Chaos Poincare section

As illustrated by Figs. 3.3(a) and (b), Poincare sections are a very powerful tool for the visual inspection and classification of the dynamics of a given Hamiltonian. The double pendulum illustrates that for autonomous systems with two degrees of fireedom a Poincare section can immediately suggest whether a given Hamiltonian allows for the existence of chaos or not. Moreover, it tells us the locations of chaotic and regular regions in phase space. 

Coexisting chaos and limit cycle) Consider the double-well oscillator (12.5,1) with parameters S = 0.15, F =0.3, and (O — 1. Show numerically that the system has at least two coexisting attractors a large limit cycle and a smaller strange attractor. Plot both in a Poincare section,... 

Chaos can further be characterized by resorting to Poincare sections. By determining, for example, the value a of the substrate corresponding to the nth peak, /3 , of product Pi in the course of aperiodic oscillations, we may construct the one-dimensional return map giving a i as a function of a (Decroly, 1987a Decroly Goldbeter, 1987). The continuous character of the curve thus obtained (fig. 4.11) denotes the deterministic nature of the chaotic behaviour. 

Thus, fig. 4.23a represents the result obtained when parameter values give rise to deterministic chaos. The continuous aspect of the curve reflects the chaotic nature of the evolution an infinity of a values can correspond to the successive maxima in /3, but these values of the substrate concentration are not disseminated at random, which fact translates into the continuous nature of the Poincare section. 

Fig. 4.23. Poincare sections obtained in system (4.1) for different values of parameter kg (in s" ) (a) 1.537 (b) 1.5 (c) 1.534 (d) 1.539 (e) 1.86. Situations (c) and (e) correspond, respectively, to figs. 4.18d and 4.21. The continuous curve in (a) corresponds to chaos the simple or complex pattern of bursting obtained in the other cases is indicated. The results are obtained by integration of eqns (4.1). The construction of the return map a +i =/[( )] is explained in fig. 4.24 and in the text (Decroly Goldbeter, 1987).

The piecewise linear map does not account, however, for the appearance of chaotic behaviour. A slight modification of the unidimensional map, taking into account some previously neglected details of the Poincare section of the differential system, shows how chaos may appear besides complex periodic oscillations of the bursting type. 

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]. 

At the time of publication of La Nouvelle Alliance the Prigoginian theory was still at the MPC stage. It is thus significant that aU general statements be illustrated only by the baker s transformation. It is only in Les Lois du Chaos (1994) and in La Fin des Certitudes (1996) that the Large Poincare Systems (LPS) show up. As stated in Section I.D 4, this concept results from the quest of real physical systems satisfying the criteria of intrinsic stochasticity. In this case, however, Prigogine and Petrosky were led to introduce a true modification... 

The helium atom is an atomic physics example of a three-body problem. On the basis of Poincare s result we have to expect that the helium atom is classically chaotic. Richter and Wintgen (1990b) showed that this is indeed the case the helium atom exhibits a mixed phase space with intermingled regular and chaotic regions (see also Wintgen et al. (1993)). Thus, conceptually, the helium atom is a close relative of the double pendulum studied in Section 3.2. Given the classical chaoticity of the helium atom we are confronted with an important question How does chaos manifest itself in the helium atom ... 

A great deal of attention has been focused in recent years by workers in classical dynamics on the geometric properties of phase space structures and their manifestation on Poincare maps (also referred to as surfaces of section). The result has been the blossoming of a huge literature on the subject of nonlinear dynamics (quasiperiodicity and dynamical chaos), which is discussed in a number of recent textbooks and articles. - ... 

Further confirmation of chaos is obtained by the Poincare s map, wherein the points on the Poincare s section represent repeated intersections at the ends of each circuit of the attractor. 

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