Mapping function chaotic

Preliminary measurements with space-resolved PMC techniques have shown that PMC images can be obtained from nanostructured dye sensitization cells. They showed a chaotic distribution of PMC intensities that indicate that local inhomogeneities in the preparation of the nanostructured layer affect photoinduced electron injection. A comparison of photocurrent maps taken at different electrode potentials with corresponding PMC maps promises new insight into the function of this unconventional solar cell type. 

Dye structures of passive tracers placed in time-periodic chaotic flows evolve in an iterative fashion an entire structure is mapped into a new structure with persistent large-scale features, but finer and finer scale features are revealed at each period of the flow. After a few periods, strategically placed blobs of passive tracer reveal patterns that serve as templates for subsequent stretching and folding. Repeated action by the flow generates a lamellar structure consisting of stretched and folded striations, with thicknesses s(r), characterized by a probability density function, f(s,t), whose... 

In order to develop the criterion more quantitatively, consider the sequence of phase-space portraits shown in Figs. 5.4(a) - (d). This sequence suggests that, as the control parameter K increases, the diameter of the resonance islands at Z = 0 mod 27t grows in action. In order to predict the touching point of the resonances, we need the widths of the resonances as a function of K. The width of the resonances is derived on the basis of the Hamiltonian (5.2.1). Since the dynamics induced by H is equivalent to the chaotic mapping (5.1.6), the Hamiltonian H itself cannot be treated analytically and has to be simplified. One way is to consider only the average effect of the periodic 6 kicks in (5.2.1). The average perturbation... 

Anomalous diffusion was first investigated in a one-dimensional chaotic map to describe enhanced diffusion in Josephson junctions [21], and it is observed in many systems both numerically [16,18,22-24] and experimentally [25], Anomalous diffusion is also observed in Hamiltonian dynamical systems. It is explained as due to power-type distribution functions [22,26,27] of trapping and untrapping times of the orbit in the self-similar hierarchy of cylindrical cantori [28]. 

Analysis of this 7feff using the techniques of nonlinear classical dynamics reveals the structure of phase space (mapped as a continuous function of the conserved quantities E, Ka, and Kb) and the qualitative nature of the classical trajectory that corresponds to every eigenstate in every polyad. This analysis reveals qualitative changes, or bifurcations, in the dynamics, the onset of classical chaos, and the fraction of phase space associated with each qualitatively distinct class of regular (quasiperiodic) and chaotic trajectories. 

Chaos can further be characterized by resorting to Poincare sections. By determining, for example, the value a of the substrate corresponding to the nth peak, /3 , of product Pi in the course of aperiodic oscillations, we may construct the one-dimensional return map giving a i as a function of a (Decroly, 1987a Decroly Goldbeter, 1987). The continuous character of the curve thus obtained (fig. 4.11) denotes the deterministic nature of the chaotic behaviour. 

Fig. 6.14. Return map for the aperiodic behaviour of fig. 6.11. The maximum of the peak in intracellular cAMP is plotted as a function of the maximum of the preceding peak. The continuous nature of the curve is an indication of the chaotic nature of the system s evolution (Goldbeter Martiel, 1987).

The above mapping of for a particular resonance onto product states requires that the resonance be isolated and assignable. However, in the case of classical chaotic motion, the resonance wave function becomes highly irregular and unassignable, so that the above mapping scheme breaks down. The dissociation of NO2 appears to fall into this latter category (Reisler et al., 1994). 

In Figure 10 the chaotic region is extremely small. However, in Figures 11 and 12 we show a second system s phase space map as a function of en-ergy.35,119 xhjs system exhibits a mode-mode resonance at low energies, with a hyperbolic fixed point located near the center of the Poincare map. Note in Figure 11 that as the energy increases, the measure of quasiperiodic phase space decreases and approaches a limit in which most of the tori are destroyed, with... 

The ultrafast initial decay of the population of the diabatic S2 state is illustrated in Fig. 16 for the first 30 fs. Since the norm of the semiclassical wave function is only approximately conserved, the semiclassical results are displayed as rough data (dashed line) and normalized data (dotted line) [i.e. pnorm P2/ Pi + P2)]. The normalized results for the population are seen to match the quantum reference data quantitatively. It should be emphasized that the deviation of the norm shown in Fig. 16 is not a numerical problem, but rather confirms the common wisdom that a two-level system as well as its bosonic representation is a prime example of a quantum system and therefore difficult to describe within a semiclassical theory. Nevertheless, besides the well-known problem of norm conservation, the semiclassical mapping approach clearly reproduces the nonadiabatic quantum dynamics of the system. It is noted that the semiclassical results displayed in Fig. 16 have been obtained without using filtering techniques. Due to the highly chaotic classical dynamics of the system, therefore, a very large number of trajectories ( 2 x 10 ) is needed to achieve convergence, even over... 

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