Stroboscopic plot

Numerical studies (Erdi Barna, 1986a, b) gave the evidence of occurrence of abnormal dynamic behaviour. Phase space plots and stroboscopic plots are obtained. The former are identified with two-dimensional projection of trajectories, the latter can be constructed by points taken at regular intervals of the period (figs 7.7 and 7.8). 

Fig. 10. Stroboscopic plot of two variables of the network, recorded at frequency Q. of the target waves in compartment I. (a) The closed curve reveals the presence of a quasi-periodic dynamics, (b) First return map associated with the stroboscopic plot. The phase 6 of the curve is defined as the angle that forms the vector joining the centre of the cycle (the cross) to the state point with a reference direction. Parameters of Equation (1) are identical to those of Figure 9. The window length is / = 8.

Fig. 13.12. Stroboscopic map plotting the value of a given variable at the end of each forcing period against its value at the end of the previous period (d) phase-locked response, giving a finite number of discrete points (b) quasi-periodic response—the points will eventually fill the complete closed curve in the plane.

FIGURE 5 Subharmonic saddle-node bifurcations, (a). The subharmonic period 3 isola for the surface model (o/o0 = 1.4, o0 = 0.001). One coordinate of the fixed points of the third iterate of the stroboscopic map is plotted vs. the varying frequency ratio oi/eio. Six such points (S, N) exist simultaneously, three of them (N) lying on the stable node period 3 and three (S) on the saddle period 3. Notice the two triple turning point bifurcations at o>/o>o = 2.9965 and 3.0286. In (b) the PFM of these trajectories on the isola Is also plotted, (c) shows another saddle-node bifurcation occurring (for a>o = 3) as this time the forcing amplitude is increased to o/o0 = I.6SS. 

Phase plane plots are not the best means of investigating these complex dynamics, for in such cases (which are at least 3-dimensional) the three (or more) dimensional phase planes can be quite complex as shown in Figures 16 and 17 (A-2) for two of the best known attractors, the Lorenz strange attractor [80] and the Rossler strange attractor [82, 83]. Instead stroboscopic maps for forced systems (nonautonomous) and Poincare maps for autonomous systems are better suited for investigating these types of complex dynamic behavior. 

Stroboscopic and Poincare maps are different from phase plane plots in that they plot the variables on the trajectory at specifically chosen and repeated time intervals. For... 

The L to M transition lends it self to modern 20 ps resolution stroboscopic investigation (Gerwert and Souvignier, 1993). Fig. 6.6-13 shows a 3-D plot of the difference spectra obtained by stroboscopic FTIR during the L to M to N/0 to BR transitions. It shows the smoothed curve fits instead of the original data. Fig. 6.6-14 shows out of this series as example one difference spectrum recorded 70 microseconds after the laser flash, which represent primarily the room temperature BR-L difference spectrum (compare also Fig. 6.6-11). 

Fig. 7.8 Stroboscopic representation of perturbed variable, (a) Quasistable butterfly shape, (b) Decay of the quasistable shape and start to tend to a new shape, (c) Tending to a new attractor, (d) Arriving at a fixed point of the Poincare plot.

The stroboscopic and Poincare maps are different from the phase plane in that they plot the variables on the trajectory at specific chosen and repeated time intervals. For example, for the forced two-dimensional system, these points are taken at every forcing period. For the Poincare map, the interval of strobing is not as obvious as in the case of the forced system and many techniques can be applied. Different planes can be used in order to get a deeper insight into the nature of strange attractors in these cases. A periodic solution (limit cycle) on the phase plane will appear as one point on the stroboscopic (or Poincare) map. When period doubling takes place, period 2... 

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