Nonlinear oscillations

Keywords optical parametric oscillators, nonlinear optics... 

Keywords Bjerknes force Bubble breakup Bubble interaction Bubble oscillation Chaotic oscillation Damping rate Droplet oscillation Nonlinear oscillation Oscillation frequency RPNNP equation Shape modes Spherical harmonics Volume oscillation... 

Similar oscillation using a single capillary in place of membrane for the first time has also been reported recently using solutions of non-electrolytes [15]. Current is the bifurcation parameter as in the previous case. Multi-periodic oscillations are observed in case of solution of urea. In the above case, the electro-osmotic flow in one direction and hydrodynamic flow in the other direction are responsible for oscillations. Nonlinearity is imposed by concentration difference across the membrane. This point will be discussed in greater depth in the subsequent section. 

Epstein I R and Pojnian J A 1998 An Introduction to Nonlinear Chemical Dynamics Oscillations, Waves, Patterns and Chaos (Oxford Oxford University Press)... 

Field R J and Burger M (eds) 1984 Oscillations and Travelling Waves in Chemical Systems (New York Wiley) Multi-author survey of nonlinear kinetics field to 1984, still a valuable introduction to researchers in this area. 

In order to illustrate some of the basic aspects of the nonlinear optical response of materials, we first discuss the anliannonic oscillator model. This treatment may be viewed as the extension of the classical Lorentz model of the response of an atom or molecule to include nonlinear effects. In such models, the medium is treated as a collection of electrons bound about ion cores. Under the influence of the electric field associated with an optical wave, the ion cores move in the direction of the applied field, while the electrons are displaced in the opposite direction. These motions induce an oscillating dipole moment, which then couples back to the radiation fields. Since the ions are significantly more massive than the electrons, their motion is of secondary importance for optical frequencies and is neglected. 

While the Lorentz model only allows for a restoring force that is linear in the displacement of an electron from its equilibrium position, the anliannonic oscillator model includes the more general case of a force that varies in a nonlinear fashion with displacement. This is relevant when tire displacement of the electron becomes significant under strong drivmg fields, the regime of nonlinear optics. Treating this problem in one dimension, we may write an appropriate classical equation of motion for the displacement, v, of the electron from equilibrium as... 

Here E(t) denotes the applied optical field, and-e andm represent, respectively, the electronic charge and mass. The (angular) frequency oIq defines the resonance of the hamionic component of the response, and y represents a phenomenological damping rate for the oscillator. The nonlinear restoring force has been written in a Taylor expansion the temis + ) correspond to tlie corrections to the hamionic... 

If we now include the anliannonic temis in equation B 1.5.1. an exact solution is no longer possible. Let us, however, consider a regime in which we do not drive the oscillator too strongly, and the anliannonic temis remain small compared to the hamionic ones. In this case, we may solve die problem perturbatively. For our discussion, let us assume that only the second-order temi in the nonlinearity is significant, i.e. 0 and b = 0 for > 2 in equation B 1.5.1. To develop a perturbational expansion fomially, we replace E(t) by X E t), where X is the expansion parameter characterizing the strength of the field E. Thus, equation B 1.5.1 becomes... 

Figure Bl.5.3 Magnitude of the second-order nonlinear susceptibility x versus frequency co, obtained from the anliannonic oscillator model, in the vicinity of the single- and two-photon resonances at frequencies cOq and coq 2> respectively.

In order to achieve a reasonable signal strength from the nonlinear response of approximately one atomic monolayer at an interface, a laser source with high peak power is generally required. Conuuon sources include Q-switched ( 10 ns pulsewidth) and mode-locked ( 100 ps) Nd YAG lasers, and mode-locked ( 10 fs-1 ps) Ti sapphire lasers. Broadly tunable sources have traditionally been based on dye lasers. More recently, optical parametric oscillator/amplifier (OPO/OPA) systems are coming into widespread use for tunable sources of both visible and infrared radiation. 

Figure B2.1.3 Output of a self-mode-locked titanium-sapphire oscillator (a) non-collinear intensity autocorrelation signal, obtained with a 100 pm p-barium borate nonlinear crystal (b) intensity spectrum.

We have encountered oscillating and random behavior in the convergence of open-shell transition metal compounds, but have never tried to determine if the random values were bounded. A Lorenz attractor behavior has been observed in a hypervalent system. Which type of nonlinear behavior is observed depends on several factors the SCF equations themselves, the constants in those equations, and the initial guess. 

Optical parametric oscillators (OPOs) represent another tunable soHd-state source, based on nonlinear optical effects. These have been under development for many years and as of this writing (ca 1994) are beginning to become commercially available. These lasers may be tuned by temperature or by rotating a crystal. Models available cover a broad wavelength range in the visible and infrared portions of the spectmm. One commercial device may be tuned from 410 to 2000 nm. 

Materials are also classified according to a particular phenomenon being considered. AppHcations exploiting off-resonance optical nonlinearities include electrooptic modulation, frequency generation, optical parametric oscillation, and optical self-focusing. AppHcations exploiting resonant optical nonlinearities include sensor protection and optical limiting, optical memory appHcations, etc. Because different appHcations have different transparency requirements, distinction between resonant and off-resonance phenomena are thus appHcation specific and somewhat arbitrary. 

Introductory Remarks.—As was mentioned in the introduction to this chapter, the quantitative part of the theory of Poincar6 was first applied in celestial mechanics.1 The two approaches the topological,2 and the analytical are unrelated in the original publications of Poincar6, and the connection between the two appeared nearly 50 years later when the theory of nonlinear oscillations was developed. 

The use of this theory in studies of nonlinear oscillations was suggested in 1929 (by Andronov). At a later date (1937) Krylov and Bogoliubov (K.B.) simplified somewhat the method of attack by a device resembling Lagrange s method of the variation of parameters, and in this form the method became useful for solving practical problems. Most of these early applications were to autonomous systems (mainly the self-excited oscillations), but later the method was extended to... 

Autonomous (A) Versus Nonautonomous (NA) Problems. Practically all nonlinear problems of the theory of oscillations reduce to the differential equation of the form... 

In 1958 N. N. Bogoliubov and Y. A. Mitropolsky (B.M.) published a treatise entitled Asymptotic Methods in the Theory of Nonlinear Oscillations,18 which presents a considerable generalization of the early K.B. theory. Since a detailed account of this work is beyond the scope of this book, we give only a few of its salient points. 

This enlarges the scope of problems that can be treated by these asymptotic methods. For example, the important problem of nonlinear resonance could otherwise be solved only in the stationary state. With this extension it is possible to determine what happens when the zone of resonance is passed at a certain rate. Likewise, with the additional extension for the slow time it is possible to attack the problem of modulated oscillations, which has previously remained outside the scope of the general theory. 

It was observed that with a linear circuit and in the absence of any source of energy (except probably the residual charges in condensers) the circuit becomes self-excited and builds up the voltage indefinitely until the insulation is punctured, which is in accordance with (6-138). In the second experiment these physicists inserted a nonlinear resistor in series with the circuit and obtained a stable oscillation with fixed amplitude and phase, as follows from the analysis of the differential equation (6-127). 

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