Chaos parameters

Fig. 4.5. Coexistence between a stable limit cycle and a regime of stable chaos. Parameter values are those of fig. 4.2, with k = 1.99 s . Initial conditions are j8 = 188.8 and y = 0.3367, with a = 29.19988 for the upper curve, and a = 29.19989 for the lower one. As in fig. 4.4, the system evolves along an unstable limit cycle before stabilizing in either of the two stable oscillatory regimes (Decroly Goldbeter, 1982).

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. 

We first examine how chaos arises in tire WR model using tire rate constant k 2 as tire bifurcation parameter. However, another parameter or set of parameters could be used to explore tire behaviour. (Independent variation of p parameters... 

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.

See chapter 4 for a general discussion of period-doubling and chaos in one-control-parameter iterative maps. 

Although (5 varies with temperature, the quantity [<5, — 5] is insensitive to temperature the solubility parameters used in Eq. (70) were therefore treated as constants. Table III gives some of the solubility parameters used by Chao and Seader. For supercritical components, the solubility parameters were back-calculated from binary-mixture data, as was also done by Shair (P2). 

Solubility Parameters and Liquid Molar Volumes in the Chao-Seader Correlation... 

The difficulties encountered in the Chao-Seader correlation can, at least in part, be overcome by the somewhat different formulation recently developed by Chueh (C2, C3). In Chueh s equations, the partial molar volumes in the liquid phase are functions of composition and temperature, as indicated in Section IV further, the unsymmetric convention is used for the normalization of activity coefficients, thereby avoiding all arbitrary extrapolations to find the properties of hypothetical states finally, a flexible two-parameter model is used for describing the effect of composition and temperature on liquid-phase activity coefficients. The flexibility of the model necessarily requires some binary data over a range of composition and temperature to obtain the desired accuracy, especially in the critical region, more binary data are required for Chueh s method than for that of Chao and Seader (Cl). Fortunately, reliable data for high-pressure equilibria are now available for a variety of binary mixtures of nonpolar fluids, mostly hydrocarbons. Chueh s method, therefore, is primarily applicable to equilibrium problems encountered in the petroleum, natural-gas, and related industries. 

Considering the similarity between Figs. 1 and 2, the electrode potential E and the anodic dissolution current J in Fig. 2 correspond to the control parameter ft and the physical variable x in Fig. 1, respectively. Then it can be said that the equilibrium solution of J changes the value from J - 0 to J > 0 at the critical pitting potential pit. Therefore the critical pitting potential corresponds to the bifurcation point. From these points of view, corrosion should be classified as one of the nonequilibrium and nonlinear phenomena in complex systems, similar to other phenomena such as chaos. 

K. K. K. Chao, M. C. Williams 1983, (The ductless siphon A useful test for evaluating dilute polymer solution elonga-tional behavior. Consistency with molecular theory and parameters), J. Rheol. 27 (5), 451 174. 

A theoretical framework for considering how the behavior of dynamical systems change as some parameter of the system is altered. Poincare first apphed the term bifurcation for the splitting of asymptotic states of a dynamical system. A bifurcation is a period-doubling, -quadrupling, etc., that precede the onset of chaos and represent the sudden appearance of a qualitatively different behavior as some parameter is varied. Bifurcations come in four basic varieties flip bifurcations, fold bifurcations, pitchfork bifurcations, and transcritical bifurcations. In principle, bifurcation theory allows one to understand qualitative changes of a system change to, or from, an equilibrium, periodic, or chaotic state. 

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. 

Figure 7. Time evolution of intensity (a) and phase portrait for the fundamental mode for 0 < < 300 (b). Parameters are the same as in Fig. 6b (BCL), but with H — 0.9. Chaos.

The emergence of order and chaos in the system of two oscillators depends on the value of the Kerr coupling constant 2- F°r the fixed parameters of damping... 

The reduction of chaos for 9 = 1.45 is presented in the intensity portraits of Fig. 39. However, as is seen in Fig. 38a, there is a small region (1.68 < 9 < 1.80) where the system behaves orderly in the classical case but the quantum correction leads to chaos. By way of an example for 9=1.75, the classical system, after quantum correction, loses its orderly features and the limit cycle settles into a chaotic trajectory. Generally, Lyapunov analysis shows that the transition from classical chaos to quantum order is very common. For example, this kind of transition appears for 9 = 3.5 where chaos is reduced to periodic motion on a limit cycle. Therefore a global reduction of chaos can be said to take place in the whole region of the parameter 0 < 9 < 7. 

Figure 39. Transition from classical chaos (a) to quantum chaos (b). The parameters are those of Fig. 38 but with fi = 1.4.

Figure 1. Comparison at identical parameter values of experimental and quantum-mechanical values for the microwave field strength for 10% ionization probability as a function of microwave frequency. The field and frequency are classically scaled, u>o = and = q6, where no is the initially excited state. Ionization includes excitation to states with n above nc. The theoretical points are shown as solid triangles. The dashed curve is drawn through the entire experimental data set. Values of no, nc are 64, 114 (filled circles) 68, 114 (crosses) 76, 114 (filled squares) 80, 120 (open squares) 86, 130 (triangles) 94, 130 (pluses) and 98, 130 (diamonds). Multiple theoretical values at the same uq are for different compensating experimental choices of no and a. The dotted curve is the classical chaos border. The solid line is the quantum 10% threshold according to localization theory for the present experimental conditions.

For a given set of parameters the period of concentration oscillations (or its average for a periodic motion) exceeds greatly the period of the correlation motion. For the slow concentration motion not only the period of the standing wave oscillations but also their amplitudes and, consequently, the amplitude in the K (t) oscillations depend on the current concentrations Na(t) and Nb(t). In other words, the oscillations of the reaction rate are modulated by the concentration motion. Respectively, the influence of the time dependence K K(t) upon the concentration dynamics has irregular, aperiodic character. A noise component modulates the autowave component (the standing waves) but the latter, in its turn, due to back-coupling causes transition to new noise trajectories. What we get as a result is aperiodic motion (chaos). The mutual influence of the concentration and correlation motions and vice versa is illustrated in Fig. 8.2, where time developments of both the concentrations and reaction rates are plotted. 

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