Oscillators chaotic

We have investigated the transitions among the types of oscillations which occur with the Belousov-Zhabotinskii reaction in a CSTR. There is a sequence of well-defined, reproducible oscillatory states with variations of the residence time [5]. Similar transitions can also occur with variation of some other parameter such as temperature or feed concentration. Most of the oscillations are periodic but chaotic behavior has been observed in three reproducible bands. The chaos is an irregular mixture of the periodic oscillations which bound it e.g., between periodic two peak oscillations and periodic three peak oscillations, chaotic behavior can occur which is an irregular mixture of two and three peaks. More recently Roux, Turner et. al. 

Figures 74 and 75 show oscillations, chaotic behavior and pattern formation observed in Ge under appropriate conditions.

Mathematical models of the reaction yield various solutions. Some of the solutions obtained are One singular point, 3 singular points, oscillating limit cycle, double periodic oscillations, chaotic oscillations. 

An urgent problem of theoretical investigations seems to be tte search for specific unsteady-state solutions oscillating, chaotic, etc., based cki rteokit tic reasoning. One of the possible alternatives for initiation of oscillating states of motion for tlieo-kinetic media the sliding — adhesion transition which was discussed in Ref. [110]. 

Keywords Bjerknes force Bubble breakup Bubble interaction Bubble oscillation Chaotic oscillation Damping rate Droplet oscillation Nonlinear oscillation Oscillation frequency RPNNP equation Shape modes Spherical harmonics Volume oscillation... 

In order to understand the complexity in oscillatory reactions, it would be worthwhile to examine their relationship with different types of chemical reactions [10], which have been summarized in Fig. 9.8 in increasing order of complexity viz., irreversible reactions -> reversible reactions parallel reaction consecutive reaction -> autocatalytic reactions damped oscillations aperiodic oscillations spatio-temporal oscillations chaotic oscillations. Further, Fig. 9.8 shows the concentration... 

The action of electric fields on structured enzyme systems leads to a great variety of behaviors it is possible to obtain regulations of enzyme activity or commutations in addition it is possible to induce oscillatory behaviors in such systems with two chemical parameters direct electric fields are able to induce periodical oscillations, whereas alternating electric fields are able to induce aperiodical oscillations (chaotic oscillations) in wide domains of the different parameters. 

Most chemically reacting systems tliat we encounter are not tliennodynamically controlled since reactions are often carried out under non-equilibrium conditions where flows of matter or energy prevent tire system from relaxing to equilibrium. Almost all biochemical reactions in living systems are of tliis type as are industrial processes carried out in open chemical reactors. In addition, tire transient dynamics of closed systems may occur on long time scales and resemble tire sustained behaviour of systems in non-equilibrium conditions. A reacting system may behave in unusual ways tliere may be more tlian one stable steady state, tire system may oscillate, sometimes witli a complicated pattern of oscillations, or even show chaotic variations of chemical concentrations. 

The existence of chaotic oscillations has been documented in a variety of chemical systems. Some of tire earliest observations of chemical chaos have been on biochemical systems like tire peroxidase-oxidase reaction [12] and on tire well known Belousov-Zhabotinskii (BZ) [13] reaction. The BZ reaction is tire Ce-ion-catalyzed oxidation of citric or malonic acid by bromate ion. Early investigations of the BZ reaction used tire teclmiques of dynamical systems tlieory outlined above to document tire existence of chaos in tliis reaction. Apparent chaos in tire BZ reaction was found by Hudson et a] [14] aiid tire data were analysed by Tomita and Tsuda [15] using a return-map metliod. Chaos was confinned in tire BZ reaction carried out in a CSTR by Roux et a] [16, E7] and by Hudson and... 

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. 

Electrons from a spark are accelerated backward and forward rapidly in the oscillating electromagnetic field and collide with neutral atoms. At atmospheric pressure, the high collision frequency of electrons with atoms induces chaotic electron motion. The electrons gain rapidly in kinetic energy until they have sufficient energy to cause ionization of some gas atoms. 

The steady-state design equations (i.e., Equations (14.1)-(14.3) with the accumulation terms zero) can be solved to find one or more steady states. However, the solution provides no direct information about stability. On the other hand, if a transient solution reaches a steady state, then that steady state is stable and physically achievable from the initial composition used in the calculations. If the same steady state is found for all possible initial compositions, then that steady state is unique and globally stable. This is the usual case for isothermal reactions in a CSTR. Example 14.2 and Problem 14.6 show that isothermal systems can have multiple steady states or may never achieve a steady state, but the chemistry of these examples is contrived. Multiple steady states are more common in nonisothermal reactors, although at least one steady state is usually stable. Systems with stable steady states may oscillate or be chaotic for some initial conditions. Example 14.9 gives an experimentally verified example. 

Steady states may also arise under conditions that are far from equilibrium. If the deviation becomes larger than a critical value, and the system is fed by a steady inflow that keeps the free energy high (and the entropy low), it may become unstable and start to oscillate, or switch chaotically and unpredictably between steady state levels. 

For chaotic or oscillating behavior the mechanism must contain an autocatalytic step ... 

Oscillations such as in Fig. 2.15 are quite regular and can be sustained for hours if the conditions are kept the same. Depending on the feed rate of the reactants, which determines how far the system deviates from equilibrium, the oscillations may become more complex, and develop into chaotic oscillations (see, for example, P.D. Cobden, J. Siera, and B.E. Nieuwenhuys, J. Vac. Sci. Technol. A10 (1992) 2487). 

Model instability is demonstrated by many of the simulation examples and leads to very interesting phenomena, such as multiple steady states, naturally occurring oscillations, and chaotic behaviour. In the case of a model which is inherently unstable, nothing can be done except to completely reformulate the model into a more stable form... 

The principal axis of the cone represents the component of the dipole under the influence of the thermal agitation. The component of the dipole in the cone results from the field that oscillates in its polarization plane. In this way, in the absence of Brownian motion the dipole follows a conical orbit. In fact the direction of the cone changes continuously (because of the Brownian movement) faster than the oscillation of the electric field this leads to chaotic motion. Hence the structuring effect of electric field is always negligible, because of the value of the electric field strength, and even more so for lossy media. 

General conditions for transition to irregular and chaotic behaviour in an oscillator under wave action have been derived using the notion about the Melnikov distance ... 

Under the condition D(ta, t0) = 0 and taking into account that sin(z/f0 + 0 < I and that 6d. > 0, the general condition for transmission to irregular (chaotic) behaviour in nonlinear oscillator under the wave action takes the form 25dsh( v w) < ttu2 Fg ev. ... 

A. Goldbeter, Biochemical Oscillations and Cellular Rhythms The Molecular Bases of Periodic and Chaotic Behaviour, Cambridge University Press, Cambridge, United Kingdom (1997). 

K. Nielsen, P. G. Sprensen, F. Hynne, and H. G. Busse, Sustained oscillations in glycolysis An experimental and theoretical study of chaotic and complex periodic behavior and of quenching of simple oscillations. Biophys. Chem. 72, 49 62 (1998). 

We demonstrate the use of Matlab s numerical integration routines (ODE-solvers) and apply them to a representative collection of interesting mechanisms of increasing complexity, such as an autocatalytic reaction, predator-prey kinetics, oscillating reactions and chaotic systems. This section demonstrates the educational usefulness of data modelling. 

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