Chaotic attractors trajectories

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.

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 C3.6.6 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.

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. 

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],... 

In the presence of weak noise there is a finite probability of noise-induced transitions between the chaotic attractor and the stable limit cycle. In Fig. 14 the filled circles show the intersections of one of the real escape trajectories with the given Poincare section. The following intuitive escape scenario can be expected in the Hamiltonian formalism. Let us consider first the escape of the system from the basin of attraction of a stable limit cycle that is bounded by an saddle cycle. In general, escape occurs along a single optimal trajectory qovt(t) connecting the two limit cycles. 

Since the basin of attraction of the CA is bounded by the saddle cycle SI, the situation near SI remains qualitatively the same and the escape trajectory remains unique in this region. However, the situation is different near the chaotic attractor. In this region it is virtually impossible to analyze the Hamiltonian flux of the auxiliary system (37), and no predictions have been made about the character of the distribution of the optimal trajectories near the CA. The simplest scenario is that an optimal trajectory approaching (in reversed time) the boundary of a chaotic attractor is smeared into a cometary tail and is lost, merging with the boundary of the attractor. 

The saddle cycles L and L2 surround the stable states P and P2 and are located at the intersection of the unstable Wu and stable Ws manifolds. The unstable manifold goes to the stable state P from one side and to the chaotic attractor from the other side. The stable manifold Ws forms a tube in the vicinity of the stable state [183]. The saddle cycles L and L2 have the multipliers (1.0000,1.0280,0.0001), and therefore trajectories will go slowly away along the unstable manifold, and they will approach quickly along the stable manifold. 

Thus, we have found that the mechanisms of escape from a nonhyperbolic attractor and a quasihyperbolic (Lorenz) attractor are quite different, and that the prehistory of the escape trajectories reflects the different structure of their chaotic attractors. The escape process for the nonhyperbolic attractor is realized via several steps, which include transitions between low-period saddle-cycles coexisting in the system phase space. The escape from the Lorenz attractor consist of two qualitatively different stages the first is defined by the stable and unstable manifolds of the saddle center point, and lies on the attractor the second is the escape itself, crossing the saddle boundary cycle surrounding the stable point attractor. Finally, we should like to point out that our main results were obtained via an experimental definition of optimal paths, confirming our experimental approach as a powerful instrument for investigating noise-induced escape from complex attractors. 

The main problem in the solution of non-linear ordinary and partial differential equations in combustion is the calculation of their trajectories at long times. By long times we mean reaction times greater than the time-scales of intermediate species. This problem is especially difficult for partial differential equations (pdes) since they involve solving many dimensional sets of equations. However, for dissipative systems, which include most applications in combustion, the long-time behaviour can be described by a finite dimensional attractor of lower dimension than the full composition space. All trajectories eventually tend to such an attractor which could be a simple equilibrium point, a limit cycle for oscillatory systems or even a chaotic attractor. The attractor need not be smooth (e.g., a fractal attractor in a chaotic system) and is in some cases difficult to compute. However, the attractor is contained in a low-dimensional, invariant, smooth manifold called the inertial manifold M which locally attracts all trajectories exponentially and is easier to find [134,135]. It is this manifold that we wish to investigate since the dynamics of the original system, when restricted to the manifold, reduce to a lower dimensional set of equations (the inertial form). The inertial manifold is, therefore, a useful notion in the field of mechanism reduction. 

Figure 6. Hypercube structure of a four-dimensional network with a chaotic attractor. One of the two cycles followed by the chaotic trajectory is marked by bold lines.

It is clear from the phase portrait of the chaotic attractor that all the periodic orbits that have lost their stability throughout the bifurcation sequence are visited during the course of time a chaotic trajectory winds its way... 

Unstable cycle is the result of a strange (chaotic) attractor. It represents a set of cycles of complicated geometry, which attract the nearby passing trajectories. To predict the behavior of trajectories for chaotic systems over a long period is impossible, because of the very high sensitivity to the initial conditions of a reaction. In experiments the initial conditions are usually given with limited accmacy. It is also very difficult to reproduce mnnerically the trajectory for the chaotic regime of a reaction due to the approximate nature of mnnerical methods. 

This model with only three variables, whose only nonlinearities are xy and xz, exhibited dynamic behavior of unexpected complexity (Fig. 7.2). It was especially surprising that this deterministic model was able to generate chaotic oscillations. The corresponding limit set was called the Lorenz attractor and limit sets of similar type are called strange attractors. Trajectories within a strange attractor appear to hop around randomly but, in fact, are organized by a very complex type of stable order, which keeps the system within certain ranges. 

Figure8.22 Phase portraits of stabilized (a) period-1 and (b) period-2 orbits embedded in a chaotic attractor in the BZ reaction. Scattered points show chaotic trajectory (delay time r = 1.3 s) before stabilization. (c) Time series showing potential of bromide-sensitive electrode. Control via change in input flow rate of cerium and bromate solutions was switched on from 27,800 s to 29,500 s to stabilize period-1 and from 30,000 s to 32,100 s to stabilize period-2. (Adapted from Petrov et al., 1993.)...

A strange (or chaotic) attractor is by definition an attractor for which the largest Lyapunov exponent is positive. Then trajectories starting from nearby points will separate exponentially fast as time evolves. Therefore, all information about the initial conditions is rapidly lost, since any uncertainty, no matter how small, will be magnified until it becomes as large as the attractor thus there is sensitive dependence on initial conditions (RUELLE [38]). Long term predictions about the state of the system are impossible. 

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