Phase space diagram

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.

X m, m = 0,..., 32, f = 1,2 with the help of a fourth order Runge-Kutta scheme (see, e.g., Milne (1970)). Every single one of the 62 resulting trajectories was followed over 200 cycles of the microwave field, and the values of I and 6 after every completion of a full cycle of the microwave field were plotted as dots in a 6,1) phase-space diagram. The result is shown in Fig. 6.5. Regular and chaotic regions are clearly visible. 

Although the phase space trajectories appear as simple curves on the two-dimensional Iz,ip phase space diagram (the 0 coordinate is suppressed) most trajectories are actually quasiperiodic. The actual trajectories he on the 2-dimensional surface of a 3-dimensional invariant torus in 4-dimensional phase space. Fig. 9.14 shows such a torus. Any point on the surface of the torus is specified by two angles, 0 and. The 0 and circuits about the torus are shown, respectively, as large and small diameter circles. The diameter of the 0... 

The fixed points on the phase space diagrams or phase spheres in Fig. 9.13 are labeled A, B, Ca, and C. Each corresponds to a periodic orbit that is said to organize the surrounding region of phase space that is filled with topologically similar quasiperiodic trajectories. 

Figure 8.11 Correspondence between (a) an energy diagram and (b) a phase space diagram for the pendulum Hamiltonian (Lichtenberg and Lieberman, 1992).

Figure 1 Phase space diagram indicating the region of accessibility for each of the three new inelastic instruments at the NIST Center for Neutron Research, The unshaded region around the NSE region indicates the effect of several planned improvements to the NSE instrument.

The sensitivity of the mass spectrometer is best expressed in terms of a phase space diagram as shown in Figure 3. This shows the initial combinations of... 

As with the pnictates, a quasi-ternary phase diagram can be developed to map out possible compounds in this composition phase space using key chalcotetre-late building blocks. We have begun to make use of the peritectic nature of the starting materials, as this has facilitated reactions between phases. 

Leaving the details, the equation describing the motion of one particle in two electrostatic waves allows perturbation methods to be applied in its study. There are three main types of behavior in the phase space - a limit cycle, formation of a non-trivial bounded attracting set and escape to infinity of the solutions. One of the goals is to determine the basins of attraction and to present a relevant bifurcation diagram for the transitions between different types of motion. 

Figure 4.23 Synthesis space diagram for a ternary system composed of tetraethylorthosilicate (TEOS), cetyltrimethylammonium bromide (CTAB), and sodium hydroxide (H, hexagonal phase [MCM-41] C, cubic phase [MCM-48] L, lamellar phase [MCM-50] H20/Si02 = 100, reaction temperature 100°C, reaction time 10 days). (Reprinted from Science, Vol. 267, A. Firouzi, D. Kumar, L.M. Bull, T. Besier, R Sieger, Q. Huo, S.A. Walker, J.A. Zasadzinski, C. Glinka, J. Nicol, D.l. Margolese, G.D. Stucky, B.F. Chmelka, Cooperative Organization of Inorganic-Surfactant and Biomimetic Assemblies, pp. 1138-1143. Copyright 1995. With permission of AAAS.)...

Fig. 4. EDS spectroscopy results of E-particles from a high bum-up LWR fuel superimposed on the isothermal section of ternary phase diagram from Kleykamp (1985) at 1700 CC. These analyses show that there is distinct heterogeneity in the composition of metallic particles in the fuels. Hence, spot analysis of an individual e-parlicle may not provide direct evidence of corrosion. The metallic system is dominated by the hexagonal close packing (e) that occupies most of the phase space. The tr-space and the body centered cubic

Figure 2.16 Temperature-composition space diagram of a ternary isomorphous system. Reprinted, by permission, from F. N. Rhines, Phase Diagrams in Metallurgy. Copyright 1956 by McGraw-Hill Book Co.

Figure 3.1 Phase-space trajectory (center) for a one-dimensional harmonic oscillator. As described in the text, at time zero the system is represented by the rightmost diagram (q = b, p = 0). The system evolves clockwise until it returns to the original point, with the period depending on the mass of the ball and the force constant of the spring...

Abstract Theoretical models and rate equations relevant to the Soai reaction are reviewed. It is found that in production of chiral molecules from an achiral substrate autocatalytic processes can induce either enantiomeric excess (ee) amplification or chiral symmetry breaking. The former means that the final ee value is larger than the initial value but is dependent upon it, whereas the latter means the selection of a unique value of the final ee, independent of the initial value. The ee amplification takes place in an irreversible reaction such that all the substrate molecules are converted to chiral products and the reaction comes to a halt. Chiral symmetry breaking is possible when recycling processes are incorporated. Reactions become reversible and the system relaxes slowly to a unique final state. The difference between the two behaviors is apparent in the flow diagram in the phase space of chiral molecule concentrations. The ee amplification takes place when the flow terminates on a line of fixed points (or a fixed line), whereas symmetry breaking corresponds to the dissolution of the fixed line accompanied by the appearance of fixed points. The relevance of the Soai reaction to the homochirality in life is also discussed. 

For a given value of a, the brute force bifurcation diagram displays all the values of the relative arteriolar radius r that the model displays when the steady state trajectory intersects a specified hyperplane (the Poincare section) in phase space. Due to the coexistence of several stable solutions, the brute force diagram must be obtained by scanning a in both directions. 

The procedure is demonstrated by a system with one degree of freedom, assumed to be conservative, with Hamiltonian H(q,p) = a. Solving for the momentum, the relation p = p(q, a.i) defines the equation of the orbit traced out by the system point in two-dimensional phase space for constant H = a.. The graphical details for two types of periodic motion are shown in the diagram. 

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