Attractor, 9 chaotic

For certain parameter values tliis chemical system can exlribit fixed point, periodic or chaotic attractors in tire tliree-dimensional concentration phase space. We consider tire parameter set... 

Figure C3.6.1 (a) WR single-handed chaotic attractor for k 2 = 0.072. This attractor is projected onto tire (c, C2) plane. The maximum value reached by c (t) is 54.1 and tire minimum reached by <7 " 2.5. The vertical line, at Cj = 8.5 for < 1, shows the position of tire Poincare section of tire attractor used later, (b) A projection, onto tire cft- ),cft2)) plane, of tire chaotic attractor reconstmcted from tire set of delayed coordinates cft),cft ),c (t2), where t = t + and t2 = t -I- T2, for 0 < t < 00, and fixed delays = 137 and T2 = 200. Note tliat botli cft ) and cftf) reach a maximum of P and a minimum of <"T""so tliat tire tliree-dimensional reconstmcted attractor is...

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.

Figure C3.6.3 The spreading of an ensemble of four points on the WR chaotic attractor, (a) The initial tight, four-point ensemble of open circles (o) at = 5.287. .., = 24.065. .. and variable = 2.884. ..,2.984. ..,3.084. .., and...

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. 

As already mentioned, the motion of a chaotic flow is sensitive to initial conditions [H] points which initially he close together on the attractor follow paths that separate exponentially fast. This behaviour is shown in figure C3.6.3 for the WR chaotic attractor at /c 2=0.072. The instantaneous rate of separation depends on the position on the attractor. However, a chaotic orbit visits any region of the attractor in a recurrent way so that an infinite time average of this exponential separation taken along any trajectory in the attractor is an invariant quantity that characterizes the attractor. If y(t) is a trajectory for the rate law fc3.6.2] then we can linearize the motion in the neighbourhood of y to get... 

Figure C3.6.4(a) shows an experimental chaotic attractor reconstmcted from tire Br electrode potential, i.e. tire logaritlim of tire Br ion concentration, in tlie BZ reaction [F7]. Such reconstmction is defined, in principle, for continuous time t. However, in practice, data are recorded as a discrete time series of measurements (A (tj) / = 1,...

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...

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. 

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.

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. 

The simulation result (Figure 4) shows that when two initial conditions are very close, after a dimensionless time of 40 units the concentration of reactant A and the reactor temperature are completely different. This means that the system has a chaotic behavior and their d3mamical states diverge from each other very quickly, i.e. the system has high sensitivity to initial conditions. This separation increases with time and the exponential divergence of adjacent phase points has a very important consequence for the chaotic attractor, i.e. 

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. 

Holmes 1983) states that when the above transversal homoclinic intersection occurs, that there is a structurally stable invariant Cantor set like the one for the Horseshoe map. It has also been shown by Holmes (1982) that this invariant set contains a countable, dense set of saddles of all periods, an uncountable set of non-periodic trajectories and a dense orbit. If nothing else is clear from the above, it is at least certain that homoclinic bifurcations for maps are accompanied by some very unusual phase portraits. Even if homoclinic bifurcations are not necessarily accompanied by the formation of stable chaotic attractors, they lend themselves to extremely long chaotic like transients before settling down to a periodic motion. Because there are large numbers of saddles present, their stable manifolds divide up the phase plane into tiny stability regions and extreme sensitivity to perturbations is expected. 

Chaotic attractors can occupy large regions of parameter space as in the... 

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. 

We then report and discuss the results of recent investigations of fluctuational escape from the basins of attraction of chaotic attractors (CAs). The question of noise-induced escape from a basin of attraction of a CA has remained a major scientific challenge ever since the first attempts to generalize the classical escape problem to cover this case [92-94]. The difficulty in solving these problems stems from the complexity of the system s dynamics near a CA and is... 

In this section the application of the optimal path approach to the problem of escape from a nonhyperbolic and from a quasihyperbolic attractor is examined. We discuss these two different types of chaotic attractor because it is known [160] that noise does not change very much the structure and properties of quasi-hyperbolic attractors, but that the structure of non-hyperbolic attractors is abruptly changed in the presence of noise, with a strong dependence on noise intensity. Note that for optical systems both types of chaotic attractor [161-163] (nonhyperbolic and quasihyperbolic) are observed, but a nonhyperbolic attractor is much more typical. 

It can be seen that the solution of the problem of the energy-optimal guiding of the system from a chaotic attractor to another coexisting attractor requires the solution of the boundary-value problem (33)-(34) for the Hamiltonian dynamics. The difficulty in solving these problems stems from the complexity of the system dynamics near a CA and is related, in particular, to the delicate problems of the uniqueness of the solution, its behaviour near a CA, and the boundary conditions at a CA. 

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],... 

Let us now formulate the problem of the energy-optimal steering of the motion from a chaotic attractor to the coexisting stable limit cycle for a simple model, a noncentrosymmetric Duffing oscillator. This is the model that, in the absence of fluctuations, has traditionally been considered in connection with a variety of problems in nonlinear optics [166]. Consider the motion of a periodically driven nonlinear oscillator under control... 

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