Regular period motion

A regularly periodic motion characteristic of a pendulum, spring, or many chemical bonds. See Hooke s Law... 

In tetra-atomic molecules that are linear, the number of degrees of freedom is F =7, which again considerably complicates the analysis. Nevertheless, the dynamics of such systems can turn out to be tractable if the anharmonicities may be considered as perturbations with respect to the harmonic zero-order Hamiltonian. In such cases, the regular classical motions remain dominant in phase space as compared with the chaotic zones, and the edge periodic orbits of subsystems again form a skeleton for the bulk periodic orbits. 

The spectrum of Lyapunov exponents provides fundamental and quantitative characterization of a dynamical system. Lyapunov exponents of a reference trajectory measure the exponential rates of principal divergences of the initially neighboring trajectories [1], Motion with at least one positive Lyapunov exponent has strong sensitivity to small perturbations of the initial conditions, and is said to be chaotic. In contrast, the principal divergences in regular motion, such as quasi-periodic motion, are at most linear in time, and then all the Lyapunov exponents are vanishing. The Lyapunov exponents have been studied both theoretically and experimentally in a wide range of systems [2-5], to elucidate the connections to the physical phenomena of importance, such as transports in phase spaces and nonequilibrium relaxation [6,7]. 

This regularity refers to the length interval of the parameter values at which periodic motion with some definite period is stable. These intervals are reduced at each reduplication of the period, the multiplier characterizing the reduction approaching the universal value ... 

Figure 2a depicts the Poincare section of the continuous flow stirrer when St = 1/4ti, and Re = 0.1. The Poincare sections are obtained by numerically tracking four passive tracer particles initially located at (0.005, -0.5), (0.005, 0.0), (0.005, 0.5) and (0.005, 1.0) during 10 convective time-scales PI/Uhs)- a quasi-periodic motion of the passive tracer particle that is initially located at (0.005, 0.0) results in a regular formation separating the upper and lower halves of the Poincare section. A zoomed image showing this KAM boundary is presented in Fig. 2c. The passive tracer particles initially located at the upper and lower halves of the channel entry cannot pass this global barrier. In addition to this, there are two unstirred zones called void zones surrounded by well stirred zone (chaotic sea) at the bottom half of the Poincare section. A zoomed image of these two void zones can be seen in Fig. 2b.

In this paper we consider the QCD counterpart of this problem. Namely, we address the problem of regular and chaotic motion in periodically driven quarkonium. Using resonance analysis based on the Chirikov criterion of stochasticity we estimate critical values of the external field strength at which quarkonium motion enters into chaotic regime. 

In practice, this model is oversimplified since the exciting wake shedding is by no means harmonic and is itself coupled with the shape oscillations and since Eq. (7-30) is strictly valid only for small oscillations and stationary fluid particles. However, this simple model provides a conceptual basis to explain certain features of the oscillatory motion. For example, the period of oscillation, after an initial transient (El), becomes quite regular while the amplitude is highly irregular (E3, S4, S5). Beats have also been observed in drop oscillations (D4). If /w and are of equal magnitude, one would expect resonance to occur, and this is one proposed mechanism for breakage of drops and bubbles (Chapter 12). 

In 1903 Dixon noted a periodic distribution of luminescence, too regular to be accidental. In a series of elegant experiments, for example, by synchronous photography in two perpendicular planes and photography from the end of the tube, Campbell showed that, in fact, it is a helical, spiral motion that occurs which only appears periodic in one projection. 

The liquid state of a material has a definite volume, but it does not have a definite shape and takes the shape of a container, unlike that of the solid state. Unlike the gas state, a liquid does not occupy the entire volume of the container if its volume is larger than the volume of the liquid. At the molecular level, the arrangement of the particles is random, unlike that of the solid state in which the molecules are regular and periodic. The molecules in the liquid state have translational motions like those in a gas state. There is short-range interparticular ordering or structure, however. 

The periodic asymmetry of the signal, (a small and a large peak which regularly alternate with the period of the motion—Fig. 10.8(a) is due to the geometry of the fabricated microband electrodes, as shown in Fig. 10.6. On one side of the electrode there is only a 20-40 pm thick layer of insulator while on the other this layer is 500 pm thick (the ceramic substrate). It has already been stated in Section 10.3.2 and shown in Fig. 10.1 that for flow impinging parallel to a wall in which an electrode is embedded, SH over the electrode will differ depending on the distance of the electrode... 

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