The stroboscopic map

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 

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V-shaped regions, but in general close up again at sufficiently large amplitude. The number marked in each of the tongues indicates how many forcing periods are required for one full oscillation or, equivalently, how many points appear in the stroboscopic map. 

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. 

One fixed point, stable or unstable, in the stroboscopic map corresponds to a periodic solution of period t. If there are two distinct points, each mapping into the other, we have a periodic solution of twice the period. Figure 30, drawn with a continuous fill-in, illustrates the way in which different kinds of periodic solution can coexist. The invariant circle (actually ovoid with a pointed... 

We are now concerned with the location of two-tori in our systems. We have formulated for that purpose a shooting on collocations iterative algorithm that computes an invariant curve of the stroboscopic map for a forced system. Part of this algorithm and its application to maps in general are described elsewhere (Kevrekidis et al., 1985). We briefly describe here its application to a two-dimensional forced system. 

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. 

As we now change stable periodic trajectories cannot lose stability through a saddle-node bifurcation, since the saddles no longer exist rather they lose stability through a Hopf bifurcation of the stroboscopic map to a torus (Marsden and McCracken, 1976). This phenomenon, as well as the torus resulting from it, is considerably different from the frequency unlocking case. One of the main differences is that the entire quasi-periodic attractor that bifurcates from a periodic trajectory lies close to it [see Figs. 9(c) and 9(d)],... 

An alternative formulation for the torus-computing algorithm is to solve for an invariant circle along with a nonlinear change of coordinates that makes the action of the stroboscopic map conjugate to a rigid rotation on the circle. This is equivalent to the parameterization... 

The effects of forced oscillations in the partial pressure of a reactant is studied in a simple isothermal, bimolecular surface reaction model in which two vacant sites are required for reaction. The forced oscillations are conducted in a region of parameter space where an autonomous limit cycle is observed, and the response of the system is characterized with the aid of the stroboscopic map where a two-parameter bifurcation diagram for the map is constructed by using the amplitude and frequency of the forcing as bifurcation parameters. The various responses include subharmonic, quasi-peri-odic, and chaotic solutions. In addition, bistability between one or more of these responses has been observed. Bifurcation features of the stroboscopic map for this system include folds in the sides of some resonance horns, period doubling, Hopf bifurcations including hard resonances, homoclinic tangles, and several different codimension-two bifurcations. 

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... 

The conditions for the existence of a pT-periodic solution to the forced system can be written in terms of a fixed point equation for the pth iterate of the stroboscopic map... 

The differential equation for M in (7) is non-antonomous and involves evaluation of the jacobian of the forced-model equations at the current value of the trajectory jc(x0, p, t) for each time step so that it must be integrated simultaneously with the system equations. Upon convergence on a fixed point, the matrix M becomes the monodromy matrix whose eigenvalues are those of the jacobian of the stroboscopic map evaluated at the fixed point and are called the Floquet multipliers of the periodic solution. 

The excitation diagram was found to contain saddle-node, Hopf, period doubling, and homoclinic bifurcations for the stroboscopic map. In addition, many of these co-dimension one bifurcation curves were found to meet at the following co-dimension two bifurcation points Bogdanov points (double +1 multipliers), points with double -1 multipliers, points with multipliers at li and H, metacritical period-doubling points, and saddle-node cusp points. 

A mathematically more precise concept associated to this property is ergodicity. The stroboscopic map is ergodic if all of its invariant sets (i.e. those that satisfy r(A) = A) have zero measure or their complement has zero measure. A stronger property is mixing that requires that for any two sets A and B the measure n (area or volume) of the intersection satisfies n B) —> /j,(A)/j,(B) for... 

Figure 3.10. (a) A schematic picture of a time-dependent potential, (b) The stroboscopic map of the flow on the phase space under the potential. 

We have worked out formulas above for the first terms in the expansion in the case of several splitting methods. We can thus view (at least locally) our numerical trajectory as the stroboscopic map (snapshots) of the evolution of a continuous Hamiltonian system. 

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