Oscillation non-linear

The bubble formed in stable cavitation contains gas (and very small amount of vapor) at ultrasonic intensity in the range of 1-3 W/cm2. Stable cavitation involves formation of smaller bubbles with non linear oscillations over many acoustic cycles. The typical bubble dynamics profile for the case of stable cavitation has been shown in Fig. 2.3. The phenomenon of growth of bubbles in stable cavitation is due to rectified diffusion [4] where, influx of gas during the rarefaction is higher than the flux of gas going out during compression. The temperature and pressure generated in this type of cavitation is lower as compared to transient cavitation and can be estimated as ... 

The excitation of oscillations with a quasi-natural system frequency and numerous discrete stationary amplitudes, depending only on the initial conditions (i.e. discretization of the processes of absorption by the system of energy, coming from the high-frequency source). A new in principle property is the possibility for excitation of oscillations with the system s natural frequency under the influence of an external high-frequency force on unperturbed linear and conservative linear and non-linear oscillating systems. 

It is well known that self-oscillation theory concerns the branching of periodic solutions of a system of differential equations at an equilibrium point. From Poincare, Andronov [4] up to the classical paper by Hopf [12], [18], non-linear oscillators have been considered in many contexts. An example of the classical electrical non-oscillator of van der Pol can be found in the paper of Cartwright [7]. Poore and later Uppal [32] were the first researchers who applied the theory of nonlinear oscillators to an irreversible exothermic reaction A B in a CSTR. Afterwards, several examples of self-oscillation (Andronov-PoincarA Hopf bifurcation) have been studied in CSTR and tubular reactors. Another... 

Tyson, J. J. (1973). Some further studies of non-linear oscillations in chemical systems. J. Chem. Phys., 58, 3919-30. 

Kai, T. and Tomita, K. (1979). Stroboscopic phase portrait of a forced non-linear oscillator. Prog. Theor. Phys., 61, 54-73. 

The principal role of diffusion in these processes could be established considering rather simple examples [2]. If the kinetic equations for a well-stirred system are able to reproduce self-oscillations (the limit cycle), the extended system could be presented as a set of non-linear oscillators continuously distributed in space. Diffusion acts to conjunct these local oscillations and under certain conditions it can result in the synchronisation of oscillations. Thus, autowave solutions could be interpreted as a result of a weak coupling (conjunction) of local oscillators when they are not synchronised completely. The stationary spatial distributions in an initially homogeneous systems can also arise due to diffusion, which makes homogeneous solutions unstable. 

N.N. Bogoljubov and U.A. Mitropolskii, Asymptotic Methods in the Theory of Non-linear Oscillations, Nauka, Moscow, 1974 (in Russian). 

H.F. De Baggis, Dynamic Systems with Stable Structures. Contributions to the Theory of Non-Linear Oscillations, Vol. 2, Princeton University Press, Princeton, 1952, p. 306. Usp. Mat. Nauk, 10 (1955) 101. 

Actively working groups are sure to include physical chemists (experimental and theoretical) and mathematicians (pure and applied). "Graphs theory , "dynamics , "non-linear oscillations , "chaos , "attractor , "synergetics , "catastrophes and finally "fractals these are the key words of modern kinetics. 

We consider the same reaction model used in previous studies as a simple model for a proton transfer reaction. [31,57,79] The subsystem consists of a two-level quantum system bilinearly coupled to a quartic oscillator and the bath consists of v — 1 = 300 harmonic oscillators bilinearly coupled to the non-linear oscillator but not directly to the two-level quantum system. In the subsystem representation, the partially Wigner transformed Hamiltonian for this system is,... 

Montgomery E. B. Dynamically Coupled, High-Frequency Reentrant, Non-linear Oscillators Embedded in Scale-Free Basal Ganglia-Thalamic-Cortical Networks Mediating Function and Deep Brain Stimulation Effects. Nonlinear Stud, 2004, 22, 385-421. 

An example simulation of this model is illustrated in Figure 3.6, using the parameter values and initial conditions indicated in the legend. Note that the model predicts sustained non-linear oscillations, which have been observed in yeast cells and in extracts from yeast and also mammalian cells [82],... 

N. Minorsky, Non-linear Oscillations, van Nostrand, Princeton, New Jersey, 1962. 

When the incoming field E consists of two frequencies, O) and dipoles non-linearly oscillating at frequencies (o) + (o ) and (o) - (o ) are Induced as well. In the sum-frequency generation (SFG) technique one focusses on the output at frequency + co and in the difference-frequency generation (DFG) technique on - a) ). As is the case with SHG, SFG and DFG are electric dipole-forbidden in centrosymmetric media. 

Figure 7. Left column. The potential energy functions V — /c, (solid lines) and their curvatures (dotted lines) for different values of c c = 2 (linear oscillator), and c — 4,6,8 (strongly non-linear oscillators). Middle column. Typical sample paths of Brownian oscillators, a = 2, with the potential energy functions shown on the left. Right column Typical sample paths of Levy oscillators, a = 1. On increasing m the potential walls become steeper, and the flights become shorter in this sense, they are confined.

N. N. Bogoliubov and Y. A. Mitropolsky, Asymptotic Methods in the Theory of Non-Linear Oscillations, Gordon Breach, New York, 1985. 

N. V. Butenin, Elements of the Theory of Non-linear Oscillations , Blaisdell Publishing Co., London, 196S. 

Asymptotic properties of the periodic solutions of systems of equations of neutral type with small lag. - In Proceedings of V Intemat. Conf. on Non-Linear Oscillations, vol. 1, 475-482, Inst. Matematiki Akad. Nauk Ukrain. SSR, Kiev, 1970. 

Figure 7.2.3. Heat transfer and rheodynamics at non-linear oscillations of a bubble in polymeric liquid. [By permission of Nauka i Tekhnica Press, from the reference 25]...

Bogoliubov NN, MitropoTskiy YA (1974) Asymptotic methods in theory of non-linear oscillations. Nauka, Moscow (in Russian)... 

In [210] the authors obtained a new analytical approximate technique for conservative non-linear oscillators. This new proposed method is a modification of the generalized harmonic balance method. With this new technique one can have not only a truly periodic solution but also the frequency of motion as a function of the amplitude of oscillation. Several examples show the efficiency of the new approach. 

If jS is taken as zero and K(r) is assumed to be positive and slowly varying over a range of r much longer than any natural period of variation of Q, then (22) reduces to the equation of motion for a forced non-linear oscillator ... 

The cases mentioned above are extremely complex because sonochemical effects require the presence of non-linearly oscillating bubbles in the propagation liquid. Before considering this situation, let us return to the case of a flat liquid and the existence of another complex process, the radiation pressure. 

J. Guckenheimer and P. Holmes, Non-linear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. 

Fosdick RL, Yu JH (1998) Thermodynamics, stability and non-linear oscillations of viscoelastic solids. 2. History type solids. Int J Non-Linear Mech 33 165-188... 

KAPRAL - There are a number of examples of such phenomena in models for chemical and biological systems, and one can consider external noise processes that induce transitions between a chaotic state and another attractor. One example that we (R. Kapral, M. Schell and S. Frazer, 3. Phys. Chem. 86, 2203, 1982) have considered is a forced non linear oscillator in the presence of external noise. The deterministic dynamics exhibits bistability between a chaotic state and a periodic state. The rates of transition between the chaotic state and the periodic state were studied as a function of the amplitude of the external noise and, in some circumstances, interesting "non-Arrhenius" behavior was observed. 

An interesting aspect of these non-linear oscillations is that they can be put to use to estimate the amplitude of the acoustic pressure both in-vivo and in-vitro. In this paper, theoretical aspects of this non-Unear resonance are first discussed. Then, experimental results showing the second harmonic using microbubbles created in the laboratory are presented. Finally, a roadmap is drawn which could eventually lead to in-vivo calibration of ultrasoimd waves intensity. 

Although tissues do emit some degree of non-linear oscillations [12], their amplitude is much less that the resonating bubble as long as the insonating pressure remains low. In order to quantify this effect, Equ. 4 describes the amplitude of the second harmonic component generated in water with traveled distance ... 

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