Poincare mapping

A convenient method for visualizing continuous trajectories is to construct an equivalent discrete-time mapping by a periodic stroboscopic sampling of points along a trajectory. One way of accomplishing this is by the so-called Poincare map (or surface-of-section) method (see figure 4.1). In general, an N — l)-dimensional surface-of-section 5 C F is chosen, and we consider the sequence of successive in-... 

Poincare maps of this form have the obvious advantage of being much simpler to study than their differential-equation counterparts, without sacrificing any of the essential behavioral properties. They may also be studied as generic systems to help abstract behaviors of more complicated systems. 

Dissipative systems whether described as continuous flows or Poincare maps are characterized by the presence of some sort of internal friction that tends to contract phase space volume elements. They are roughly analogous to irreversible CA systems. Contraction in phase space allows such systems to approach a subset of the phase space, C P, called an attractor, as t — oo. Although there is no universally accepted definition of an attractor, it is intuitively reasonable to demand that it satisfy the following three properties ([ruelle71], [eckmanSl]) ... 

The Poincare map method was applied by linxiang et al. [140] to study mixing in curved channels. 

Fig. 4.4.7 (a) Reconstruction of the Stationary Helical Vortex (SHV) mode from MRI data acquired with the spin-tagging spin-echo sequence [27], The axial flow is upwards and the inner cylinder is rotating clockwise. The two helices represent the counter-rotating vortex streamtubes. (b) Construction of Poincare map for SHV [41]. The orbit of a typical particle is... 

In order to identify the periodic orbits (POs) of the problem, we need to extract the periodic points (or fixed points) from the Poincare map. Adopting the energy F = 0.65 eV, Fig. 31 displays the periodic points associated with some representative POs of the mapped two-state system. The properties of the orbits are collected in Table VI. The orbits are labeled by a Roman numeral that indicates how often trajectory intersects the surfaces of section during a cycle of the periodic orbit. For example, the two orbits that intersect only a single time are labeled la and lb and are referred to as orbits of period 1. The corresponding periodic points are located on the p = 0 axis at x = 3.330 and x = —2.725, respectively. Generally speaking, most of the short POs are stable and located in... 

Figure 31. Periodic points of the Poincare map at the energy E = 0.65 eV. The Roman numerals indicate how often the corresponding orbit intersects the surface of section. Panel (b) shows an enlargement of the main regular island around x,p) = (3.3,0). The thin hnes represent various tori of the system.

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 5 shows the 3-dimensional reconstructed attractors and their projections on canonical planes. The reconstructed phase portraits do not exhibit a defined structure, i.e., it is not toroidal or periodic. As matter of fact, the oscillatory structure is only observed in the Poincare map. The Poincare map is often used to observe the oscillatory structure in dynamical systems. The... 

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... 

Fig. 6. Poincare maps. The section was chosen I (z) = zs = 0 and the crosses indicate no periodic oscillation. Once again, the smallest attractor corresponds to experiment E2.b. zi,Z2,zs are also dimensionless.

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]. 

Fig. 14. Poincare map at = 0. (a) For the disperse regime, the attractor crosses the Poincare section in a short line, (b) for the four-flow region regime, the width of the short line increases, and (c) for the three-flow region regime, the attractor contains two regions (i) winding orbits and (ii) disperse orbits.

Since the estimated MLE is negative, Am < 0, we can say that this case displays as essential regular dynamical behavior. The Poincare map displays an orbit set contained in a short line. The above results can be imputed to the following fact The bubbles rise in an almost-linearly pathway and the liquid phase falls downward between bubbles streams (see scheme in the Figure 10a). This means that the bubbles interactions are feeble. In this way, the modes induced by one bubble stream can not affect another one. 

Mapping techniques, and the associated bifurcation analyses, are also of great importance when applied with the Poincare map described in the appendix to chapter 5. These are used to establish local stability, and changes... 

A technique for distinguishing between phase-locked and quasi-periodic responses, and which is particularly useful when m and n are large numbers, is that of the stroboscopic map. This is essentially a special case of the Poincare map discussed in the appendix of chapter 5. Instead of taking the whole time series 0p(r), for all t, we ask only for the value of this concentration at the end of each forcing period. Thus at times t = 2kn/a>, with k = 1, 2, etc., we measure the surface concentrations of one of our species. If the system is phase locked on to a closed path with a>/a>0 = m/n, then the stroboscopic map will show the measured values moving in a sequence between m points, as in Fig. 13.12(a). If the system is quasi-periodic, the iterates of 0p will never repeat and, eventually, will draw out a closed cycle (Fig. 13.12(b)) in the... 

Moreover, the Poincare mappings of (3.14) at values of Pc) fixed by the existence of (5, 6) and I4 + L5 = 0 show the presence of classically chaotic motions with a bifurcation at E - 6900 cm-1 (see Fig. 6). At this bifurcation, the periodic orbit (5, 6) becomes unstable because one of its Lyapunov exponents turns positive, as shown in Fig. 7. The periodic orbit (6, 7) destabilizes by a similar scenario around E - 7200 cm-1. These results show that the interaction between the bending modes leads to classically chaotic behaviors that destabilize successively the periodic orbits. For the bulk peri-... 

FIGURE 28 A three-dimensional limit-cycle and Poincare map. 

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