Subharmonic bifurcation

This route should already be familiar to us from our discussion of the logistic map in chapter 4, Prom that chapter, we recall that the Feigenbaum route calls for a sequence of period-doubling bifurcations pitchfork bifurcations versus the Hopf bifurcations of the Landau-Hopf route) such that if subharmonic bifurcations are observed at Reynolds numbers TZi and 7 2, another can be expected at TZ determined by... 

Figure C3.6.6 The figure shows the C2 coordinate, for < 0, of the family of trajectories intersecting the ( 2, 3) Poineare surfaee at = 8.5 as a function of bifurcation parameter k 2- As the ordinate k 2 decreases, the first subharmonie easeade is visible between k 2 0.1, the value of the first subharmonic bifurcation to k 2 0.083, the subharmonie limit of the first cascade. Periodic orbits that arise by the tangent bifurcation mechanism associated with type-I intermittency (see the text for references) can also be seen for values of k 2 smaller than this subharmonie limit. The left side of the figure ends at k 2 = 0.072, the value corresponding to the chaotic attractor shown in figure C3.6.1(a). Other regions of chaos can also be seen.

The present study suggests that the probability of encountering smooth onedimensional maps with non-quadratic maxima should not be ignored as nongeneric for real physical systems. The anomalous bifurcation sequence discussed above is a consequence of a system s symmetry with respect to spatial inversion this kind, or possibly other kinds, of symmetry are commonly present in real physical systems. Experimentally, such a bifurcation sequence could easily be distinguished from the usual subharmonic bifurcations. This is because considerable elongation in period (measured in the continuous time /, and not the step number n) is expected to occur each time a closed orbit is being transformed into a homoclinic orbit. 

We now consider the phenomenon of entrainment (the development of resonances) on the torus (Meyer, 1983). When (and if) the off-diagonal band in Fig. 6 crosses the diagonal [Figs. 6(c) and 6(f)], there exist points whose images fall on themselves they are fixed points of the map we study. These points lie on periodic trajectories that are locked on the torus. Such trajectories appear in pairs in saddle-node bifurcations and are usually termed subharmonics . When this occurs there is no quasi-periodic attractor winding around the torus surface, but the basic structure of the torus persists the invariant circle is patched up from the unstable manifolds of the periodic saddle-points with the addition of the node-periodic point (Arnol d, 1973, 1982). As we continue changing some system parameter the periodic points may come to die in another saddle-node bifurcation (see Fig. 5). Periodic trajectories thus... 

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. 

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. 

Forced oscillation is a well-known technique for the characterization of linear systems and is referred to as a frequency response method in the process control field. By contrast, the response of nonlinear systems to forcing is much more diverse and not yet fully understood. In nonlinear systems, the forced response can be periodic with a period that is some integer multiple of the forcing period (a subharmonic response), or quasi-periodic (characterized by more than one frequency) or even chaotic, when the time series of the response appears to be random. In addition, abrupt transitions or bifurcations can occur between any of these responses as one or more of the parameters is varied and there can be more than one possible response for a given set of parameters depending on the initial conditions or recent history of the system. 

E. This double -1 point is yet another codimension-two bifurcation, which will be discussed in detail later. Another period 1 Hopf curve extends from point F through points G and H. F is another double -1 point and, as one moves away from F along the Hopf curve, the angle at which the complex multipliers leave the unit circle decreases from it. The points G and H correspond to angles jt and ixr respectively and are hard resonances of the Hopf bifurcation because the Floquet multipliers leave the unit circle at third and fourth roots of unity, respectively. Points G and H are both important codimension-two bifurcation points and will be discussed in detail in the next section. The Hopf curves described above are for period 1 fixed points. Subharmonic solutions (fixed points of period greater than one) can also bifurcate to tori via Hopf bifurcations. Such a curve exists for period 2 and extends from point E to K, where it terminates on a period 2 saddle-node curve. The angle at which the complex Floquet multipliers leave the unit circle approaches zero at either point of the curve. 

Among all of the points on the period 1 Hopf curve, some will have complex Floquet multipliers A with a phase angle 6 of mln)2u with n = 3 or 4 (i.e. third or fourth roots of unity) and are called hard reasonances. Because these points are fixed points for Fn that have multipliers equal to A" = 1, it is not surprising to find that subharmonic fixed points of period n are involved in addition to the bifurcating period 1 fixed point. 

Figure C3.6.5 The first two periodie orbits in the main subharmonie sequence are shown projected onto the (cj,C2) plane. This sequence arises from a Hopf bifurcation of the stable fixed point for the parameters given in the text. The arrows indicate the direction of motion, (a) The limit cyele or period-1 orbit at k 2 = 0.11. (b) The first subharmonic or period-2 orbit at k 2 = 0.095.

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