Saddle cycle

The need to be able to control chaos has attracted considerable attention. Methods already available include a variety of minimal forms of interaction [150-155] and methods of strong control [156,157] that necessarily require a large modification of the system s dynamics, for at least a limited period of time. For example, in Refs. 158 and 159, the procedure of controlling chaos by means of minimal forms of interaction (saddle cycle stabilization) is realized for different laser systems. 

A simplified parameter space diagram obtained numerically [168] is shown in Fig. 13. The dashed lines bound the region in which both the linear and nonlinear responses of period 1 coexist. The upper line marks the boundary of the linear response, and the lower line marks that for the nonlinear responses. The boundaries of hysteresis for the period 1 resonance are shown by solid lines. The region in which linear response coexists with one or two nonlinear responses of period 2 is bounded by dotted lines. This region is similar to the one bounded by dashed lines. The region of coexistence of the two resonances of period 2 is bounded by the dashed-dotted line. Chaotic states are indicated by small dots. The chaotic state appears as the result of period-doubling bifurcations, and thus corresponds to a nonhyperbolic attractor [167]. Its boundary of attraction Sfl is nonfractal and is formed by the unstable manifold of the saddle cycle of period 1 (SI). 

Figure 14. The basins of attraction of the SC (shaded) and CA (white) for a Poincare cross section with

In the presence of weak noise there is a finite probability of noise-induced transitions between the chaotic attractor and the stable limit cycle. In Fig. 14 the filled circles show the intersections of one of the real escape trajectories with the given Poincare section. The following intuitive escape scenario can be expected in the Hamiltonian formalism. Let us consider first the escape of the system from the basin of attraction of a stable limit cycle that is bounded by an saddle cycle. In general, escape occurs along a single optimal trajectory qovt(t) connecting the two limit cycles. 

The trajectory q pl (t) is determined by minimizing S in (20) on the set of all classical deterministic trajectories determined by the Hamiltonian H (37), that start on a stable limit cycle as t — —oo and terminate on a saddle cycle as t > oo. That is, qopt(t) is a heteroclinic trajectory of the system (37) with minimum action, where the minimum is understood in the sense indicated, and the escape probability assumes the form P exp( S/D). We note that the existence of optimal escape trajectories and the validity of the Hamiltonian formalism have been confirmed experimentally for a number of nonchaotic systems (see Refs. 62, 95, 112, 132, and 172 and references cited therein). 

Since the basin of attraction of the CA is bounded by the saddle cycle SI, the situation near SI remains qualitatively the same and the escape trajectory remains unique in this region. However, the situation is different near the chaotic attractor. In this region it is virtually impossible to analyze the Hamiltonian flux of the auxiliary system (37), and no predictions have been made about the character of the distribution of the optimal trajectories near the CA. The simplest scenario is that an optimal trajectory approaching (in reversed time) the boundary of a chaotic attractor is smeared into a cometary tail and is lost, merging with the boundary of the attractor. 

To find the boundary conditions on the CA, we analyze the prehistory probability distribution Ph(q, t qf, f/) of the escape trajectories. The corresponding distribution is shown in the Fig. 16. It can be inferred by the inspection of how the ridge of the most probable escape path merges the CA that most of the escape trajectories pass close to the saddle cycle of the period 5 embedded into the CA. 

Figure 17. Escape trajectories for the parameters as in Fig. 16. The squares and circles show one period of the saddle cycle S3 and one period of S5, respectively.

This hypothesis can be elaborated further using a statistical analysis of the trajectories arriving a small tube around S3 with the noise intensity reduced by a few orders of magnitude up to D = 1.5 x 10 6, see Fig. 17 [173]. The analysis reveals that the energetically favorable way to move the system from the CA to the stable limit cycle starts at the saddle cycle of period 5 (S5) embedded in the CA, passes through saddle cycle S3 and finishes at the saddle cycle SI at the boundary of the basin of attraction of the CA. Subsequent motion of the system towards the stable limit cycle does not require external action. 

It can be seen from the figure that the optimal force switches on at the moment when the system leaves S5 along its unstable manifold. The optimal force returns to zero when the system reaches the saddle cycle SI. 

Thus we conclude that the solution u t) and the corresponding boundary conditions can be found using our new experimental method. Moreover the problem of escape from the CA of a periodically driven nonlinear oscillator can essentially be reduced to the analysis of a transition between three saddle cycles... 

There are also two local bifurcations. The first one takes place for r 13.926..., when a homoclinic tangency of separatrixes of the origin O occurs (it is not shown in Fig. 20) and a hyperbolic set appears, which consists of a infinite number of saddle cycles. Beside the hyperbolic set, there are two saddle cycles, L and L2, around the stable states, Pi and P2. The separatrices of the origin O reach the saddle cycles Li and L2, and the attractors of the system are the states Pi and P2. The second local bifurcation is observed for r 24.06. The separatrices do not any longer reach to the saddle cycles L and L2. As a result, in the phase space of the system a stable quasihyperbolic state appears— the Lorenz attractor. The chaotic Lorenz attractor includes separatrices, the saddle point O and a hyperbolic set, which appears as a result of homoclinic tangency of the separatrices. The presence of the saddle point in the chaotic... 

The saddle cycles L and L2 surround the stable states P and P2 and are located at the intersection of the unstable Wu and stable Ws manifolds. The unstable manifold goes to the stable state P from one side and to the chaotic attractor from the other side. The stable manifold Ws forms a tube in the vicinity of the stable state [183]. The saddle cycles L and L2 have the multipliers (1.0000,1.0280,0.0001), and therefore trajectories will go slowly away along the unstable manifold, and they will approach quickly along the stable manifold. 

It is clear that all of the escape trajectory from the Lorenz attractor lies on the attractor itself. The role of the fluctuations is, first, to bring the trajectory to a seldom-visited area in the neighborhood of the saddle cycle L, and then to induce a crossing of the cycle L. So we may conclude that the role of the fluctuations is different in this case, and the possibility of applying the Hamiltonian formalism will require a more detailed analysis of the crossing process. 

Thus, we have found that the mechanisms of escape from a nonhyperbolic attractor and a quasihyperbolic (Lorenz) attractor are quite different, and that the prehistory of the escape trajectories reflects the different structure of their chaotic attractors. The escape process for the nonhyperbolic attractor is realized via several steps, which include transitions between low-period saddle-cycles coexisting in the system phase space. The escape from the Lorenz attractor consist of two qualitatively different stages the first is defined by the stable and unstable manifolds of the saddle center point, and lies on the attractor the second is the escape itself, crossing the saddle boundary cycle surrounding the stable point attractor. Finally, we should like to point out that our main results were obtained via an experimental definition of optimal paths, confirming our experimental approach as a powerful instrument for investigating noise-induced escape from complex attractors. 

The fixed point is stable. It is encircled by a saddle cycle, a new type of unstable limit cycle that is possibleonly in phase spaces of three ormore dimensions. Thecycle has... 

Fig. 8.3.2. A structurally unstable heteroclinic trajectory connecting a saddle-focus and a saddle periodic orbit with positive multipliers, i.e. both manifolds of the saddle cycle are homeomorphic to a cylinder.

Hence as the parameter approaches a strong resonance, the saddle cycle shrinks continuously to the point O from the outside of a neighborhood of the origin. At the precise moment of resonance it collapses into O so that the latter becomes unstable. Upon passing through the resonance the cycle distances anew from the point O and as e is further changed the cycle leaves the (small) neighborhood of O. The case where ReC i > 0 is identical but applied to the inverse map Tf in this case the point O is completely unstable for 7 0. 

Fig. 11.4.4. The case l > 0. The stable node (—) at the origin is surrounded by a period-two saddle cycle (+, +). When the cycle collapses at the origin, the latter becomes unstable.

---