Phase-space analysis

VII. Phase-Space Analysis and Vibronic Periodic Orbits... 

VII. PHASE-SPACE ANALYSIS AND VIBRONIC PERIODIC ORBITS... 

Because the mapping approach treats electronic and nuclear dynamics on the same dynamical footing, its classical limit can be employed to study the phase-space properties of a nonadiabatic system. With this end in mind, we adopt a onemode two-state spin-boson system (Model IVa), which is mapped on a classical system with two degrees of freedom (DoF). Studying various Poincare surfaces of section, a detailed phase-space analysis of the problem is given, showing that the model exhibits mixed classical dynamics [123]. Furthermore, a number of periodic orbits (i.e., solutions of the classical equation of motion that return to their initial conditions) of the nonadiabatic system are identified and discussed [125]. It is shown that these vibronic periodic orbits can be used to analyze the nonadiabatic quantum dynamics [126]. Finally, a three-mode model of nonadiabatic photoisomerization (Model III) is employed to demonstrate the applicability of the concept of vibronic periodic orbits to multidimensional dynamics [127]. 

The control of a stirred tank reactor with hysteresis in the control element-I Phase space analysis (with J.C. Hyun). Chem. Eng. Sci 27, 1341-1359 (1972). 

From the earlier section on Molecular Dynamics and Equilibrium Statistical Mechanics, we hope that we have made clear that a conserved quantity is the starting point for phase space analysis and the derivation of a probability distribution function. Following the same analysis that led to the distribution functions for NVE, NVT, and NPT dynamics, the new distribution /(q, p,, I) for GSLLOD coupled to a Nose-Hoover thermostat is given by... 

Generally speaking, the degrees of freedom in many-body systems, such as Ar7, are too many to analyze the phase-space dynamics, and only limited methods originally developed to investigate chaotic systems with a few degrees of freedom can be applicable for the analysis. Seko et al. calculated the phase volume—that is, the configuration entropy—of Ar7 and proposed a new concept of the temperature in micro-clusters based on this phase volume [17], A phase-space analysis seems to be prospective even for many-body systems, such as Ar7. However, most of the currently available methods concern statistical properties. The methods and quantities that are directly related to the dynamics are expected for a detailed analysis. 

The phase space analysis visualizes the behavior of the radical concentrations and leads rather directly to the conditions for equilibria, but these can also be derived by other means.1517 It has also been applied to explore the effects of an additional radical generation50 and of a direct or indirect decay of the dormant chains.57 Examples are discussed below.50... 

IV. PHASE-SPACE ANALYSIS OF UNIMOLECULAR REACTION DYNAMICS... 

By reason of their stability or regularity, most biological rhythms correspond to oscillations of the limit cycle type rather than to Lotka-Volterra oscillations. Such is the case for the periodic phenomena in biochemical and cellular systems discussed in the following chapters. Similarly, the phase space analysis of two-variable models indicates that the oscillatory dynamics of neurons corresponds to the evolution towards a limit cycle (Fitzhugh, 1961 Kokoz Krinskii, 1973 Krinskii Kokoz, 1973 Hassard, 1978 Rinzel, 1985 Av-Ron, Parnas Segel, 1991). A similar evolution is predicted (May, 1972) by models for predator-prey interactions in ecology. 

Phase space analysis provides an explanation for the existence of a domain of sustained oscillations bounded by two critical values of the substrate injection rate. As indicated by relation (2.12), the steady state, located at the intersection of the two nullclines, moves towards the right when the substrate injection rate increases, since yo is proportional to v. Since parameter v enters only into eqn (2.21a) of the substrate nullcline and not into eqn (2.21b), the product nullcline remains unchanged... 

R. T. Skodje and M. J. Davis,/. Chem. Phys., 88,2429 (1988). A Phase-Space Analysis of the Collinear I -I- HI Reaction. R. T. Skodje and M. J. Davis, Chem. Phys. Lett., 175,92 (1990). Statistical Rate Theory for Transient Chemical Species Classical Lifetimes from Periodic Orbits. 

Phase-Space Analysis of Water Dripping Intervals... 

Figure 8, Example of the phase-space analysis of water dripping intervals from the HeWs Half Acre infiltration test (Test 8, Dripping Point 6) (a) time-series of dripping water intervals, (b) autocorrelation function, (c) mutual information junction used to determine the time delay, (d) false nearest neighbors used to determine the global embedding dimension, (e) local embedding dimension, and (f) Lyapunov exponents and Lyapunov dimension.

The residts of the phase-space analysis of time-series data for capillary pressure for one of the tests show that the time intervals (x) between pressure spikes (pulses) can be described using a sinq>le exponential equation, given as a difference equation by ... 

The phase space analysis of both inlet and outlet capillary pressures produce a zero Lyapunov exponent (I), itr lying that this dynamic system can be described by differential equations (50). A conq>arison of the pseudo-phase-space three-dunensional attractors for the inlet and outlet capillary pressures shows (Figures 14b and 14c) that these attractors are analogous to those described using the solution of the Kuramoto-Sivashinsky equation (Figures 14d and 14e) discussed below. 

Aldiough direct measurements of variables characterizing (he individual flow and chemical transport processes under field condidons are not technically feasible, their cumulative effect can be characterized by the phase-space analysis of time-series data for the infiltration and outflow rates, capillary pressure, and dripping-water frequency. The tune-series of low-frequency fluctuadons (assumed to represent intrafracture flow) are described by three-dimensional attractors similar to those fi m die sohidon of the Kuramoto-Sivashinsky equadon. These attractoia demonstrate die stretching and folding of fluid elements, followed by diffusion. 

Phase-Space Analysis. Having determined the time lag r for decorrelating the components of the phase space and the embedding dimension, a line joining the dimensional points given by... 

Periodicity and aperiodicity in neurochemical systems CCD Numerical integration, phase space analysis... 

The first one of these techniques may be called a geometrical phase space analysis and has been utilized already in section 6.3, Fig. 12. Let us consider the case of a network with two capacitive variables X and X2 and let the equations of motion be... 

A second weak point of the geometrical phase space analysis is the fact that it will be very difficult if not impossible to detect the onset of a limit cycle at a critical or bifurcation point as defined in Section 6.4 for two variables by... 

Perform the phase space analysis as described in Section 7.7 for the network (6.27). Make plausible that the directional field in this case may possibly give rise to a limit cycle oscillation. Is it possible to determine the bifurcation point form the phase space analysis ... 

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