Orbitally stable periodic trajectory

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. 

It is sometimes very usefiil to look at a trajectory such as the synnnetric or antisynnnetric stretch of figure Al.2.5 and figure A1.2.6 not in the physical spatial coordinates (r. . r y), but in the phase space of Hamiltonian mechanics [16, 29], which in addition to the coordinates (r. . r ) also has as additional coordinates the set of conjugate momenta. . pj. ). In phase space, a one-diniensional trajectory such as the aiitisymmetric stretch again appears as a one-diniensional curve, but now the curve closes on itself Such a trajectory is referred to in nonlinear dynamics as a periodic orbit [29]. One says that the aihiamionic nonnal modes of Moser and Weinstein are stable periodic orbits. 

In order to identify the periodic orbits (POs) of the problem, we need to extract the periodic points (or fixed points) from the Poincare map. Adopting the energy F = 0.65 eV, Fig. 31 displays the periodic points associated with some representative POs of the mapped two-state system. The properties of the orbits are collected in Table VI. The orbits are labeled by a Roman numeral that indicates how often trajectory intersects the surfaces of section during a cycle of the periodic orbit. For example, the two orbits that intersect only a single time are labeled la and lb and are referred to as orbits of period 1. The corresponding periodic points are located on the p = 0 axis at x = 3.330 and x = —2.725, respectively. Generally speaking, most of the short POs are stable and located in... 

Upon convergence, the eigenvalues of dF/dx (the characteristic or Floquet multipliers FMt) are independent of the particular point on the limit cycle (i.e. the particular Poincare section or anchor equation used). One of them, FMn, is constrained to be unity (Iooss and Joseph, 1980) and this may be used as a numerical check of the computed periodic trajectory the remaining FMs determine the stability of the periodic orbit, which is stable if and only if they lie in the unit circle in the complex plane ( FM, < 1,1 i = n - 1). The multiplier with the largest absolute value is usually called the principal FM (PFM). When (as a parameter varies) the PFM crosses the unit circle, the periodic orbit loses stability and a bifurcation occurs. 

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. 

Further insights into reaction dynamics can be obtained by analyzing the stability of classical trajectories. Presumably, stable periodic orbits will be restricted to KAM tori and therefore be nonreactive and unstable periodic orbits will provide information about the location of resonances and therefore some quahtative features of the intramolecular energy flow. 

Proof of Theorem 5.3. Condition (3.4) makes E locally asymptotically stable. By the Poincare-Bendixson theorem, it is necessary only to show that with condition (3.4) there are no limit cycles. Suppose there were a limit cycle. However, there is at most a finite number of limit cycles and each must contain < in its interior. Hence there is a periodic trajectory V that contains no other periodic trajectory in its interior. Intuitively speaking, r is the trajectory closest to the rest point. The constant term in the formula given in Lemma 5.1 is negative. The corollary shows that P is asymptotically stable. This is a contradiction, since the rest point is asymptotically stable - that is, between the two there must be an unstable periodic orbit. ... 

Proof. Let 7 = (a (/), (/), 0) be the orbitally asymptotically stable periodic orbit of period T given by Theorem 5.4. (We have already noted that if there are several orbits then one must be asymptotically stable, by our assumption of hyperbolicity.) Let the Floquet multipliers of 7, viewed as a solution of (3.1), be 1 and p, where 0periodic orbit, define p( 3) by... 

There is a regular closed stable trajectory near a shear rate 7 = 0.8 which must have no positive Lyapunov exponents. For this window of periodic orbits the period varies smoothly with 7 and the results are summarised in the Table(l). The periodic orbit is similar to that for 7 = 1.0 shown in Fig. (5) but with many less loops associated with wagging s. 

Quasiperiodic trajectories are a special case of Poisson-stable trajectories. The latter plays one of the leading roles in the theory of dynamical systems as they form a large class of center motions in the sense of Birkhoff (Sec. 7.2). Birkhoff had partitioned the Poisson-stable trajectories into a number of subclasses. This classification is schematically presented in Sec. 7.3. Having chosen this scheme as his base, as early as in the thirties, Andronov had undertaken an attempt to collect and correlate all known types of dynamical motions with those observable from physical experiments. Since his arguments were based on the notion of stability in the sense of Lyapunov for an individual trajectory, Andronov had soon come to the conclusion that all possible Lyapunov-stable trajectories are exhausted by equilibrium states, periodic orbits and almost-periodic trajectories (these are quasiperiodic and limit-quasiperiodic motions in the finite-dimensional case). 

In the preceding sections, we have discussed the set of center motions. In essence, we have found that it is the closure of the set of Poisson-stable trajectories. It does not exclude the case where the latter ones may simply be periodic orbits. But if there is a single Poisson-stable unclosed trajectory, then by virtue of Birkhoff s theorem in Sec. 1.2, there is a continuum of Poisson-stable trajectories. As for the rest of the trajectories in the center, it is known that the set of points which are not Poisson-stable is the union of not more... 

So, one can see that the original system with a Poisson-stable imclosed trajectory will possess infinitely many periodic orbits (p t, xjb), where (/ (0, Xk) = Xk k = 1, 2,...) with periods r, such that Xk xq and Tk +oo as A -> +oo. 

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. 

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.

If li > 0, then for /x < 0, there exists a stable fixed point O at the origin as well as an unstable orbit of period two bounding the attraction basin of O at /i = 0, the period two orbit merges into O the latter becomes unstable and all trajectories, except O, leave a neighborhood of the origin for /x > 0, see Fig. 11.4.2. 

Theorem 11.1. If the first Lyapunov value Li in (11.5.3) is negative, then for small /i < 0, the equilibrium state O is stable and all trajectories in some neighborhood U of the origin tend to O. When fx > 0, the equilibrium state becomes unstable and a stable periodic orbit of diameter y/Ji emerges see Fig. 11.5.1) such that all trajectories from U, excepting O, tend to it. 

Let us examine next the bifurcations of the system (11.5.1) in the multidimensional case. If Li < 0 (Fig. 11.5.4), then when // < 0, the equilibrium state O is stable (rough focus when p < 0, and a weak focus aX p = 0) and it attracts all trajectories in a small neighborhood of the origin. When > 0 the point O becomes a saddle-focus with a two-dimensional unstable manifold and an m-dimensional stable manifold. The edge of the unstable manifold is the stable periodic orbit which now attracts all trajectories, except those in the stable manifold of O. One multiplier of the periodic orbit was calculated in Theorem 11.1, this is po p) = 1 — 47r /a (0) -h o p). To find the others we... 

Inside the resonant wedge all trajectories on the invariant torus tend to a stable periodic orbit, which means that the dominating regimes here is a periodic one. Outside the wedge either a quasiperiodic regime or a periodic one of a very long period is established on the torus. Both are practically indistinguishable. Therefore, a transition over the boundary of a resonant... 

Since the contraction in the local map can be made arbitrarily strong and the derivative of the global map is bounded, the superposition T = To oTi inherits the contraction of the local map for all small p as well. It then follows from the Banach principle of contracting mappings (Sec. 3,15) that the map T has a unique stable fixed point on So- As this is a map defined along the trajectories of the system, it follows that the system has a stable periodic orbit in V which attracts all trajectories in V. The period of this orbit is the sum of two times the dwelling time t of local transition from Sq to S and the flight time from Si to Sq. The latter is always finite for all small p. It now follows from (12.1.4) that the period of the stable periodic orbit increases asymptotically of order tt/x/a This completes the proof. 

Theorem 12.9. Consider a one-parameter family of dynamical systems which has a saddle-node periodic orbit L at = 0 such that all orbits in the global unstable set tend to L as t -foo, but do not lie in W[. Let the essential map satisfy m = 0 and fo (p) < 1 for all (p. Then after disappearance of the saddle-node for /i > 0, the system has a stable periodic orbit non-homotopic to L in U) which is the only attractor for all trajectories in U. 

Starting with any (x, y), a trajectory of system (12.4.8) converges typically to an attractor of the fast system corresponding to the chosen value of x. This attractor may be a stable equilibrium, or a stable periodic orbit, or of a less trivial structure — we do not explore this last possibility here. When an equilibrium state or a periodic orbit of the fast system is structurally stable, it depends smoothly on x. Thus, we obtain smooth attractive invariant manifolds of system (12.4.8) equilibrium states of the fast system form curves Meq and the periodic orbits form two-dimensional cylinders Mpo, as shown in Fig. 12.4.6. Locally, near each structurally stable fast equilibrium point, or periodic orbit, such a manifold is a center manifold with respect to system (12.4.8). Since the center manifold exists in any nearby system (see Chap. 5), it follows that the smooth attractive invariant manifolds Meqfe) and Mpo( ) exist for all small e in the system (12.4.7) [48]. 

When the system has a symmetry such that 0 is symmetric with respect to O2 and the separatrix Fi is symmetric with respect to the separatrix F2, then the bifurcations are rather simple a stable periodic orbit is born when the separatrices split inwards when the separatrices split outwards, all trajectories (next to 0x 2) leave U. 

Definition 14.2. A point eo on the stability boundary of a periodic trajectory Le is said to be safe if L q is asymptotically orbitally stable. 

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. 

The classical counterpart of resonances is periodic orbits [91, 95, 96, 97 and 98]. For example, a purely classical study of the H+H2 collinear potential surface reveals that near the transition state for the H+H2 H2+H reaction there are several trajectories (in R and r) that are periodic. These trajectories are not stable but they nevertheless affect strongly tire quantum dynamics. A study of tlie resonances in H+H2 scattering as well as many other triatomic systems (see, e.g., [99]) reveals that the scattering peaks are closely related to tlie frequencies of the periodic orbits and the resonance wavefiinctions are large in the regions of space where the periodic orbits reside. 

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