation function for this model when barrier-crossing is included. The results are used to calculate the frequency dependence of the polarizability. They indicate that (i) the far-infrared absorption peak is centred at a lower frequency than in the harmonic approximation, and (ii) the bandwidth of the far-infrared absorption is increased significantly over the harmonic approximation leading to additional contributions at frequencies below the far-infrared peak, in qualitative agreement with recent experimental observations.

Observation of the "Brownian motion" of the electric field in a laser.

Critical dynamics of the laser field near the threshold of oscillation is investigated and observation of the Brownian motion of the electric field amplitude driven by quantum noise is reported. Experimental results agree well with the nonlinear oscillator model of the laser.

A Stochastic Model for Nonlinear Oscillations of Duffing Type

A nonlinear model for random oscillators, which includes a stochastic version of the Duffing oscillator in both cases of hard and soft spring, is studied in the diffusion limit. This is an asymptotic limit involving random fluctuations of small size, and long time. The nonlinearity is also assumed to be small. One of the main results is that the effect of the type and size of nonlinearity under investigation does not affect appreciably the qualitative long-time behavior of the moments of the displacement of the oscillator, in both the hard and the soft spring models. We also prove that there are no distributions. Finally, a numerical study for the first two moments is performed and plots are given.

Oscillator models for multiphoton optical bistability and phase conjugacy

Nonlinear oscillator models are constructed to treat the bistability in situations involving elementary excitations in solids. Such models are shown to be useful not only in describing single photon but also multiphoton bistability. The resulting bistability both with and without cavity is considered. The two-photon excitonic bistability in CuCl is in detail. The effect of local field corrections can also be incorporated, in a simple manner, in such models.

A nonlinear solar cycle model with potential for forecasting on a decadal time scale

This paper describes a novel non-linear oscillator model of the sunspot cycle which accurately reproduces several of the observed qualitative and quantitative characteristics of the real cycle including the long term amplitude modulation pattern. The model accounts for 96% of cycle peak height variance over the period 1859 to 1980. The aim of this work is to assess the potential of such models for forecasting solar activity on decadal and possibly longer time scales. Longer term forecasts may have practical economic significance because of the growing evidence for relationships between solar cycle variations and terrestrial weather and climatic variations (Bandeen and Moran, 1975; Currie, 1980; Williams, 1981). The model predicts that cycle 22 will have an annual mean peak amplitude in the range 25 to 45, the lowest peak activity for 260 yr.

Forced-vibration study of the aeroelastic instability of a square-section cylinder near vortex resonance

Force measurements made on a square-section cylinder forced to oscillate in a direction normal to a stream are presented. These measurements are used to estimate the values that the combined mass-damping parameter of an equivalent spring-mounted cylinder should take if it is to perform oscillation at prescribed values of amplitude ratio and reduced velocity. The mass-damping parameter appears to be predicted well by the quasisteady theory of galloping when the reduced velocity is well above lock-in. Below resonance, the parameter is negative, contrary to the predictions of quasisteady theory, and this indicates why galloping is not usually observed. It is found from the forced-vibration measurements that at a given value of reduced velocity the combined mass-damping parameter may have the same value for more than one amplitude of oscillation, and this observation explains why a spring-mounted cylinder can vibrate at more than one amplitude. An attempt is made to fit the measurements of force and phase angle around vortex resonance with predictions of the nonlinear lift oscillator model proposed by Hartlen and Currie [5]. It is found that the phase angle between the lift and the cylinder displacement is not predicted well by the theory.

Mechanism of collisionless dissociation of polyatomic molecules

A detailed investigation of the effect of collisionless dissociation of polyatomic molecules was carried out in a cycle of experiments [4], in which collisionless dissociation of polyatomic molecules was revealed to have a distinguishing feature of exceeding importance for laser photochemistry and laser isotope separation, viz., isotopic selectivity of the dissociation process. The experimentally observed features of collisionless dissociation were theoretically explained in [5-6]. The explanation was based on the subdivision of the energy spectrum of the vibrational states into two qualitatively different regions. The first part of the spectrum includes the lower vibrational levels down to the energies of 4-5 laser photons. As shown in [5-6], the remaining vibrational levels, located in regions of higher energies and contained in the second part of the spectrum, are so closely spaced that they form a quasicontinuum of vibrational states. This division of the spectrum into two parts is generally speaking arbitrary, since the quasicontinuum does not have an abrupt boundary on the lowenergy side; nevertheless, this division reflects correctly the specifics of the spectrum of polyatomic molecules. As a result, in contrast to the one-dimensional nonlinear oscillator model, satisfactory qualitative agreement was obtained between the theoretical and the experimental results. The process of excitation of polyatomic molecules was correspondingly subdivided in [5-6], in accord with the subdivision of the spectrum, into two stages. The first stage is resonant excitation of several low-lying levels of a mode active in the IR region, up to states lying in the quasicontinuum. Use was made in this case of the system of vibrational levels of a one-dimensional nonlinear oscillator. This made it possible to explain the resonant character of the energy acquisition during the first stage, and consequently the isotopic selectivity of the collisionless dissociation of polyatomic molecules~ During the second stage, the molecules landing in the quasicontinuum region were resonantly excited, in view of the high level density, in the region of the quasicontinuum, up to dissociation states or even ionization states [7]. It is precisely the second stage which strongly distinguishes the process of excitation of polyatomic molecules from the excitation of diatomic molecules, and permits a qualitatively correct identification of the distinguishing features of collisionless dissociation of polyatomic molecules.

Non-linear oscillator models in bluff body aero-elasticity

A critique of non-linear oscillator models is offered and a systematic and self-consistent procedure for constructing such a model is proposed based upon theoretical and experimental fluid mechanical information. Numerical studies are conducted for the proposed model as well as for the earlier one of Skop and Griffin. A comparison with Jones' experimental results is also made.

A nonlinear oscillator model for an acoustic resonance chamber

The exact nonlinear acoustic wave equation, derived from conservation of momentum, mass, and the equation of state, was used to average out the spatial dependence and obtain a forced anharmonic oscillator equation for the time-dependent part. The non-linearity is expressed by a resistive-restoring coupling term ɑxẋ, rather than the restoring term αx2 + βx3 in the modified Duffing equation. A solution is sought in the form of a Fourier series, including all harmonics, and the equation is solved by a perturbation technique, retaining the dominant term in each harmonic. This results in an infinite system of coupled, nonlinear algebraic equations and leads to non-Lorentzian response curves for a sinusoidally driven resonator. These response curves are evaluated numerically and compared with predictions of the Duffing equation and with recent measurements of response curves and harmonic generation in an acoustic resonance chamber (M. Barmatz, N. Jacobi, and J. Stoneburner, 10th International Congress on Acoustics, Sydney, 1980). [Work supported by NASA.]

The solar-cycle period-amplitude relation as evidence of hysteresis of the solar-cycle nonlinear magnetic oscillation and the long-term /55 year/ cyclic modulation

A new dynamical model of the solar cycle has predicted that the cycle should have a hysteretic nature: the behavior of each 11 year cycle should depend on previous cycles. In the light of this new understanding of the dynamical mechanism of the solar cycle, Waldmeier's (hypothetical) law was examined as a yet unexplained characteristic of the cycle by studying the observed sunspot frequency curve. Contrary to this hypothetical law, however, it was found that sunspot cycle curves did not form a single-parameter family characterized by the maximum amplitude of the cycle. The evolutionary trajectories in period-amplitude phase space verified the hysteretic nature of the observed cycle and revealed long-term (55 year instead of the previously claimed 80 year) periodic modulations, called here 55 year grand cycles. Each 55 year grand cycle forms a loop in the phase space, and the characteristics of each 11 year cycle depend on its position in the ascending or descending phase of the grand cycle. This new law was analyzed by the nonlinear multiple-period dynamo oscillation model which has predicted the hysteretic nature. The era from cycle 11 to cycle 15 turned out to be an anomalous one characterized by alternating amplitudes for oddmore » and even cycles. Cycles 16--20 seem to constitute one grand cycle. If this is true, cycle 21 would be the beginning of another grand maximum and the model predicts that its duration would be short.« less

Dynamic polarisation echoes in KDP powder

A brief outline of the theory of dynamic powder echoes (polarisation echoes) is given on the basis of the nonlinear oscillator model proposed and discussed by Kajimura et al. (1976) and by Fossheim et al. (1978). The theoretical description is mainly confined to the low signal limit. Some properties of the high signal case are also pointed out, although a complete analytical solution cannot be given here. Experimental results on KDP show a build-up and decay of the type discussed in detail for several substances by Fossheim et al. It is observed that although the decay time T2 is field dependent the temperature dependence of the decay constant is relatively unaffected by field strength. Thus the critical damping near ferroelectric and other phase transitions may be investigated even when the low signal limit cannot be reached experimentally. Such measurements were performed on KDP.