Saddle periodic trajectories

The most important characteristic in our test cases, however, is that within the 1/1 and the 2/1 resonance horns the torus will break as FA increases. In all models this happens when the unstable source period 1 that existed within the torus hits the saddle-periodic trajectories that lie on the torus. This occurs through a saddle-node bifurcation in the 1/1 resonance horn [Fig. 8(d)], and through an unstable period doubling in the 2/1 resonance [Fig. 8(c)]. After these bifurcations the basic structure of the torus has collapsed, and we are left only with the stable entrained periodic trajectories. 

Theorem 10.6. Let ReC i < 1. Then for any small e 7 0 the point 0 w = 0) is stable. Moreover the map T possesses a saddle periodic trajectory 01,02,03) of period three which lies apart from the point O over a distance 0 e). One of the two unstable separatrices of each point Oi tends to O, the other unstable separatrix leaves a neighborhood of the origin. The stable separatrices of the periodic trajectory form a boundary of the basin of attraction of the point O see Fig. 10.6.1). 

This is similar to Case 1, but with Li(0) >0. As e —> —0, a saddle periodic trajectory shrinks into a stable point O. Upon moving through e = 0, the equilibrium state becomes a saddle-focus it spawns a two-dimensional unstable invariant manifold (i.e. the boundary Ss is dangerous). 

This is the same as Case 2 but with l > 0. The instability occurs because a period-two saddle periodic trajectory merges with a stable periodic orbit. When e > 0, the latter becomes a saddle so that its unstable manifold is homeomorphic to a Mobius band. 

A homoclinic bifurcation is a composite construction. Its first stage is based on the local stability analysis for determining whether the equilibrirun state is a saddle or a saddle-focus, as well as what the first and second saddle values are, and so on. On top of that, one deals with the evolution of a -limit sets of separatrices as parameters of the system change. A special consideration should also be given to the dimension of the invariant manifolds of saddle periodic trajectories bifurcating from a homoclinic loop. It directly correlates with the ratio of the local expansion versus contraction near the saddle point, i.e. it depends on the signs of the saddle values. 

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. 

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)],... 

Holmes 1983) states that when the above transversal homoclinic intersection occurs, that there is a structurally stable invariant Cantor set like the one for the Horseshoe map. It has also been shown by Holmes (1982) that this invariant set contains a countable, dense set of saddles of all periods, an uncountable set of non-periodic trajectories and a dense orbit. If nothing else is clear from the above, it is at least certain that homoclinic bifurcations for maps are accompanied by some very unusual phase portraits. Even if homoclinic bifurcations are not necessarily accompanied by the formation of stable chaotic attractors, they lend themselves to extremely long chaotic like transients before settling down to a periodic motion. Because there are large numbers of saddles present, their stable manifolds divide up the phase plane into tiny stability regions and extreme sensitivity to perturbations is expected. 

The period of the anti-symmetric stretch periodic trajectory does not correspond, however, to any of the three recurrences we see in Figure 8.4. This is not at all surprising in order to come back to the FC region, which in this case is considerably displaced from the anti-symmetric stretch orbit, the trajectory must necessarily couple to the symmetric stretch mode. If we were to launch the wavepacket at the outer slope of the saddle point, the anti-symmetric stretch periodic orbit would support recurrences by itself without coupling to the symmetric stretch mode. An example is the dissociation of IHI discussed in Section 7.6.2. 

A more vivid characteristics of systems with complex behaviors is the presence of a Poincare homoclinic trajectory, i.e. a trajectory which is biasymptotic to a saddle periodic orbit as t —> oo. The existence of a homoclinic orbit which lies at the transverse intersection of the stable and unstable... 

Let us suppose next that system (7.5.1) has a periodic trajectory L x = d t), of period r. The periodic orbit L is structurally stable if none of its (n — 1) multipliers lies on the unit circle. Recall that the multipliers of L are the eigenvalues of the (n — 1) x (n — 1) matrix A of the linearized Poincare map at the fixed point which is the point of intersection of L with the cross-section. The orbit L is stable (completely unstable) if all of its multipliers lie inside (outside of) the unit circle. Here, the stability of the periodic orbit may be understood in the sense of Lyapunov as well as in the sense of exponential orbital stability. In the case where some multipliers lie inside and the others lie outside of the unit circle, the periodic orbit is of saddle type. 

For example, the following theorem shows that a Morse-Smale system cannot have a homoclinic trajectory to a saddle periodic orbit. 

Theorem 7.11. Let L be a saddle periodic orbits and let P be its homoclinic trajectory along which Wf and intersect transversely. Then, any small neighborhood of L JT contains infinitely many saddle periodic orbits. 

The value 9 is also a modulus of topological equivalence in the case of a three-dimensional fiow which has two saddle periodic orbits such that an unstable manifold of one periodic orbit has a quadratic tangency with a stable manifold of another orbit along a heteroclinic trajectory. 

There are some other occurrences of moduli in structurally unstable three-dimensional systems of codimension-one with simple dynamics. For example, consider a three-dimensional system with a saddle-focus O and a saddle periodic orbit L. Let i 2 = p iu), and A3 be the characteristic roots at O such that /o < 0, cj > 0, A3 > 0, i.e. assume the saddle-focus has type (2,1) let i/ < 1 and I7I > 1 be the multipliers of the orbit L. Let one of the two sepa-ratrices P of O tend to L as t -> +00, i.e. T W[, as shown in Fig. 8.3.2. This condition gives the simplest structural instability. All nearby systems with similar trajectory behavior form a surface B of codimension-one. Belogui [28] had found that the value... 

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.

We can now describe the behavior of trajectories in a small neighborhood of the periodic trajectory L to which the fixed point O of the Poincare map corresponds. In the two-dimensional case the behavior of trajectories is shown in Fig. 10.2.4, and a higher-dimensional case in Fig. 10.2.5. The invariant strongly stable manifold Wff (the imion of the trajectories which start from the points of Wq on the cross-section) partitions a neighborhood of L into a node and a saddle region. In the node region all trajectories wind towards L... 

As for the original map (10.3.1) the fixed point O is asymptotically stable when Ik < 0 and is a saddle when Ik > 0. In the latter case the stable and unstable manifolds of O are the manifolds and, respectively. In terms of the Poincare map of the system of differential equations, the corresponding periodic trajectory L is stable when Ik < 0, or a saddle when Ik > 0. Note that in the saddle case the two-dimensional unstable manifold W L) is, in a neighborhood of the periodic trajectory, a Mobius band. 

Point O is stable the function (2i — SR sin 3(f) is a Lyapunov function for small R. Clearly, the stable separatrices of the saddle points tend to infinity as t —> — 00. Otherwise, they had to tend to a completely unstable periodic trajectory or a completely unstable equilibrium state but there is none. Another possibility is that a stable separatrices of one saddle might coincide with the unstable one of the other saddle thereby forming a separatrix cycle as that shown in Fig. 10.6.2, but with four saddles however this hypothesis contradicts to the negative divergence condition. The unstable separatrices cannot tend to infinity as t -> +cx). To prove this, check that when R is large, V <0 for the function V in (10.6.12). Therefore, all trajectories of the system, as t increases, must get inside some closed curve V == C with C sufficiently large, where they remain forever. The same behavior applies to the separatrices of the saddle. Thus, the only option for the unstable separatrices of the point Oi is that one tends to O and the other to 0 as shown in Fig. 10.6.6. 

Fig. 11.3.7. Scenario of a saddle-node bifurcation of periodic orbits in The stable periodic orbit and the saddle periodic orbit in (a) coalesce at /i = 0 in (b) into a saddle-node periodic orbit, and then vanishes in (c). The unstable manifold of the saddle-node orbit in (c) is homeomorphic to a semi-cylinder. A trajectory following the path in (b) slows down along a transverse direction near the virtual saddle-node periodic orbit (i.e., the ghost of the saddle-node orbit in (b)) so that its local segment is similar to a compressed spring.

In this case the limit of a periodic trajectory as e —0 is a homoclinic cycle r composed of a simple saddle-node equilibrium state and its... 

In this case, the topological limit of the bifurcating periodic motion as —> — 0 contains no equilibrium point, but a periodic trajectory of the saddle-node type which disappears when e < 0. The trajectory is a simple saddle-node in the sense that it has only one multiplier, equal to 1, and the first Lyapunov value is not equal to zero. 

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. 

The point q = p = 0 (or P = Q = 0) is a fixed point of the dynamics in the reactive mode. In the full-dimensional dynamics, it corresponds to all trajectories in which only the motion in the bath modes is excited. These trajectories are characterized by the property that they remain confined to the neighborhood of the saddle point for all time. They correspond to a bound state in the continuum, and thus to the transition state in the sense of Ref. 20. Because it is described by the two independent conditions q = 0 and p = 0, the set of all initial conditions that give rise to trajectories in the transition state forms a manifold of dimension 2/V — 2 in the full 2/V-dimensional phase space. It is called the central manifold of the saddle point. The central manifold is subdivided into level sets of the Hamiltonian in Eq. (5), each of which has dimension 2N — 1. These energy shells are normally hyperbolic invariant manifolds (NHIM) of the dynamical system [88]. Following Ref. 34, we use the term NHIM to refer to these objects. In the special case of the two-dimensional system, every NHIM has dimension one. It reduces to a periodic orbit and reproduces the well-known PODS [20-22]. 

The transition state concept, once understood in static terms only, as the saddle point separating reactants and products, may be fruitfully expanded to encompass the transition region, a landscape in several significant dimensions, one providing space for a family of trajectories and for a significant transition state lifetime. The line between a traditional transition structure and a reactive intermediate thus is blurred The latter has an experimentally definable lifetime comparable to or longer than some of its vibrational periods. 

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