Torus attractor

Figure 4.11 shows sample trajectories starting from various initial conditions for four different values of q. (Keep in mind that these are not plots of attractors each trajectory is unique to a particular starting position on the unit torus). For q = 0,... 

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... 

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)],... 

At interesting phenomenon occurs in the case of other resonance horns we have studied it for the case of the 3/1 resonance. The torus pattern breaks when the subharmonic periodic trajectories locked on it for small FA decollate from the torus as FA increases. We are left then with two attractors a stable period 3 and a stable quasi-periodic trajectory. This is a spectacular case of multistability (co-existence of periodic and quasi-periodic oscillations). The initial conditions will determine the attractor to which the system will eventually converge. This decollation of the subharmonics from the torus was predicted by Greenspan and Holmes (1984). They also predicted chaotic trajectories close to the parameter values where the subharmonic decollation occurs. 

The profile and phase plots show oscillatory behavior of Figure 4.56, but no pattern or periods can be seen therein. This is called a strange attractor in modern nonlinear dynamics theory. A strange attractor can be chaotic or nonchaotic (high-dimensional torus). Differentiating between chaotic and nonchaotic strange attractors is beyond the scope of this undergraduate book. 

The trajectories of dissipative dynamic systems, in the long run, are confined in a subset of the phase space, which is called an attractor [32], i.e., the set of points in phase space where the trajectories converge. An attractor is usually an object of lower dimension than the entire phase space (a point, a circle, a torus, etc.). For example, a multidimensional phase space may have a point attractor (dimension 0), which means that the asymptotic behavior of the system is an equilibrium point, or a limit cycle (dimension 1), which corresponds to periodic behavior, i.e., an oscillation. Schematic representations for the point, the limit cycle, and the torus attractors, are depicted in Figure 3.2. The point attractor is pictured on the left regardless of the initial conditions, the system ends up in the same equilibrium point. In the middle, a limit cycle is shown the system always ends up doing a specific oscillation. The torus attractor on the right is the 2-dimensional equivalent of a circle. In fact, a circle can be called a 1-torus,... 

It is necessary to emphasize one principal peculiarity of the copolymerization dynamics which arises under the transition from the three-component to the four-component systems. While the attractors of the former systems are only SPs and limit cycles (see Fig. 5), for the latter ones we can also expect the realization of other more complex attractors [202]. Two-dimensional surfaces of torus on which the system accomplishes the complex oscillations (which are superpositions of the two simple oscillations with different periods) ate regarded to be trivial examples of such attractors. Other similar attractors are fitted by the superpositions of few simple oscillations, the number of which is arbitrary. And, finally, the most complicated type of dynamic behavior of the system when m 4 is fitted by chaotic oscillations [16], for which a so-called strange attractor is believed to be a mathematical image [206]. 

Chaotic behavior in nonlinear dissipative systems is characterized by the existence of a new type of attractor, the strange attractor. The name comes from the unusual dimensionality assigned to it. A steady state attractor is a point in phase space, whereas a limit cycle attractor is a closed curve. The steady state attractor, thus, has a dimension of zero in phase space, whereas the limit cycle has a dimension of one. A torus is an example of a two-dimensional attractor because trajectories attracted to it wind around over its two-dimensional surface. A strange attractor is not easily characterized in terms of an integer dimension but is, perhaps surprisingly, best described in terms of a fractional dimension. The strange attractor is, in fart, a fractal object in phase space. The science of fractal objects is, as we will see, intimately connected to that of nonlinear dynamics and chaos. 

Another route to chaos that is important in chemical systems involves a torus attractor which arises via bifurcation from a limit cycle attractor. Again chaos is found to be associated with periodic behavior and to arise from it through a sequence of transformations and associated bifurcations of a periodic state of the system. The specific sequence is different in this case, however, and somewhat more complex. 

As mentioned in a previous section, a limit cycle can, at times, undergo a Hopf bifurcation. This would be revealed in a stability analysis of the cross-sectional point in the Poincare section of the limit cycle. A Hopf bifurcation would correspond to an associated pair of eigenvalues whose real part passes from negative to positive while all other eigenvalues remain negative. In physical terms, a Hopf bifurcation means a second frequency becomes available to the system, and this is reflected in the disappearance of the limit cycle attraaor and the appearance of a torus attractor. (The limit cycle still exists actually, but the bifurcation renders it unstable so that all trajectories are repelled from it.)... 

The torus attractor has associated with it two distinct kinds of behavior quasiperiodic and periodic. These two kinds of behavior are, in turn, associated with two distinct relationships between the two natural periods (or frequencies) in the system. The first frequency corresponds to the now unstable limit cycle... 

Chaos does not occur as long as the torus attraaor is stable. As a parameter of the system is varied, however, this attractor may go through a sequence of transformations that eventually render it unstable and lead to the possibility of chaotic behavior. An early suggestion for how this happens arose in the context of turbulent fluid flow and involved a cascade of Hopf bifurcations, each of which generate additional independent frequencies. Each additional frequency corresponds to an additional dimension in phase space the associated attractors are correspondingly higher dimensional tori so that, for example, two independent frequencies correspond to a two-dimensional torus (7 ), whereas three independent frequencies would correspond to a three-dimensional torus (T ). The Landau theory suggested that a cascade of Hopf bifurcations eventually accumulates at a particular value of the bifurcation parameter, at which point an infinity of modes becomes available to the system this would then correspond to chaos (i.e., turbulence). 

Figure 26 Generation of a torus attractor via two Hopf bifurcations. The first Hopf bifurcation converts a stable fixed point (a focus) into an unstable focus. A stable limit cycle generally originates at this bifurcation point. A second Hopf bifurcation occurs, rendering the limit cycle unstable, and giving rise to a stable torus. Each Hopf bifurcation results in one additional frequency of oscillation in the system.

Figure 27 The torus attractor generated by quasiperiodic trajectories. In (a) a quasiperiodic trajectory is shown making slightly more than one pass around the torus. As can be seen on the front left side of the torus attractor, the quasiperiodic trajectory does not exactly match up with its first pass the eventual result is that the surface of the torus will be covered completely by the quasiperiodic trajectory. TTie Poincare surface of section shown in (b) results in a circular or ellipsoidal cross section (c).

Figure 28 Definition of the angle 6 from the Poincare section of a torus attractor derived from experimental data. The index labels the order with which points appear in this section as the trajectory winds its way over the surface of the torus. This definition can be generalized to a wrinkled or fractal torus.

Figure 33 Evolution of a torus attractor found in simulations with the DOP model of the peroxidase-oxidase reaction, equations [115]. Parameter values used are k2 = 1250, fea = 0.046875, = 1. 104, = 0.001,...

This example shows that mixed-mode oscillations, while arising from a torus attractor that bifurcates to a fractal torus, give rise to chaos via the familiar period-doubling cascade in which the period becomes infinite and the chaotic orbit consists of an infinite number of unstable periodic orbits. Mixedmode oscillations have been found experimentally in the Belousov-Zhabotin-skii (BZ) reaction 2.84 and other chemical oscillators and in electrochemical systems, as well. Studies of a three-variable autocatalator model have also provided insights into the relationship between period-doubling and mixedmode sequences. Whereas experiments on the peroxidase-oxidase reaction have not been carried out to determine whether the route to chaos exemplified by the DOP model occurs experimentally, the DOP simulations exhibit a route to chaos that is probably widespread in the realm of nonlinear systems and is, therefore, quite possible in the peroxidase reaction, as well. 

The reconstructed attractor is shown in figure 3 (for clarity only few trajectories are plotted). Five intersections by planes nearly perpendicular to most of the trajectories (Poincare sections) are depicted. Clearly these sections delimit closed curves the attractor is a torus as e2q>ected for quasiperiodic dynamics. 

A more general case is also considered in [139] concerning the disappearance of a saddle-node torus and followed by the appearance of Anosov attractors and multi-dimensional solenoids. 

In such a case we have the so-called soft loss of stability. The newly established regime inside the attracting spot may be either a new equilibrium state, a periodic trajectory, a non-resonant torus, or even a strange attractor (a situation generally referred as instant chaos). The latter option is possible when Oeo bas three zero eigenvalues (see [18], or [129] for systems with symmetry). 

The first example illustrates one of the most typical bifurcations which occur in dissipative systems namely a stable periodic orbit L adheres to the homoclinic loop of a saddle. Denote the unstable separatrices of the saddle by Fi and F2. Let Fi form a homoclinic loop at the bifurcation point. Denote the limit set of the second separatrix by D(F2). In the general case fI(F2) is an attractor for instance, a stable equilibrium state, a stable periodic trajectory, or a stable torus, etc. Since inunediately after bifurcation a representative point will follow closely along F2, it seems likely that fl(F2) will become its new attractor. 

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