Periodic attractor

For certain parameter values tliis chemical system can exlribit fixed point, periodic or chaotic attractors in tire tliree-dimensional concentration phase space. We consider tire parameter set... 

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. 

At the th period doubling the period of the oscillation is In the limit —> co we arrive at the strange attractor where the time variation of the concentrations is no longer periodic. This is the period-doubling route to chaos. 

Figure C3.6.6 The figure shows tire coordinate, for < 0, of tire family of trajectories intersecting tire Poincare surface at cq = 8.5 as a function of bifurcation parameter k 2- As tire ordinate k 2 decreases, tire first subhannonic cascade is visible between k 2 0.1, tire value of tire first subhannonic bifurcation to k 2 0.083, tire subhannonic limit of tire first cascade. Periodic orbits tliat arise by tire tangent bifurcation mechanism associated witli type-I intennittency (see tire text for references) can also be seen for values of k 2 smaller tlian tliis subhannonic limit. The left side of tire figure ends at k 2 = 0.072, tire value corresponding to tire chaotic attractor shown in figure C3.6.1(a). Otlier regions of chaos can also be seen.

Class cl Small number of attractors with short periods. The number of attractors typically remains very small ( 1 — 3), even for large N. 

Class c3 Small number of attractors with long periods. The number of attractors changes irregularly as N increases, and, at least for some rules, appears to depend on some number-theoretic properties of the size and rule. A singular behavior, for example, frequently occurs near N = 2 — i, where i = 0,1 or — 1, depending on the rule. 

Class c4 Large number of attractors with possibly long periods. Kaneko has found that this class is characterized by an exponentially increasing number of attractors, although the increase is irregular. 

The simplest possible attraetor is a fixed point, for which all trajectories starting from the appropriate basin-of-attraction eventually converge onto a single point. For linear dissipative dynamical systems, fixed-point attractors are in fact the only possible type of attractor. Non-linear systems, on the other hand, harbor a much richer spectrum of attractor-types. For example, in addition to fixed-points, there may exist periodic attractors such as limit cycles for two-dimensional flows or doubly periodic orbits for three-dimensional flows. There is also an intriguing class of attractors that have a very complicated geometric structure called strange attractors [ruelleSO],... 

Universality in Unimodal Maps A seminal work on the 2-symbol dynamics of one-dimensional unimodal mappings due to Metropolis, Stein Stein [metro73]. Specifically, they studied the iterates of various mappings within periodic windows, labeling the attractor sequences by strings of the form RLLRL , where R and L indicate whether f xo) falls to the right or left of xq, respectively. Each periodic sequence therefore corresponds to a unique finite length word made up of R s and L s. 

The typical strategy employed in studying the behavior of nonlinear dissipative dynamical systems consists of first identifying all of the periodic solutions of the system, followed by a detailed characterization of the chaotic motion on the attractors. 

The spatial and temporal dimensions provide a convenient quantitative characterization of the various classes of large time behavior. The homogeneous final states of class cl CA, for example, are characterized by d l = dll = dmeas = dmeas = 0 such states are obviously analogous to limit point attractors in continuous systems. Similarly, the periodic final states of class c2 CA are analogous to limit cycles, although there does not typically exist a unique invariant probability measure on... 

The lower a graph is more interesting. While initially the Poincar6 phase portrait looks the same as before (point E, inset 2c) an interval of hysteresis is observed. The saddle-node bifurcation of the pericxiic solutions occurs off the invariant circle, and a region of two distinct attractors ensues a stable, quasiperiodic one and a stable periodic one (Point F, inset 2d). The boundary of the basins of attraction of these two attractors is the one-dimensional (for the map) stable manifold of the saddle-type periodic solutions, SA and SB. One side of the unstable manifold will get attract to one attractor (SC to the invariant circle) while the other side will approach die other attractor (SD to die periodic solution). 

As we further change the parameter R, the hysteresis interval ends (the invariant circle stops existing) and the only attractor is the stable periodic frequency locked solution N. Both sides of the unstable manifold of the sad e-type frequency locked solution are attracted to N (Point G, inset 2e). 

A class of kick-excited self-adaptive dynamical systems is formed and proposed. The class is characterized by a nonlinear (inhomogeneous) external periodic excitation (as regards the coordinates of the excited system) and is remarkable for the occurrence of the following objective regularities the phenomenon of discrete oscillation excitation in macro-dynamical systems having multiple branch attractors and strong self-adaptive stability. 

In the present chapter, steady state, self-oscillating and chaotic behavior of an exothermic CSTR without control and with PI control is considered. The mathematical models have been explained in part one, so it is possible to use a simplified model and a more complex model taking into account the presence of inert. When the reactor works without any control system, and with a simple first order irreversible reaction, it will be shown that there are intervals of the inlet flow temperature and concentration from which a small region or lobe can appears. This lobe is not a basin of attraction or a strange attractor. It represents a zone in the parameters-plane inlet stream flow temperature-concentration where the reactor has self-oscillating behavior, without any periodic external disturbance. 

It is important to remark that this behavior is similar to that previously considered by Eqs.(9), when two external periodic disturbances are applied. Nevertheless, this behavior can be very difficult to obtain, because the lobe of Figure 8 is small. Figures 10 and 11 shows chaotic oscillations and a new strange attractor. By simulation it is possible to obtain plots similar to those in Figures 2, 4, 5 and 6. 

Figure 5 shows the 3-dimensional reconstructed attractors and their projections on canonical planes. The reconstructed phase portraits do not exhibit a defined structure, i.e., it is not toroidal or periodic. As matter of fact, the oscillatory structure is only observed in the Poincare map. The Poincare map is often used to observe the oscillatory structure in dynamical systems. The... 

Fig. 6. Poincare maps. The section was chosen I (z) = zs = 0 and the crosses indicate no periodic oscillation. Once again, the smallest attractor corresponds to experiment E2.b. zi,Z2,zs are also dimensionless.

Essentially, MLE is a measure on time-evolution of the distance between orbits in an attractor. When the dynamics are chaotic, a positive MLE occurs which quantifies the rate of separation of neighboring (initial) states and give the period of time where predictions are possible. Due to the uncertain nature of experimental data, positive MLE is not sufficient to conclude the existence of chaotic behavior in experimental systems. However, it can be seen as a good evidence. In [50] an algorithm to compute the MLE form time series was proposed. Many authors have made improvements to the Wolf et al. s algorithm (see for instance [38]). However, in this work we use the original algorithm to compute the MLE values. 

Controlled chaos may also factor into the generation of rhythmic behavior in living systems. A recently proposed modeL describes the central circadian oscillator as a chaotic attractor. Limit cycle mechanisms have been previously offered to explain circadian clocks and related phenomena, but they are limited to a single stable periodic behavior. In contrast, a chaotic attractor can generate rich dynamic behavior. Attractive features of such a model include versatility of period selection as well as use of control elements of the type already well known for metabolic circuitry. 

The second simple particular case on the way to general case is a reaction network with components A, ..., A whose auxiliary discrete dynamical system has one attractor, a cycle with period t > 1 A +i A - +x. ., A ... 

In general case, let the system have several attractors that are not fixed points, but cycles Ci, C2,... with periods ti, T2,... >1. By gluing these cycles in points, we transform the reaction network if into if. The dynamical system is transformed into Eor these new system and network, the connection... 

The case of a frequency mismatch between laser pumps and cavity modes was investigated [83], and for the first time, chaos in SHG was found. When the pump intensity is increased, we observe a period doubling route to chaos for Ai = 2 = 1. Now, for/i = 5.5, Eq. (3) give aperiodic solutions and we have a chaotic evolution in intensities (Fig. 5a) and a chaotic attractor in phase plane (Imaj, Reai) (Fig. 5b). 

The frequency of modulation il is now the main parameter, and we are able to switch the system of SHG between different dynamics by changing the value of il. To find the regions of where a chaotic motion occurs, we calculate a Lyapunov spectrum versus the knob parameter il. The first Lyapunov exponent A,j from the spectrum is of the greatest importance its sign determines the chaos occurrence. The maximal Lyapunov exponent Xj as a function of is presented for GCL in Fig. 6a and for BCL in Fig. 6b. We see that for some frequencies il the system behaves chaotically (A-i > 0) but orderly Ck < 0) for others. The system in the second case is much more damped than in the first case and consequently much more stable. By way of example, for = 0.9 the system of SHG becomes chaotic as illustrated in Fig. 7a, showing the evolution of second-harmonic and fundamental mode intensities. The phase point of the fundamental mode draws a chaotic attractor as seen in the phase portrait (Fig. 7b). However, the phase point loses its chaotic features and settles into a symmetric limit cycle if we change the frequency to = 1.1 as shown in Fig. 8b, while Fig. 8a shows a seven-period oscillation in intensities. To avoid transient effects, the evolution is plotted for 450 < < 500. 

Fig. 13.11. A quasi-periodic trace with co/co0 1.7 the attractor in the phase space (inset) has...

Fig. 13.20. The variation in the periodicity of the attractor wth the Newtonian cooling time rN for the consecutive exothermic reaction model in a CSTR. (Adapted and reprinted with permission from Jorgensen, D. V. and Aris, R. (1983). Chem. Eng. Sci., 38, 45-53.)...

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). 

Sincic and Bailey (1977) relaxed the assumption of only one stable attractor for a given set of operating conditions and showed examples of some possible exotic responses in a CSTR with periodically forced coolant temperature. They also probed the way in which multiple steady states or sustained oscillations in the dynamics of the unforced system affect its response to periodic forcing. Several theoretical and experimental papers have since extended these ideas (Hamer and Cormack, 1978 Cutlip, 1979 Stephanopoulos et al., 1979 Hegedus et al., 1980 Abdul-Kareem et al., 1980 Bennett, 1982 Goodman et al., 1981, 1982 Cutlip et al., 1983 Taylor and Geiseler, 1986 Mankin and Hudson, 1984 Kevrekidis et al., 1984). 

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