Phase-space plot

Because the degrees of freedom decouple in the linear approximation, it is easy to describe the dynamics in detail. There is the motion across a harmonic barrier in one degree of freedom and N — 1 harmonic oscillators. Phase-space plots of the dynamics are shown in Fig. 1. The transition from the reactant region at q <0 to the product region at q >0 is determined solely by the dynamics in (pi,qi), which in the traditional language of reaction dynamics is called the reactive mode. 

Fig. 10.6. 2D CALDER simulations for UHC experimental parameters FWD and BWD proton energy distributions (a) and related electron phase space plot (b) at the laser peak...

Thus, the resonance angular momenta are given by = 2mTV. The resonance angular momenta correspond exactly to the large elliptical structures seen in the phase-space plots shown in Fig. 5.4. These phase-space structures, by inference called resonances too, are of prime importance for Chirikov s criterion. 

Can chaotic systems be differentiated from random fluctuations Yes, even though the dynamics are complex and resemble a stochastic system, they can be differentiated from a truly stochastic system. Figure 11.15 compares the plots of N = 10,001 and a selection of points chosen randomly from 13,000 to 0. Note that after approximately 10 time intervals, the dynamics of both are quite wild and it would be difficult to distinguish one from another as far as one is deterministic and the other chaotic. However, there is a simple way to differentiate these two alternatives the phase-space plot. 

Experimenting with the B-Z reaction in a CSTR at various residence times for the same inlet concentrations and temperature, Hudson and Mankin (1981) obtained and reported horatian oscillations. The measurements of the bromide ion electrode potential and the platinum electrode potential were recorded. Calculating the time derivative of Pt, or the time delay, Pt(t-lp), three dimensional phase space plots were presented. With the time delay as the third variable a horatian solution was obtained, Fig. III.C.2. 

The position and angular trajectory of an electron, and hence of the emitted photons, are correlated parameters. The phase space plot relates the position (x or y) of an electron and its angular trajectory (x or y )... 

Figure 4.11 Idealised electron beam emittance phase space plots (a) vertical (b) horizontal (radial direction). The dotted line is a lower emittance mode than the full line. From Winick (1980) with permission.

Numerical studies (Erdi Barna, 1986a, b) gave the evidence of occurrence of abnormal dynamic behaviour. Phase space plots and stroboscopic plots are obtained. The former are identified with two-dimensional projection of trajectories, the latter can be constructed by points taken at regular intervals of the period (figs 7.7 and 7.8). 

It is very convenient to utilize a phase space consisting of the ray position and ray slope to express a mode spectrum of a multimode waveguide. Figure 27 shows a phase space plot of the output end of a branching waveguide. 

FIGURE 27 Phase space plot of a waveguide branch. [From Kokubun, Y, Suzuki, S., Fuse, T. and Iga, K. (1986). Appl. Opt]... 

Figure 7.4 Three-dimensional phase space plot of the dividing surfaces and the example trajectories. The two non-reacting trajectories shown in Figure 7.3 are now plotted in the phase space with one axis representing the momenta. While they cross the naively taken dividing surface (qi = 0) more than once, they do not cross the true transition state taken as a surface in the phase space. Note the momentum dependence of the true TS.

Fig. 12. Three periodic orbits (one unstable, two stable) shown for the above-resonance case in a 1 1 resonance. The (u,p ) surface of section shows the characteristic figure eight of a double well phase space plot, and three fixed points, one for each periodic orbit. If we push the double well analogy, we should see (i) decay rate F at top of barrier (this is the width of the clusters of lines seen near E = 8,9,... in Fig. 11). (ii) Splitting AE due to motion at the bottom of each well. This is confirmed in the full set of eigenvalues (not shown). (iii) Symmetric and antisymmetric pairs of states very closely spaced in energy. These are seen in Fig. 13.

Fig. 5. Langevin trajectories for a harmonic oscillator of angular frequency u = 1 and unit mass simulated by a Verlet-like method (extended to Langevin dynamics) at a timestep of 0.1 (about 1/60 the period) for various 7. Shown for each 7 are plots for position versus time and phase-space diagrams.

Fig. 6.3. To ensure the accuracy of a nonequilibrium work free energy calculation, the switching paths should go down the funnel. The important phase space regions for the intermediate states along the ideal funnel paths are illustrated in this plot, for the case where r0 and / are partially overlapped. Two funnel paths need to be constructed to transfer the systems from both 0 and 1 to a common intermediate M where rm is inside the r0 and J overlap region. The construction of such paths is discussed in Sect. 6.6...

Figure 4.13 shows R-space plots (not corrected for phase shift) for the 1-con-nected and 3-connected catalysts just described. We are fortunate that the V=0,... 

Fig. 5.2. Contour plots of two representative Wigner distribution functions PW(R,P) for two harmonic oscillators in their ground vibrational states, Equation (5.15), in the two-dimensional phase-space (R,P). The widths in the R-and in the P-directions are inversely related.

To compare reactions with different time constants it is useful to plot them as trajectories in a multi-dimensional phase space whose coordinates are the species concentrations and the temperature. Fig. 2 shows trajectories projected onto the temperature vs. [r] plane for reactions with identical initial fuel and air concentrations but different initial radical concentrations and temperature. Trajectories beginning at the left had no initial radicals, and the trajectory starting at 1200 K is represented in Fig. 1. The exponential increase of [r] to [r]e is isothermal so it appears horizontal in Fig. 2. The knee of the curve represents the relatively flat portion of Fig. 1 where [r] is approximately [R]e. As the temperature increases [r] remains approximately equal to [R]e, which lies to the left of the dashed line due to consumption of fuel and oxygen. 

Pharmacokinetic studies are in general less variable than pharmacodynamic studies. This is so since simpler dynamics are associated with pharmacokinetic processes. According to van Rossum and de Bie [234], the phase space of a pharmacokinetic system is dominated by a point attractor since the drug leaves the body, i.e., the plasma drug concentration tends to zero. Even when the system is as simple as that, tools from dynamic systems theory are still useful. When a system has only one variable a plot referred to as a phase plane can be used to study its behavior. The phase plane is constructed by plotting the variable against its derivative. The most classical, quoted even in textbooks, phase plane is the c (f) vs. c (t) plot of the ubiquitous Michaelis-Menten kinetics. In the pharmaceutical literature the phase plane plot has been used by Dokoumetzidis and Macheras [235] for the discernment of absorption kinetics, Figure 6.21. The same type of plot has been used for the estimation of the elimination rate constant [236]. 

Figure 10.3 Indirect link model with bolus intravenous injection. (A) The classical time profiles of the two variables c(t) (solid line) and E (t) (dashed line) for dose qo = 0.5. (B) A two-dimensional phase space for the concentration c(t) vs. effect E (t) plot using three doses 0.5, 0.75, and 1 (solid, dashed, and dotted lines, respectively).

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