Transient chaos

Under some conditions, it is observed that complex oscillatory sequences develop even in batch systems, typically towards the end of the oscillatory phase of the reaction. Transient chaos —see section A3.14.3.3— appears to be established [18]. 

Show numerically that the Lorenz equations can exhibit transient chaos when r = 21 (with <7 = 10 and b = as usual). 

Other names used for transient chaos are metastable chaos (Kaplan and Yorke... 

Transient chaos) Example 9.5.1 shows that the Lorenz system can exhibit transient chaos for r = 21, <7 = 10, b = j. However, not all trajectories behave this way. Using numerical integration, find three different initial conditions for which there is transient chaos, and three others for which there isn t. Give a rule of thumb which predicts whether an initial condition will lead to transient chaos or not. 

Even when (1) has no strange attractors, it can still exhibit complicated dynamics (Moon and Li 1985). For i nstance, consider a regime in which two or more stable limit cycles coexist. Then, as shown in the next example, there can be transient chaos before the system settles down. Furthermore the choice of final state depends sensitively on initial conditions (Grebogi et al. 1983b). 

For F = 0.25, find two nearby trajectories that both exhibit transient chaos before finally converging to different periodic attractors. 

Solution To find suitable initial conditions, we could use trial and error, or we could guess that transient chaos might occur near the ghost of the strange attractor of Figure 12.5.6. For instance, the point (Xq, )= (0.2,0.1) leads to the time series shown in Figure 12.5.8a. 

Transient chaos does not occur if the trajectory starts close enough to... 

Grebogi, C.,Ott,E., and Yorke, J. A. (1983a) Crises, sudden changes in chaotic attractors and transient chaos. Physica DI, 181. 

D. Reznik and E. Scholl Oscillation modes, transient chaos and its control in a modulation-doped semiconductor double-deterostructure, Z. Phys. B 91, 309 (1993). 

Stability, Bifurcations, Limit Cycles Some aspects of this subject involve the solution of nonlinear equations other aspects involve the integration of ordinaiy differential equations apphcations include chaos and fractals as well as unusual operation of some chemical engineering eqmpment. Ref. 176 gives an excellent introduction to the subject and the details needed to apply the methods. Ref. 66 gives more details of the algorithms. A concise survey with some chemical engineering examples is given in Ref. 91. Bifurcation results are closely connected with stabihty of the steady states, which is essentially a transient phenomenon. 

This set of first-order ODEs is easier to solve than the algebraic equations where all the time derivatives are zero. The initial conditions are that a ut = no, bout = bo,... at t = 0. The long-time solution to these ODEs will satisfy Equations (4.1) provided that a steady-state solution exists and is accessible from the assumed initial conditions. There may be no steady state. Recall the chemical oscillators of Chapter 2. Stirred tank reactors can also exhibit oscillations or more complex behavior known as chaos. It is also possible that the reactor has multiple steady states, some of which are unstable. Multiple steady states are fairly common in stirred tank reactors when the reaction exotherm is large. The method of false transients will go to a steady state that is stable but may not be desirable. Stirred tank reactors sometimes have one steady state where there is no reaction and another steady state where the reaction runs away. Think of the reaction A B —> C. The stable steady states may give all A or all C, and a control system is needed to stabilize operation at a middle steady state that gives reasonable amounts of B. This situation arises mainly in nonisothermal systems and is discussed in Chapter 5. 

In such cases, we could try to reduce the high-frequency output noise by suppressing it at the input. So that could be a valid reason to place a small ceramic capacitor at the input of an older-generation switcher IC (i.e., one with a BJT switch). Its primary purpose is then not to ensure that the control does not go into chaos because of switch transient noise, but to reduce the output noise in noise-sensitive applications. 

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. 

As observed in Figs.3.14-4, all patterns generated by the C3-C1, C3-C2 and C2-C1 representations, remind, in one way or another, a butterfly. The latter stands for a basic phenomenon in the chaos model known as the butterfly effect, after the title of a paper by Edward N.Lorenz Can the flap of a butterfly s wing stir up a tornado in Texas An additional point may be summarized as follows, i.e., How come that relatively simple mathematical models create very complicated dynamic behaviors, on the one hand, and how Order, followed by esthetics patterns, may be created by the specific representation of the transient behavior, on the other ... 

Chao, B. T., Transient heat and mass transfer to translating droplet, Trans. ASME, J. Heat Transfer, Vol. 91, No. 2, pp. 273-291, 1969. 

Wang J, Sorensen P G and Hynne F 1994 Transient period doublings, torus oscillations and chaos in a closed chemical system J. Phys. Chem. 98 725-7... 

X values (see fig. 10.17). However, irregular oscillations of cdc2 kinase have been observed in the model, at least in the transient phase. The possibility of chaotic dynamics resulting from the periodic stimulation of the mitotic oscillator by pulses of growth factor remains very hypothetical. The occurrence of chaos in relation to the cell cycle has been discussed by Mackey (1985), and by Lloyd, Lloyd Olsen (1992), who used an abstract model of the mitotic oscillator subjected to forcing by a sinusoidal input with much shorter period. 

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