Chaotic problems

A particular class of problems whose solutions are sensitive to initial conditions is known as chaotic problems. The phenomenon of chaos has been observed in fluid mechanics, chemical reactions, and biological systems (Cvitanocic, 1987). A special feature of these systems is unpredictability. The chaotic solutions are so sensitive to initial conditions that two systems with minute differences in their initial states can eventually diverge from each other. Thus their long-term dynamics are unpredictable. 

Similarly, divergence will also occur if we have an infinitely precise computer to solve the chaotic problem, but the balloon experiences an unaccounted-for, infinitesimal fluctuation between finite time steps of the trajeaory calculation. The accumulation of errors resulting from the extreme sensitivity of a chaotic trajectory to its instantaneous environment was called the butterfly effect by Lorenz. The butterfly effect arises from the fact that the balloon s trajectory is dynamically unstable, which means that it is so sensitive to changes in its instantaneous environment that the perturbations caused by the fluttering of a butterfly s wing thousands of miles away are sufficient to cause the trajectory of the balloon to change from what it would otherwise have been. 

Even with these complications due to anliannonicity, tlie vibrating diatomic molecule is a relatively simple mechanical system. In polyatomics, the problem is fiindamentally more complicated with the presence of more than two atoms. The anliannonicity leads to many extremely interestmg effects in tlie internal molecular motion, including the possibility of chaotic dynamics. 

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. 

The Hemian-Kluk method has been developed further [153-155], and used in a number of applications [156-159]. Despite the formal accuracy of the approach, it has difficulties, especially if chaotic regions of phase space are present. It also needs many trajectories to converge, and the initial integration is time consuming for large systems. Despite these problems, the frozen Gaussian approximation is the basis of the spawning method that has been applied to... 

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. 

H. Waalkens, A. Burbanks, and S. Wiggins, A computational procedure to detect a new type of high-dimensional chaotic saddle and its application to the 3D Hill s problem, J. Phys. A 37, L257 (2004). 

In fact, with the help of Krein s trace formula, the quantum field theory calculation is mapped onto a quantum mechanical billiard problem of a point-particle scattered off a finite number of non-overlapping spheres or disks i.e. classically hyperbolic (or even chaotic) scattering systems. 

In this paper we consider the QCD counterpart of this problem. Namely, we address the problem of regular and chaotic motion in periodically driven quarkonium. Using resonance analysis based on the Chirikov criterion of stochasticity we estimate critical values of the external field strength at which quarkonium motion enters into chaotic regime. 

More recently, the problem of self-oscillation and chaotic behavior of a CSTR with a control system has been considered in others papers and books [2], [3], [8], [9], [13], [14], [20], [21], [27]. In the previously cited papers, the control strategy varies from simple PID to robust asymptotic stabilization. In these papers, the transition from self-oscillating to chaotic behavior is investigated, showing that there are different routes to chaos from period doubling to the existence of a Shilnikov homoclinic orbit [25], [26]. It is interesting to remark that in an uncontrolled CSTR with a simple irreversible reaction A B it does not appear any homoclinic orbit with a saddle point. Consequently, Melnikov method cannot be applied to corroborate the existence of chaotic dynamic [34]. 

This approach is of course limited by the availability of reliable data and the resolution of the data. An inherent problem in the np-scaling process is the interaction between variance in input parameters and non-linearity in models. This may prodnce chaotic behaviour, van Bodegom et al. (2002) discuss this in relation to CH4 emission from rice. The point at which inpnt data are averaged before making model runs may also be limited by the available computing... 

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