Chaos control

R. Femat, J. Capistran-Tobias, and G. Solis-Perales. Laplace domain controllers for chaos control. Phys. Lett. A, 252(1-2) 27-36, 1999. 

The energy of the control functions obtained by these methods varies from 0.14 to 0.6 and thus it is more then one order of magnitude larger then the energy of the optimal control function u(t) found by our new technique, (see Fig. 19). Similar results were obtained using the algorithm for adaptive chaos control [151] for the migration of the nonlinear oscillator from the CA to SC (see Fig. 19). 

Chaos control of spatio-temporal oscillations in resonant tunneling diodes. . 158... 

In order to control the noised-induced patterns, we will now use the method of time-delayed feedback which was previously applied successfully in deterministic chaos control of this particular system [47] as well as for control of noise-induced oscillations in simple models [7-9] without spatial degrees of freedom. 

H. G. Schuster (Editor) Handbook of Chaos Control (Wiley-VCH, Wein-heim, 1999). 

Glanz, J. 1994. Do chaos-control techniques offer hope for epilepsy Science 265 1174. 

It should be noted that chaos control can only be obtained if deterministic chaos is involved. In case of (i) chaotic laser (ii) diode (iii) hydrodynamic and magneto-elastic systems and (iv) more recently myocardial tissue, feedback algorithm has been successfully applied to stabilize periodic oscillations. Quite recently, in order to stabilize periodic behaviour in the chaotic regime of oscillatory B-Z reaction, Showalter [14] and co-workers (1998) applied proportional feedback mechanism. Feedback was applied to the system by perturbing the flow rate of cesium-bromate solutions in the reactor keeping the flow rate of malonic acid fixed in these experiments. This experimental arrangement helped the stabilization of periodic behaviour within the chaotic regime. 

Ditto W L and Showalter K (eds) 1997 Control and synchronization of chaos focus issue Chaos 7 509-687... 

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. 

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

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. 

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. 

To make sense out of this chaos, a method is needed to pick out the rules that will be useful in controlling the system from among the large number that are worthless. The process of identifying productive classifiers relies on a mechanism that provides rewards to those rules that are helpful with a penalty for those that are not. The better rules then gradually emerge from the background noise. 

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. 

McCullough ML, Robertson AS, Chao A, Jacobs EJ, Stampfer MJ, Jacobs DR, Diver WR, Calle EE and Thun MJ. 2003. A prospective study of whole grains, fruits, vegetables and colon cancer risk. Cancer Causes Control 14(10) 959-970. 

Heat conduction in one dimensional systems Fourier law, chaos, and heat control... 

While in the previous sections we have discussed the relation between dynamical chaos and heat conductivity, in the following we will turn our attention to the possibility to control heat flow. Actually a model of thermal rectifier has been recently proposed(Terrano et al, 2002) in... 

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