Quasi-periodic Trajectories

The cores of the spiral waves need not be stationary and can move in periodic, quasi-periodic or even chaotic flower trajectories [42, 43]. In addition, spatio-temporal chaos can arise if such spiral waves break up and the spiral wave fragments spawn pairs of new spirals [42, 44]. 

Although the phase space of the nonadiabatic photoisomerization system is largely irregular, Fig. 36A demonstrates that the time evolution of a long trajectory can be characterized by a sequence of a few types of quasi-periodic orbits. The term quasi-periodic refers here to orbits that are close to an unstable periodic orbit and are, over a certain timescale, exactly periodic in the slow torsional mode and approximately periodic in the high-frequency vibrational and electronic degrees of freedom. In Fig. 36B, these orbits are schematically drawn as lines in the adiabatic potential-energy curves Wo and Wi. The first class of quasi-periodic orbits we wish to consider are orbits that predominantly... 

Figure 36, (A) Classical time evolution of the reaction coordinate (p as obtained for a representative trajectory describing nonadiabatic photoisomerization. (B) The vibronic motion of the system can be characterized by various quasi-periodic orbits, which are schematically drawn here as lines in the adiabatic potentials Wo and Wi. (C) As an example of a mixed orbit, the time evolution between t — 2.7 ps and t2 =3.9 ps is shown here in more detail.

If the quotient o>/a>0 is irrational, the path across the toroidal surface will return to a different point on the completion of each cycle. Eventually the trajectory will pass over every point on the surface of the torus without ever forming a closed loop. This is quasi-periodicity , and an example is shown in Fig. 13.11. The corresponding concentration histories do not necessarily give complex waveforms, as can be seen from the figure. However, the period of the oscillations is neither simply that of the natural cycle nor just that of the forcing term, but involves both. 

In between the resonance horns are regions of the parameter plane for which the response is quasi-periodic. Note that it is even possible for the frequencies to have a simple ratio and yet for the system to lie outside the corresponding resonance horn if the amplitude is raised. Figure 13.15 shows two time series for forcing with oj/oj0 = 10/1. At low forcing amplitude, rr = 0.005, we have phase locking and a simple if rather crumpled limit cycle. With rf = 0.01, however, the response is quasi-periodic a few cycles are shown and demonstrate quite well how the trajectory begins to wind around the torus. 

An obvious map to consider is that which takes the state (x(t), y(t) into the state (x(t + r), y(t + t)), where r is the period of the forcing function. If we define xn = x(n t) and y = y(nr), the sequence of points for n = 0,1,2,... functions in this so-called stroboscopic phase plane vis-a-vis periodic solutions much as the trajectories function in the ordinary phase plane vis-a-vis the steady states (Fig. 29). Thus if (x , y ) = (x +1, y +j) and this is not true for any submultiple of r, then we have a solution of period t. A sequence of points that converges on a fixed point shows that the periodic solution represented by the fixed point is stable and conversely. Thus the stability of the periodic responses corresponds to that of the stroboscopic map. A quasi-periodic solution gives a sequence of points that drift around a closed curve known as an invariant circle. The points of the sequence are often joined by a smooth curve to give them more substance, but it must always be remembered that we are dealing with point maps. 

This approach needs modification as soon as multiple attracting periodic trajectories exist for a particular set of operating parameters. A conceptually different modification will be necessary to account for attractors which are not simply periodic. Quasi-periodic solutions, characterized by multiple frequencies, are the first type one should expect these are by no means exotic but occur generally in several periodically forced systems. Deterministic chaotic situations, arising from the system nonlinearities (and not the stochastic responses due to random noise) need not be discarded as intractable (Wolf et al., 1986 Shaw, 1981). 

In many cases ordinary differential equations (ODEs) provide adequate models of chemical reactors. When partial differential equations become necessary, their discretization will again lead to large systems of ODEs. Numerical methods for the location, continuation and stability analysis of periodic and quasi-periodic trajectories of systems of coupled nonlinear ODEs (both autonomous and nonautonomous) are extensively used in this work. We are not concerned with the numerical description of deterministic chaotic trajectories where they occur, we have merely inferred them from bifurcation sequences known to lead to deterministic chaos. Extensive literature, as well as a wide choice of algorithms, is available for the numerical analysis of periodic trajectories (Keller, 1976,1977 Curry, 1979 Doedel, 1981 Seydel, 1981 Schwartz, 1983 Kubicek and Hlavacek, 1983 Aluko and Chang, 1984). 

A quasi-periodic solution to a system of ODEs is characterized by at least two frequencies that are incommensurate (their ratio is an irrational number) (Bohr, 1947 Besicovitch, 1954). Several such frequencies may be present on high-order tori, but for the two-dimensional forced systems we examine, we may have no more than two distinct frequencies (a two-torus, T2). A quasi-periodic solution is typically bora when a pair of complex conjugate FMs of a periodic trajectory leave the unit circle at some angle , where /2ir is irrational. Such a solution is also expected when we periodically perturb an autonomously oscillating system with a frequency incommensurate to its natural frequency. 

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... 

Small-order resonance horns (p, q small) and particularly those with 1 p, <7 4 are comparatively wide and easier to locate computationally through algorithms that will locate the periodic entrained trajectories. These algorithms, however, will be inadequate for a complete analysis of our systems since (at least as FA — 0) periodic trajectories appear in disconnected isolas. The motivation behind the construction of our torus-computing algorithm is to provide a means of study of this two-parameter bifurcation diagram that can continue smoothly both within the resonance horns and in the region of quasi-periodicity that separates or—from another point of view—unites them. 

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. 

Quasi-Newton methods may be used instead of our full Newton iteration. We have used the fast (quadratic) convergence rate of our Newton algorithm as a numerical check to discriminate between periodic and very slowly changing quasi-periodic trajectories the accurate computed elements of the Jacobian in a Newton iteration can be used in stability computations for the located periodic trajectories. There are deficiencies in the use of a full Newton algorithm, such as its sometimes small radius of convergence (Schwartz, 1983). Several other possibilities for continuation methods also exist (Doedel, 1986 Seydel and Hlavacek, 1986). The pseudo-arc length continuation was sufficient for our calculations. 

So, apart from the regular behavior, which is either steady-state, periodic, or quasi-periodic behavior (trajectory on a torus, Figure 3.2), some dynamic systems exhibit chaotic behavior, i.e., trajectories follow complicated aperiodic patterns that resemble randomness. Necessary but not sufficient conditions in order for chaotic behavior to take place in a system described by differential equations are that it must have dimension at least 3, and it must contain nonlinear terms. However, a system of three nonlinear differential equations need not exhibit chaotic behavior. This kind of behavior may not take place at all, and when it does, it usually occurs only for a specific range of the system s control parameters 9. 

In most cases of 0 < d < 4, there are a set of bounded trajectories surrounding the stable fixed point and forming the main quasi-periodic islands. These regular trajectories are bounded by the largest invariant island. Outside the largest island there also exist smaller quasi-periodic islands, forming an invariant set of positive Lebesgue measure in the two-dimensional phase space. Besides, there exists a Cantor-like invariant set of unstable trajectories... 

The 3-phospholene potential energy surface has an energy barrier between isomers with height Ej, = 5083 cm. Results from direct trajectory calculations by De Leon and Marston [23,63] are available for one energy, namely, 5133 cm . The PSS for this energy is shown in Fig. 35. It is seen that although overall the system displays characteristics of chaotic motion, a considerable portion of the PSS supports quasi-periodic motion. 

A Poincare surface of section may be used to identify the chaotic and quasi-periodic regions of phase space for a two-dimensional Hamiltonian. An ensemble of trajectories, chosen to randomly sample the phase space, are calculated and for each trajectory a point is plotted in the (9i,Pi)-plane every time Q2 = 0 for p2 > 0. A quasi-periodic trajectory lies on an invariant curve, while the points are scattered for a chaotic trajectory with no pattern. Figure 44 shows an example for a two-dimensional model for HOCl the HO bond distance is frozen in these calculations [351]. It clearly illustrates how the phase space becomes gradually more chaotic as the energy increases. 

Poincare surfaces of section are difficult to construct and to interpret for Hamiltonians with more than two degrees of freedom and other procedures must be used to identify a trajectory as quasi-periodic or chaotic. One approach is to calculate the power spectrum of a trajectory given by the Fourier analysis [352] according to... 

For quasi-periodic trajectories, like those for the normal-mode Hamiltonian in Eq. (69), I to) consists of a series of lines at the frequencies for the normal modes of vibration. In contrast, a Fourier analysis of a chaotic trajectory results in a multitude of peaks, without identifiable frequencies for particular modes. An inconvenience in this approach is that for a large molecule with many modes, a trajectory may have to be integrated for a long time T to resolve the individual lines in a power spectrum for a quasi-periodic trajectory. Moreover, in the presence of a resonance between different modes, the interpretation of the power spectrum may become misleading. 

The nature of the intramolecular motion may also be identified by studying the way the separation of two trajectories evolves with time [353]. If the motion is regular (quasi-periodic) the separation is linear with time, but exponential if the motion is irregular (chaotic). If the separation is exponential, the rate of the separation — called the Lyapunov characteristic exponent — provides qualitative information concerning the IVR rate for the chaotic trajectories. This type of analysis has been reported, for example, for NO2 [271] and the Cl CHsBr complex [354]. 

The simplest approach [338] to describe a non-ergodic unimolecular system is to assume that the reactant s phase space only consists of quasi-periodic and chaotic trajectories, whose numbers are ATqp and Nch- If a micro-canonical ensemble is prepared at t = 0 and if it is assumed that a restricted micro-canonical ensemble is maintained within the chaotic region, while no trajectory dissociates from the quasi-periodic region, the number of reactant molecules versus time is... 

This model is clearly incomplete, since it does not account for vague tori [355] and the complex Arnold web [357, 358] structure of a multidimensional phase space with both chaotic and quasi-periodic trajectories. However, Eq. (74) does properly describe that, with non-ergodic dynamics, the lifetime distribution will have an initial component that decays faster than the RRKM prediction as found in the simulations by Bunker [323,324] and the more recent study of HCO dissociation [51]. Additionally, there will be a component to the classical rate, which is slower than /srrkm, for example, in the dissociations of NO2 and O3 this component cannot be described by an expression as simple as the one in Eq. (74). 

---