Stable periodic trajectory

As we now change 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)],... 

For Stable periodic trajectories the two eigenvalues are complex numbers conjugate to each other, and the corresponding eigenvectors correspond to a simple rotation around the fixed point qo,po). By contrast, provided that... 

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. 

The phase plane has to give place to the three-dimensional state-space when there are three dimensions. If we have a stable periodic solution such as is shown in Fig. 27, all trajectories in its neighborhood tend toward it, but this is harder to show in three dimensions than it is in two and showing that x(T + t) = x(t) when you do not know T is virtually impossible. Poincare solved this difficulty by setting up a plane transversal to the limit cycle, as in Fig. 28. The limit cycle penetrates this surface at P0 and, if a nearby trajectory penetrates it at a succession of points that, as shown, converge on P0, then... 

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. 

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. 

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. 

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. 

Chaotic solutions are those which are neither periodic nor asymptotic to a periodic solution but are characterized by extreme sensitivity to initial conditions. A solution that is asymptotic to a stable periodic solution is not sensitive to starting point, for, if we start from two nearby values, the trajectories will both converge on the same periodic solution and get closer and closer together. With a chaotic solution, the trajectories starting at two nearby points ultimately diverge no matter how close they may have been at the beginning. If /( )( ) denotes the nth iterate,... 

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... 

The central region of the local mode representation of the phase space trajectories is called the resonance region. In this resonance region of phase space the trajectories ( 3 and 4) are not free to explore the full 0 < ip < n range and are threfore classified as normal mode trajectories. Points A and B are fixed points which he at the maximum and minimum E extremes, Ea(I) and Eb(I), of the resonance region for a particular value of I. Point A at Iz = 0 (vr = vi) and ip = 7t/2 (out-of-phase motion of the R and L oscillators) is stable and corresponds to a pure antisymmetric stretch. Point B at Iz = 0 (vr = vl) and ip = 0 and 7r (in-phase motion) is unstable (because it lies on the separatrix) and corresponds to a pure symmetric stretch. Quasiperiodic trajectories that circulate about a stable fixed point resemble the fixed point periodic trajectory. At E > Ea(I) no trajectories of any type can exist. At E < Er(I) the B-like trajectories vanish and are replaced by trajectories that circulate about the Ca, Cb fixed points and are therefore C-like. The Ca and Cb lines (Iz = / = 2, 0 < ip < 7r) are actually the north and south poles on the local mode polyad phase sphere (Fig. 9.13(c)). The stable fixed points he near the poles and trajectories la and 2a circulate about the fixed point near Ca and trajectories lb... 

Fig. 4.4. Birhythmicity two stable limit cycles coexist for the same parameter values. The curves are obtained by numerical integration of eqns (4.1) for the parameter values of fig. 4.2, with = 1.8 s. The evolution towards either one of the two stable limit cycles depends on initial conditions these differ in the fifth decimal place for a 32.02223 for the upper curve, and 32.02222 for the bottom one, while )3 = 250 and y=0.25 for the two curves. These initial conditions are close to an unstable periodic trajectory that the system transiently follows before settling on one or the other stable cycle (Decroly Goldbeter, 1982).

The main difference between the Hamiltonian and dissipative systems arises from the conservation condition that applies to the former. In Hamiltonian systems, the total energy is fixed. A trajectory with a given initial condition and energy will continue with that same energy for the remainder of the trajectory. In the phase space representation, this will result in a stable trajectory that does not pull in toward an attractor. A periodic trajectory in a Hamiltonian system will have an amplitude and position in the phase space that is determined by the initial conditions. In fact, the phase space representation of a Hamiltonian system often includes many choices of initial conditions in the same phase space portrait. The Poincare section, to be described below, likewise contains many choices of initial conditions in one diagram. 

We observe that the medium tends to the homogeneous bulk oscillation when 0 is small. The smooth curves in Fig.7 A are the wave trajectories for the numerical solution of (1) in the case I = O.O36 and 0 = 0.2. (The bulk reaction dynamics of (1) has a stable periodic oscillation with period Tp = 159 when I = O.O36.) The local disturbance around x = 0 initiates the bulk oscillation in this region which transiently acts as a pacemaker. The leading wave propagates with nearly constant velocity Ca, = 0.57 as it advances into the medium which is lingering for a long time near the (barely) unstable rest state. The next several succeeding waves initially... 

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. 

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... 

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