Mapping chaotic

Figure C3.6.2 (a) The (fi2,cf) Poincare surface of a section of the phase flow, taken at ej = 8.5 with cq < 0, for the WR chaotic attractor at k = 0.072. (b) The next-amplitude map constmcted from pairs of intersection coordinates. ..,(c2(n-l-l),C2(n-l-2),C2(n-l-l)),...j. The sequence of horizontal and vertical line segments, each touching the diagonal B and the map, comprise a discrete trajectory. The direction on the first four segments is indicated.

The existence of chaotic oscillations has been documented in a variety of chemical systems. Some of tire earliest observations of chemical chaos have been on biochemical systems like tire peroxidase-oxidase reaction [12] and on tire well known Belousov-Zhabotinskii (BZ) [13] reaction. The BZ reaction is tire Ce-ion-catalyzed oxidation of citric or malonic acid by bromate ion. Early investigations of the BZ reaction used tire teclmiques of dynamical systems tlieory outlined above to document tire existence of chaos in tliis reaction. Apparent chaos in tire BZ reaction was found by Hudson et a] [14] aiid tire data were analysed by Tomita and Tsuda [15] using a return-map metliod. Chaos was confinned in tire BZ reaction carried out in a CSTR by Roux et a] [16, E7] and by Hudson and... 

Figure C3.6.4 Single-handed chaotic attractor and next-amplitude map reconstmcted from experimental data for tire BZ reaction, (a) The reconstmcted attractor projected onto tire + x)) plane (see tire text for a...

Strange Attractors The motion on strange attractors exhibits many of the properties normally associated with completely random or chaotic behavior, despite being well-defined at all times and fully deterministic. More formally, a strange attractor S is an attractor (meaning that it satisfies properties (i)-(iii) above) that also displays sensitivity to initial conditions. In the case of a one-dimensional map, Xn+i = for example, this means that there exists a <5 > 0 such that for... 

Note that while the attractor for a o is a fractal, it is not a chaotic attractor, since initially neighboring points do not undergo the same kind of exponential divergence that we observed earlier in the Bernoulli map. 

This is strictly true only for two-dimensional area-preserving maps in dimensions iV > 4, chaotic orbits may leak through KAM surfaces by a process called Arnold diffusion (see [licht83j). 

At the risk of oversimplifying, there are essentially three different dynamical regimes of the one-dimensional circle map (we have not yet formed our CML) (I) j A < 1 - for which we find mode-locking within the so-called AmoW Tongues (see section 4.1.5) and the w is irrational (11) k = 1 - for which the non mode-locked w intervals form a self-similar Cantor set of measure zero (111) k > 1 - for which the map becomes noninvertible and the system is, in principle, ripened for chaotic behavior (the real behavior is a bit more complicated since, in this regime, chaotic and nonchaotic behavior is actually densely interwoven in A - w space). 

Intermlttency Manneville [mann80] showed that for the special case of a generic intermittency threshold in which the tangent point lies at the endpoint of the interval (in the case of a one dimensional iterative map of an interval to itself), the resulting chaotic dynamics has a power spectrum S f) 1/ (/(log/) ) for low /. Miracky, et. al. were able to modify the map to obtain an exact 1// behavior [mirack87]. Because the result depends on the fine-tuning of an external parameter, however, it does not so mucdi explain the generic appearance of flicker noise phenomena as beg the obvious question, why do systems typically sit at whatever... 

Turbulence is generally understood to refer to a state of spatiotemporal chaos that is to say, a state in which chaos exists on all spatial and temporal scales. If the reader is unsatisfied with this description, it is perhaps because one of the many important open questions is how to rigorously define such a state. Much of our current understanding actually comes from hints obtained through the study of simpler dynamical systems, such as ordinary differential equations and discrete mappings (see chapter 4), which exhibit only temporal chaosJ The assumption has been that, at least for scenarios in which the velocity field fluctuates chaotically in time but remains relatively smooth in space, the underlying mechanisms for the onset of chaos in the simpler systems and the onset of the temporal turbulence in fluids are fundamentally the same. 

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. 

Linxiang, W, Yueun, F., Ying, C., Animation of chaotic mixing by a backward Poincare cell-map method, Int. J. Bifurcation Chaos 11, 7 (2001) 1953-1960. 

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 fact, with the help of Krein s trace formula, the quantum field theory calculation is mapped onto a quantum mechanical billiard problem of a point-particle scattered off a finite number of non-overlapping spheres or disks i.e. classically hyperbolic (or even chaotic) scattering systems. 

In order to analyze both systems, some techniques from nonlinear science are burrowed. Firstly, a phase portrait is constructed from delay coordinates, a Poincare map is also computed, FFT is exploited to derive a Power Spectrum Density (PSD) Maximum Lyapunov Exponents (MLE) are also calculated from time series. Although we cannot claim chaos, the evidence in this chapter shows the possible chaotic behavior but, mostly important, it exhibits that the oscillatory behavior is intrinsically linked to the controlled systems. The procedures are briefly described before discuss each study case. 

Figure 6 displays the Poincare maps for all experiments. Note that even the projections in canonical planes (see Figure 5) seem ordered in layers. That is, a toroidal structure can be seen form the Poincare surface. That is, small amplitude oscillations were detected in time series (see Figures 3 and 4) for all experiments. The t3rpical behavior of aperiodic (possibly chaotic) oscillations can be confirmed is one takes a look at the corresponding Poincare section... 

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