Oscillator nonuniform

Biology (oscillating neurons, firefly flashing rhythm, human sleep-wake cycle) 

Condensed-matter physics (Josephson junction, charge-density waves) Mechanics (Overdamped pendulum driven by a constant torque) 

Some of these applications will be discussed later in this chapter and in the exercises. 

Use linear stability analysis to classify the fixed points of (1) for a ft). Solution The fixed points 6 satisfy 

Thus the fixed point with cos 0 0 is the stable one, since 0. This 

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We now consider a simple mechanical example of a nonuniform oscillator an overdamped pendulum driven by a constant torque. Let 0 denote the angle between the pendulum and the downward vertical, and suppose that d increases counterclockwise (Figure 4.4.1). 

We are not yet prepared to analyze (6) in general. For now, let s restrict ourselves to the overdamped limit fi 1. Then the term Ptfi" may be neglected after a rapid initial transient, as discussed in Section 3.5, and so (6) reduces to a nonuniform oscillator ... 

The oscillation period for the nonuniform oscillator is given by the inte-f dd... 

The electroconformational coupling method measures the force acting on a cell by a nonuniform oscillating electric field. Depending on the frequency... 

In order to accurately describe such oscillations, which have been the center of attention of modern liquid state theory, two major requirements need be fulfilled. The first has already been discussed above, i.e., the need to accurately resolve the nonlocal interactions, in particular the repulsive interactions. The second is the need to accurately resolve the mechanisms of the equation of state of the bulk fluid. Thus we need a mechanistically accurate bulk equation of state in order to create a free energy functional which can accurately resolve nonuniform fluid phenomena related to the nonlocality of interactions. So far we have only discussed the original van der Waals form of equation of state and its slight modification by choosing a high-density estimate for the excluded volume, vq = for a fluid with effective hard sphere diameter a, instead of the low-density estimate vq = suggested by van der Waals. These two estimates really suggest... 

Organic semiconductors such as K-TCNQ and TEA(TCNQ)2 have anomalous infrared (IR) absorption lines at frequencies corresponding the totally symmetric (ag) vibration modes of the TCNQ molecule. Normally, these modes would not be IR active, but because they modulate the electron overlap integrals, they induce charge oscillations along the (nonuniform) TCNQ stacks [108]. This is perhaps the clearest experimental evidence for... 

In real systems, especially in heterogeneous catalytic and biological sys terns, the reactants are often arranged irregularly in space. Therefore, an arising instability may cause simultaneous diffusion of substances from one point to another inside the system to make the reactant concentration oscillations arranged in a certain manner in space during the occurrence of nonlinear chemical transformations. As a result, a new dissipative structure arises with a spatially nonuniform distribution of certain reac tants. This is a consequence of the interaction between the process of diffusion, which tends to create uniformity of the system composition, and local processes of the concentration variations in the course of nonlinear... 

Next, we focus on a portion of the film that is small in lateral extent compared with the SAW wavelength. As the wave passes a fixed point, the lower surface of the film oscillates in response to the sinusoidal SAW surface displacement. If the film is acoustically thick (/7 1), the upper portions of the film tend to lag behind the driven substrate/film interface, inducing strains across the thickness of the film. This inertial deformation of the film results in nonuniform displacement across the film. 

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