Chaos attractor

Actively working groups are sure to include physical chemists (experimental and theoretical) and mathematicians (pure and applied). "Graphs theory , "dynamics , "non-linear oscillations , "chaos , "attractor , "synergetics , "catastrophes and finally "fractals these are the key words of modern kinetics. 

There are many examples in nature where a system is not in equilibrium and is evolving in time towards a thennodynamic equilibrium state. (There are also instances where non-equilibrium and time variation appear to be a persistent feature. These include chaos, oscillations and strange attractors. Such phenomena are not considered here.)... 

Chaotic attractors are complicated objects with intrinsically unpredictable dynamics. It is therefore useful to have some dynamical measure of the strength of the chaos associated with motion on the attractor and some geometrical measure of the stmctural complexity of the attractor. These two measures, the Lyapunov exponent or number [1] for the dynamics, and the fractal dimension [10] for the geometry, are related. To simplify the discussion we consider tliree-dimensional flows in phase space, but the ideas can be generalized to higher dimension. 

The next problem to consider is how chaotic attractors evolve from tire steady state or oscillatory behaviour of chemical systems. There are, effectively, an infinite number of routes to chaos [25]. However, only some of tliese have been examined carefully. In tire simplest models tliey depend on a single control or bifurcation parameter. In more complicated models or in experimental systems, variations along a suitable curve in the control parameter space allow at least a partial observation of tliese well known routes. For chemical systems we describe period doubling, mixed-mode oscillations, intennittency, and tire quasi-periodic route to chaos. 

At the th period doubling the period of the oscillation is In the limit —> co we arrive at the strange attractor where the time variation of the concentrations is no longer periodic. This is the period-doubling route to chaos. 

Figure C3.6.6 The figure shows tire coordinate, for < 0, of tire family of trajectories intersecting tire Poincare surface at cq = 8.5 as a function of bifurcation parameter k 2- As tire ordinate k 2 decreases, tire first subhannonic cascade is visible between k 2 0.1, tire value of tire first subhannonic bifurcation to k 2 0.083, tire subhannonic limit of tire first cascade. Periodic orbits tliat arise by tire tangent bifurcation mechanism associated witli type-I intennittency (see tire text for references) can also be seen for values of k 2 smaller tlian tliis subhannonic limit. The left side of tire figure ends at k 2 = 0.072, tire value corresponding to tire chaotic attractor shown in figure C3.6.1(a). Otlier regions of chaos can also be seen.

Several important topics have been omitted in this survey. We have described only a few of the routes by which chaos can arise in chemical systems and have made no attempt to describe in detail the features of the different kinds of chemical strange attractor seen in experiments. A wide variety of chemical patterns have been observed and while the many aspects of the mechanisms for their appearance are understood, some features like nonlinear... 

The values could be almost repeating but not quite so. In chaos theory, these are called Lorenz attractor systems. 

Behavior for a > aoo- What happens for a > Qoo The simple answer is that the logistic map exhibits a transition to chaos, with a variety of different attractors for Qoo < a < 4 exhibiting exponential divergence of nearby points. To leave it at that, however, would surely bo a great disservice to the extraordinarily beautiful manner in which this trairsition takes place. 

A set of experiments on gas-liquid motion in a vertical column has been carried out to study its d3mamical behavior. Fluctuations volume fraction of the fluid were indirectly measured as time series. Similar techniques that previous section were used to study the system. Time-delay coordinates were used to reconstruct the underl3ung attractor. The characterization of such attractor was carried out via Lyapunov exponents, Poincare map and spectral analysis. The d3mamical behavior of gas-liquid bubbling flow was interpreted in terms of the interactions between bubbles. An important difference between this study case and former is that gas-liquid column is controlled in open-loop by manipulating the superficial velocity. The gas-liquid has been traditionally studied in the chaos (turbulence) context [24]. 

Controlled chaos may also factor into the generation of rhythmic behavior in living systems. A recently proposed modeL describes the central circadian oscillator as a chaotic attractor. Limit cycle mechanisms have been previously offered to explain circadian clocks and related phenomena, but they are limited to a single stable periodic behavior. In contrast, a chaotic attractor can generate rich dynamic behavior. Attractive features of such a model include versatility of period selection as well as use of control elements of the type already well known for metabolic circuitry. 

The case of a frequency mismatch between laser pumps and cavity modes was investigated [83], and for the first time, chaos in SHG was found. When the pump intensity is increased, we observe a period doubling route to chaos for Ai = 2 = 1. Now, for/i = 5.5, Eq. (3) give aperiodic solutions and we have a chaotic evolution in intensities (Fig. 5a) and a chaotic attractor in phase plane (Imaj, Reai) (Fig. 5b). 

The frequency of modulation il is now the main parameter, and we are able to switch the system of SHG between different dynamics by changing the value of il. To find the regions of where a chaotic motion occurs, we calculate a Lyapunov spectrum versus the knob parameter il. The first Lyapunov exponent A,j from the spectrum is of the greatest importance its sign determines the chaos occurrence. The maximal Lyapunov exponent Xj as a function of is presented for GCL in Fig. 6a and for BCL in Fig. 6b. We see that for some frequencies il the system behaves chaotically (A-i > 0) but orderly Ck < 0) for others. The system in the second case is much more damped than in the first case and consequently much more stable. By way of example, for = 0.9 the system of SHG becomes chaotic as illustrated in Fig. 7a, showing the evolution of second-harmonic and fundamental mode intensities. The phase point of the fundamental mode draws a chaotic attractor as seen in the phase portrait (Fig. 7b). However, the phase point loses its chaotic features and settles into a symmetric limit cycle if we change the frequency to = 1.1 as shown in Fig. 8b, while Fig. 8a shows a seven-period oscillation in intensities. To avoid transient effects, the evolution is plotted for 450 < < 500. 

A more complicated behavior of the MLE is observed for higher values of 7j. Varying the length of the pulse 7j, we observe regions of order and chaos. By way of an example, the phase portrait Reoti versus Imai for a chaotic attractor is shown in Fig. 15. 

FIGURE 10 Example of chaos for AlAo 1.45, cu/stable fixed points have been found, (b) The time series for a chaotic trajectory after 150 periods of forced oscillations. The arrows indicate a near periodic solution with period 21. The periodicity eventually slips into short random behaviour followed by long near period behaviour. This near periodicity reflects the fact that the chaotic attractor forces the trajectory to eventually pass near the stable manifolds of the period 21 saddle located around the perimeter of the chaotic attractor. 

Below we show how the energy-optimal control of chaos can be solved via a statistical analysis of fluctuational trajectories of a chaotic system in the presence of small random perturbations. This approach is based on an analogy between the variational formulations of both problems [165] the problem of the energy-optimal control of chaos and the problem of stability of a weakly randomly perturbed chaotic attractor. One of the key points of the approach is the identification of the optimal control function as an optimal fluctuational force [165],... 

Here u(t) is the control function. It is a system where chaos can be observed at relatively small values h m 0.1 of the driving force amplitude and the chaotic attractor is a nonhyperbolic attractor or a quasiattractor [167]. 

C. Sparrow, The Lorenz Equations Bifurcations, Chaos and Strange Attractors, Springer, New-York, 1982. 

The Lorenz and Rossler models are deterministic models and their strange attractors are therefore called deterministic chaos to emphasize the fact that this is not a random or stochastic behavior. 

There are many other interesting and complex dynamic phenomena besides oscillation and chaos which have been observed but not followed in depth both theoretically and experimentally. One example is the wrong directional behavior of catalytic fixed-bed reactors, for which the dynamic response to input disturbances is opposite of that suggested by the steady-state response [99, 100], This behavior is most probably connected to the instability problems in these catalytic reactors as shown crudely by Elnashaie and Cresswell [99]. Recently Elnashaie and co-workers [102-105] have also shown rich bifurcation and chaotic behavior of an anaerobic fermentor for producing ethanol. They have shown that the periodic and chaotic attractors may give higher ethanol yield and productivity than the optimal steady states. These results have been confirmed experimentally [105],... 

Mathematical models that describe and predict the inanimate world quite well are actually of little value in the system of deterministic chaos that governs biology. The answers one can expect from mathematical approaches to evolution (in contrast to my earlier perception) cannot be narrowed to less than the surface of the chaotic attractor of the system which is a little like watching evolution on earth from a satellite.1 The limits of the attractor surface are given by the initial conditions which are not knowable in sufficient detail.2 Empiricism can help, after all our laws of science by and large are the results of repeated observations. 

V. Belykh, I. Belykh, and E. Mosekilde Hyperbolic Plykin attractor can exist in neuron models. Int. J. Bifurcation and Chaos 2005,15 3567-3578. 

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