Poincare sections

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 2. Schematic of typical data and consistent Poincare sections from the quasiperiodic regime of Rayleigh-B nard convection. The rotation number W (in arbitrary units) is plotted versus Rayleigh number R for two different values...

The function P can be computed from either an analytical or a numerical representation of the flow field. In such a way, a 3-D convection problem is essentially reduced to a mapping between two-dimensional Poincare sections. In order to analyze the growth of interfacial area in a spatially periodic mixer, the initial distri-... 

Bouwmans (1992 see also Bouwmans et al., 1997) used a particle tracking technique in a RANS flow field to estimate trajectories of neutral and buoyant additions, to construct Poincare sections of additions crossing specific horizontal cross-sectional planes, to predict probabilities of surfacing for buoyant additions, and to mimic the temporal response of conductivity probes. 

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... 

In a similar manner than previous case, we have chosen the plane S(Z) = 23 = 0 to construct a Poincare section of the attractors in Figure 13. We tracked the orbit z(t) and recorded the pairs (21,22) at which 23 = 0. Due to... 

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.

Fig. A5.1. Construction of a Poincare section in the phase plane illustrating the shooting method of locating limit cycle solutions.

Figure 9. Supercritical antipitchfork bifurcation scenario for symmetric XYX molecules on the left, bifurcation diagram in the plane of energy versus position in the center, typical phase portraits in some Poincare section in the different regimes on the right, the fundamental periodic orbits in position space.

The cases of hyperbolic-without-reflection and hyperbolic-with-reflection stability have to be distinguished. In both cases, the trajectories in the neighborhood of the periodic orbit trace out hyperbolic paths in the Poincare section, but if the stability is hyperbolic with reflection, the trajectories cross over between the branches of the hyperbola on each iteration. 

Upon convergence, the eigenvalues of dF/dx (the characteristic or Floquet multipliers FMt) are independent of the particular point on the limit cycle (i.e. the particular Poincare section or anchor equation used). One of them, FMn, is constrained to be unity (Iooss and Joseph, 1980) and this may be used as a numerical check of the computed periodic trajectory the remaining FMs determine the stability of the periodic orbit, which is stable if and only if they lie in the unit circle in the complex plane ( FM, < 1,1 i = n - 1). The multiplier with the largest absolute value is usually called the principal FM (PFM). When (as a parameter varies) the PFM crosses the unit circle, the periodic orbit loses stability and a bifurcation occurs. 

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. 

The perpendicular slice through the phase portrait provides the stroboscopic phase portrait or Poincare section (e.g. [3]). This is in the case of the harmonic oscillator one point in the phase portrait. A further powerful method for the analysis of nonlinear dynamical systems is the determination of the Fourier spectrum of the response function >2. 

Fig. 7.8 Poincare sections after 2000 cycles. Initially nine marker points were placed along the y axis and six along the x axis. The dimensionless amplitude was 0.5, as in Fig. 7.7. The parameter was the dimensionless period (a) 0.05 (h) 0.10 (c) 0.125 (d) 0.15 (e) 0.20 (f) 0.35 (g) 0.50 (h) 1.0 (i) 1.5. For the smallest values of the time period we see that the virtual marker points fall on smooth curves. The general shape of these curves would he the streamlines of two fixed continuously operating agitators. As the time period increases the virtual marker particles fall erratically and the regions indicate chaotic flow. With increasing time periods larger and larger areas become chaotic. [Reprinted by permission from H. Aref, Stirring Chaotic Advection, J. Fluid Meek, 143, 1-21 (1984).]...

Fig. 7.10 Poincare sections of viscous Newtonian flow in alternately turning eccentric cylinders. The inner cylinder turned counterclockwise for a given time, and then the outer cylinder was turned clockwise for 800 periods. There were 11 initial particles. [Reprinted by permission from J. Chaiken, R. Chevray, M. Tabor, and Q. M. Tan, Experimental Study of Lagrangian Turbulence in Stokes Flow, Proc. R. Soc. London A, 408, 165-174 (1986).]...

Fig. 7.44 Concentration distributional and Poincare sections after eight elements side by side, (a) and (b) right-left 180° elements (c) and (d) right-right 180° elements. [Reprinted by permission from O. S. Galaktionov, P. D. Anderson, G. W. M. Peters, and H. E. H. Meijer, Analysis and Optimization of Kenics Mixers, Int. Polym. Process., 18, 138-150 (2003).]...

For a given value of a, the brute force bifurcation diagram displays all the values of the relative arteriolar radius r that the model displays when the steady state trajectory intersects a specified hyperplane (the Poincare section) in phase space. Due to the coexistence of several stable solutions, the brute force diagram must be obtained by scanning a in both directions. 

To determine Tsiow and Tjast in our numerical simulations we have calculated the mean return times of the trajectory to two appropriately chosen Poincare sections... 

Figure 7. The two Poincare sections, (a) S+ and (b) at a total energy E for an uncoupled bound two-mode system, Eq. (12). The region where > q is referred as to region A, and that where q < q as region B hereinafter. (E) (E)) denotes unstable (stable) invariant cyhnder...

Figure 8. The two Poincare sections, (a) and (b) S+ at a total energy E same as Fig. 7...

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