Phase-plane stroboscopic

FIGURE 29 The development of the stroboscopic phase-plane. Segments (a) and (d) show the trajectory settling down to a limit-cycle through a sequence of points at times that are multiples of t. This is drawn out in the time dimension in (b) and shown in its regularity in (c). If there is a unique periodic solution, the stroboscopic plane will show a sequence of states converging on (e). 

An obvious map to consider is that which takes the state (x(t), y(t) into the state (x(t + r), y(t + t)), where r is the period of the forcing function. If we define xn = x(n t) and y = y(nr), the sequence of points for n = 0,1,2,... functions in this so-called stroboscopic phase plane vis-a-vis periodic solutions much as the trajectories function in the ordinary phase plane vis-a-vis the steady states (Fig. 29). Thus if (x , y ) = (x +1, y +j) and this is not true for any submultiple of r, then we have a solution of period t. A sequence of points that converges on a fixed point shows that the periodic solution represented by the fixed point is stable and conversely. Thus the stability of the periodic responses corresponds to that of the stroboscopic map. A quasi-periodic solution gives a sequence of points that drift around a closed curve known as an invariant circle. The points of the sequence are often joined by a smooth curve to give them more substance, but it must always be remembered that we are dealing with point maps. 

FIGURE 7 Typical shapes of subharmonic trajectories. A subharmonic period 4 within the 4/3 resonance horn is a three-peaked oscillation in time (a), has three loops in its phase plane projection (b), and four loops in its x-cos 0 projection (c) (Brusselator, a = 0.0072, o = 4/3). The subharmonic period 4 within the 4/1 resonance horn has one loop in its phase plane projection (e), four loops in the x-cos projection (f) and is a one-peaked oscillation in time (d). Stroboscopic points are denoted by O. Try to imagine them winding around the doughnut in three-dimensional space An interesting shape shows up at the period 2 resonance in the 2/3 resonance horn (surface model >/aio = 2/3, alao = 0.1, o0 = 0.001) (g, h). These shapes are comparatively simple because of the shape of the unperturbed limit cycle which for all cases was a simple closed curve. 

The mathematical method that will be used to characterize the system response is the stroboscopic map that was used by Kai Tomita (1979) and Kevrekidis et al. (1984,1986). If a point in the phase plane is used as an initial condition for the integration of the ordinary differential equations (odes) (4), a trajectory will be traced out. After integrating for one forcing period (from t = 0 to t = T), the trajectory will arrive at a new point in the phase palne. This new point is defined as the stroboscopic map of the original point and integration for... 

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 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... 

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.

The advection problem is thus described by a periodically driven non-autonomous Hamiltonian dynamical system. In such case, besides the two spatial dimensions an additional variable is needed to complete the phase space description, which is conveniently taken to be the cyclic temporal coordinate, r = t mod T, representing the phase of the periodic time-dependence of the flow. In time-dependent flows ip is not conserved along the trajectories, hence trajectories are no longer restricted to the streamlines. The structure of the trajectories in the phase space can be visualized on a Poincare section that contains the intersection points of the trajectories with a plane corresponding to a specified fixed phase of the flow, tq. On this stroboscopic section the advection dynamics can be defined by the stroboscopic Lagrangian map... 

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