Chaotic zone

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. 

At still higher energies, the elliptic island undergoes a typical cascade of bifurcations in which subsidiary elliptic islands of periods 6, 5, 4, 3 are successively created, which leads to the global destruction of the main elliptic island to the benefit of the surrounding chaotic zone. The cascade ends with a period-doubling bifurcation at Ed, above which the periodic orbit 0 is hyperbolic with reflection, and the main elliptic island has disappeared... 

Even when great care was taken to ensure that the liquid feed was introduced to the disc in an axisymmetric manner with the minimum disturbance, the smooth inner him always broke down into an array of spiral ripples, as shown in Figures 5 and 6. These spiral structures then broke down further until the wave pattern became utterly chaotic, provided that the disc was big enough. It is known that liquid him how over a surface is intrinsically unstable, and the phenomenon has been studied by several workers (3-7). It appears to be qualitatively equivalent to the breakdown of a smoke plume rising from a lighted cigarette, where a chaotic zone is generated about 20 cm above the source. The behavior can also be observed when a liquid him hows over a stationary surface such as a windowpane or a dam spillway. 

Different methods have been developed either for a rapid computation of the LCIs (Cincotta and Simo 2000) or for detecting the structure of the phase space (chaotic zones, weak chaos, regular resonant motion, invariant tori). Especially for this last purpose we quote the frequency map analysis (Laskar 1990, Laskar et al. 1992, Laskar 1993, Lega and Froeschle 1996), the sup-map method (Laskar 1994, Froeschle and Lega 1996), and more recently the fast Lyapunov indicator (hereafter FLI, Froeschle et al. 1997, Froeschle et al. 2000) and the Relative Lyapunov Indicator (Sandor et al. 2000). The definitions and comparisons between different methods including a preliminary version of the FLI have been discussed in Froeschle and Lega (1998, 1999). 

Figure 2,a shows the variation of the FLI with time for 4 different kinds of orbits. The upper curve, with initial conditions (10-4, 0) in the chaotic zone just described, shows an exponential variation of the FLI with time. The upper value of 20 is a computational threshold that allows to avoid floating overflow. 

Values slightly greater than logT indicate either very thin chaotic layers or invariant tori close to very thin chaotic zones. By making a zoom around some of these orbits we have checked that this is indeed the case. 

Using the frequency map analysis we visualized (Lega and Froeschle 1996) the predicted result on the topology of the neighborhood of noble tori for values of the perturbing parameter well above the ones allowed by the mathematical demonstration. Moreover, we have measured the size of the complementary set of tori, showing that the size of islands and chaotic zones decreases exponentially when the distance to the noble torus goes to zero. 

Laskar, J. (1990). The chaotic motion of the Solar System. A numerical estimate of the size of the chaotic zones. Icarus, 88 266-291. 

Finally it should be emphasized, that Luzzati et al. have pointed out [69], that the topological properties of the micellar phases are profoundly different from those of the bicontinuous phases. Introducing the term chaotic zones for the regions of highest disorder it could be shown, that these zones occupy special crystallographic positions, which belong either to symmetry elements or to the IPMS. 

So, the results of Theorems 12.3, 12.5 and 12.7 are summarized as follows IfW is a smooth toruSy then a smooth attracting invariant torus persists after the disappearance of the saddle-node L. If is homeomorphic to a torus but it is non-smoothy then chaotic dynamics appears after the disappearance of L, Herey either the torus is destroyed and chaos exists for all small /i > 0 the big lobe condition is sufficient for that)y or chaotic zones on the parameter axis alternate with regions of simple dynamics. 

Catalytic reactors can roughly be classified as random and structured reactors. In random reactors, catalyst particles are located in a chaotic way in the reaction zone, no matter how carefully they are packed. It is not surprising that this results in a nonuniform fiow over the cross-section of the reaction zone, leading to a nonuniform access of reactants to the outer catalyst surface and, as a consequence, undesired concentration and temperature profiles. Not surprisingly, this leads, in general, to lower yield and selectivity. In structured reactors, the catalyst is of a well-defined spatial structure, which can be designed in more detail. The hydrodynamics can be simplified to essentially laminar, well-behaved uniform fiow, enabling full access of reactants to the catalytic surface at a low pressure drop. 

In the present chapter, steady state, self-oscillating and chaotic behavior of an exothermic CSTR without control and with PI control is considered. The mathematical models have been explained in part one, so it is possible to use a simplified model and a more complex model taking into account the presence of inert. When the reactor works without any control system, and with a simple first order irreversible reaction, it will be shown that there are intervals of the inlet flow temperature and concentration from which a small region or lobe can appears. This lobe is not a basin of attraction or a strange attractor. It represents a zone in the parameters-plane inlet stream flow temperature-concentration where the reactor has self-oscillating behavior, without any periodic external disturbance. 

In the FF regime the solid movement in the lower region of the vessel becomes less chaotic and seems to settle to a lean core surrounded by a denser annulus or wall zone. 

In the no-barrier zone, the cross-sectional flow field helical flow is induced by the grooves and shows non-linear rotation with only one elliptic point [58], In the barrier zone, a spatially periodic perturbation on the helical flow is imposed and thereby two co-rotating flows form, characterized by a hyperbolic point and two elliptic points. By periodic change of the two flow fields, a chaotic flow can be generated. 

In the following section we discuss the classical and quantum dynamics of a Csl molecule scattered off a reaction zone consisting of an arrangement of inhomogenous fields. This system shows classical and quantum chaotic scattering. It can, at least in principle, be built as a laboratory experiment, which would enable the experimenter to check the theoretical predictions advanced in the following sections. 

When the Lewis number is nonunity, the mass diffusivity can be greater than the thermal diffusivity. This discrepancy in diffusivities is important with respect to the reactant that limits the reaction. Ignoring the hydrodynamic instability, consider again the condition between a pair of streamlines entering a wrinkle in a laminar flame. This time, however, look more closely at the flame stmcture that these streamlines encompass, noting that the limiting reactant will diffuse into the flame zone faster than heat can diffuse from the flame zone into the unbumed mixture. Thus, the flame temperature rises, the flame speed increases, and the flame wrinkles bow further in the downstream direction. The result is a flame that looks very much like the flame depicted for the hydrodynamic instability in Fig. 45. The flame surface breaks up continuously into new cells in a chaotic... 

As soon as this Dream-I sinks to the lowest threshold of perception, it enters into a danger zone, where the personal and collective subconsciousness flow together, and out of the latter, dark chaotic figures emerge (encounter with the Guardian of the Threshold). 

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.

For this value of the perturbing parameter a lot of orbits are still invariant tori. Some resonant curves are displayed surrounding some elliptic points and a chaotic, though well confined, zone is generated by the existence of the hyperbolic point at the origin. 

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