Quasi periodic systems

The next problem to consider is how chaotic attractors evolve from tire steady state or oscillatory behaviour of chemical systems. There are, effectively, an infinite number of routes to chaos [25]. However, only some of tliese have been examined carefully. In tire simplest models tliey depend on a single control or bifurcation parameter. In more complicated models or in experimental systems, variations along a suitable curve in the control parameter space allow at least a partial observation of tliese well known routes. For chemical systems we describe period doubling, mixed-mode oscillations, intennittency, and tire quasi-periodic route to chaos. 

Figure 1,8, for example, plots the probability that a cell has value 1 at time t4-l - labeled Pt+i - versus the probability that a cell had value 1 at time t -labeled Pt - for a particular four dimensional cellular automaton rule. The rule itself is unimportant, as there are many rules that display essentially the same kind of behavior. The point is that while the behavior of this rule is locally featureless - its space-time diagram would look like noise on a television screen - the global density of cells with value 1 jumps around in quasi-periodic fashion. We emphasize that this quasi-periodicity is a global property of the system, and that no evidence for this kind of behavior is apparent in the local dynamics. 

Landau proposed in 1944 that turbulence arises essentially through the emergence of an ever increasing number of quasi-periodic motions resulting from successive bifurcations of the fluid system [landau44]. For small TZ, the fluid motion is, as we have seen, laminar, corresponding to a stable fixed point in phase space. As Ti is... 

At finite velocity kinetic friction behaves quite differently in the sense that the commensurability plays a less significant role. Besides, the system shows rich dynamic properties since Eq (16) may lead to periodic, quasi-periodic, or chaotic solutions, depending on damping coefficient y and interaction strength h. Based on numerical results of an incommensurate case [18,19], we outline a force curve of F in Fig. 23 asafunction ofv, in hopes of gaining a better understanding of dynamic behavior in the F-K model. 

A striking example of the so formed class of kick-excited self-adaptive dynamical phenomena and systems is the model of a pendulum influenced by quasi-periodic short-term actions, as considered in papers (Damgov, 2004) - (Damgov and Trenchev, 1999). 

Since the first report of oscillation in 1965 (159), a variety of other nonlinear kinetic phenomena have been observed in this reaction, such as bi-stability, bi-rhythmicity, complex oscillations, quasi-periodicity, stochastic resonance, period-adding and period-doubling to chaos. Recently, the details and sub-systems of the PO reaction were surveyed and a critical assessment of earlier experiments was given by Scheeline and co-workers (160). This reaction is beyond the scope of this chapter and therefore, the mechanistic details will not be discussed here. Nevertheless, it is worthwhile to mention that many studies were designed to explore non-linear autoxidation phenomena in less complicated systems with an ultimate goal of understanding the PO reaction better. 

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.

The experiments and the simulation of CSTR models have revealed a complex dynamic behavior that can be predicted by the classical Andronov-Poincare-Hopf theory, including limit cycles, multiple limit cycles, quasi-periodic oscillations, transitions to chaotic dynamic and chaotic behavior. Examples of self-oscillation for reacting systems can be found in [4], [17], [18], [22], [23], [29], [30], [32], [33], [36]. The paper of Mankin and Hudson [17] where a CSTR with a simple reaction A B takes place, shows that it is possible to drive the reactor to chaos by perturbing the cooling temperature. In the paper by Perez, Font and Montava [22], it has been shown that a CSTR can be driven to chaos by perturbing the coolant flow rate. It has been also deduced, by means of numerical simulation, that periodic, quasi-periodic and chaotic behaviors can appear. 

A technique for distinguishing between phase-locked and quasi-periodic responses, and which is particularly useful when m and n are large numbers, is that of the stroboscopic map. This is essentially a special case of the Poincare map discussed in the appendix of chapter 5. Instead of taking the whole time series 0p(r), for all t, we ask only for the value of this concentration at the end of each forcing period. Thus at times t = 2kn/a>, with k = 1, 2, etc., we measure the surface concentrations of one of our species. If the system is phase locked on to a closed path with a>/a>0 = m/n, then the stroboscopic map will show the measured values moving in a sequence between m points, as in Fig. 13.12(a). If the system is quasi-periodic, the iterates of 0p will never repeat and, eventually, will draw out a closed cycle (Fig. 13.12(b)) in the... 

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. 

Let us imagine a scenario for which a supercritical Hopf bifurcation occurs as one of the parameters, fi say, is increased. For fi < fi, the stationary state is locally stable. At fi there is a Hopf bifurcation the stationary state loses stability and a stable limit cycle emerges. The limit cycle grows as ft increases above fi. It is quite possible for there to be further bifurcations in the system if we continue to vary fi. With three variables we might expect to have period-doubling sequences or transitions to quasi-periodicity such as those seen with the forced oscillator of the previous section. Such bifurcations, however, will not be signified by any change in the local stability of the stationary state. These are bifurcations from the oscillatory solution, and so we must test the local stability of the limit cycle. We now consider how to do this. 

When the time dependence takes the form of a periodic perturbation of some parameter, we speak of this as periodic forcing.19 The response will obviously not be a steady state, but can be periodic, quasi-periodic, or chaotic. If the response is periodic, it may be with a period that is a multiple of the period of forcing. It is quasi-periodic if the response winds itself onto the cylinder in a helix whose pitch is an irrational multiple of the forcing period, so that it is never quite truly periodic. An example20 of a forced system is the Gray-Scott autocatalator with the feed concentration sinusoidally perturbed ... 

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). 

Beyond its applicability in the simplification of the computations, the stroboscopic representation greatly simplifies the recognition of patterns in the transient response of periodically forced systems. A sustained oscillation appears as a finite number of repeated points, while a quasi-periodic response appears as an invariant circle (see Figs. 3, 4, 6 and 9). 

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. 

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. 

Forced oscillation is a well-known technique for the characterization of linear systems and is referred to as a frequency response method in the process control field. By contrast, the response of nonlinear systems to forcing is much more diverse and not yet fully understood. In nonlinear systems, the forced response can be periodic with a period that is some integer multiple of the forcing period (a subharmonic response), or quasi-periodic (characterized by more than one frequency) or even chaotic, when the time series of the response appears to be random. In addition, abrupt transitions or bifurcations can occur between any of these responses as one or more of the parameters is varied and there can be more than one possible response for a given set of parameters depending on the initial conditions or recent history of the system. 

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