Nonlinear chaos

For the experienced practitioner of atomic physics there appears to be an enigma right at this point. What does nonlinear chaos theory have to do with linear quantum mechanics, so successful in the classification of atomic states and the description of atomic dynamics The answer, interestingly, is the enormous advances in atomic physics itself. Modern day experiments are able to control essentially isolated atoms and molecules to unprecedented precision at very high quantum numbers. Key elements here are the development of atomic beam techniques and the revolutionary effect of lasers. Given the high quantum numbers, Bohr s correspondence principle tells us that atoms are best understood on the basis of classical mechanics. The classical counterpart of most atoms and molecules, however, is chaotic. Hence the importance of understanding chaos in atomic physics. 

Tabor M 1989 Chaos and Integrability in Nonlinear Dynamics An Introduction (New York Wiley)... 

Epstein I R and Pojman J A 1999 Overview nonlinear dynamios related to polymerio systems Chaos 9 255-9... 

Epstein I R and Pojnian J A 1998 An Introduction to Nonlinear Chemical Dynamics Oscillations, Waves, Patterns and Chaos (Oxford Oxford University Press)... 

Several important topics have been omitted in this survey. We have described only a few of the routes by which chaos can arise in chemical systems and have made no attempt to describe in detail the features of the different kinds of chemical strange attractor seen in experiments. A wide variety of chemical patterns have been observed and while the many aspects of the mechanisms for their appearance are understood, some features like nonlinear... 

A mathematician would classify the SCF equations as nonlinear equations. The term nonlinear has different meanings in different branches of mathematics. The branch of mathematics called chaos theory is the study of equations and systems of equations of this type. 

S. H. Strogatz, Nonlinear Dynamics and Chaos With Applications to Phy.dcs, Biology, CJiemistry and Engineering Addison Wesley, Reading (1994). 

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. 

Iaml91j Lam, L. and H.C. Morris, editors. Nonlinear Structures in Physical Systems Pattern Formation, Chaos and Waves, Springer- Verlag (1991). 

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. 

Michael Thompson is currently Editor of the Royal Society s Philosophical Transactions (Series A). He graduated from Cambridge with first class honours in Mechanical Sciences in 1958, and obtained his PhD in 1962 and his ScD in 1977. He was a Fulbright researcher in aeronautics at Stanford University, and joined University College London (UCL) in 1964. He has published four books on instabilities, bifurcations, catastrophe theory and chaos, and was appointed professor at UCL in 1977. Michael Thompson was elected FRS in 1985 and was awarded the Ewing Medal ofthe Institution of Civil Engineers. He was a senior SERC fellow and served on the IMA Council. In 1991 he was appointed director of the Centre for Nonlinear Dynamics. 

Abstract. The vast majority of the literature dealing with quantum dynamics is concerned with linear evolution of the wave function or the density matrix. A complete dynamical description requires a full understanding of the evolution of measured quantum systems, necessary to explain actual experimental results. The dynamics of such systems is intrinsically nonlinear even at the level of distribution functions, both classically as well as quantum mechanically. Aside from being physically more complete, this treatment reveals the existence of dynamical regimes, such as chaos, that have no counterpart in the linear case. Here, we present a short introductory review of some of these aspects, with a few illustrative results and examples. 

The ar tide is organized as follows. We will begin with a discussion of the various possibilities of dynamical description, clarify what is meant by nonlinear quantum dynamics , discuss its connection to nonlinear classical dynamics, and then study two experimentally relevant examples of quantum nonlinearity - (i) the existence of chaos in quantum dynamical systems far from the classical regime, and (ii) real-time quantum feedback control. 

We stress that the chaos identified here is not merely a formal result - even deep in the quantum regime, the Lyapunov exponent can be obtained from measurements on a real system. Quantum predictions of this type can be tested in the near future, e.g., in cavity QED and nanomechanics experiments (H. Mabuch et.al., 2002 2004). Experimentally, one would use the known measurement record to integrate the SME this provides the time evolution of the mean value of the position. From this fiducial trajectory, given the knowledge of the system Hamiltonian, the Lyapunov exponent can be obtained by following the procedure described above. It is important to keep in mind that these results form only a starting point for the further study of nonlinear quantum dynamics and its theoretical and experimental ramifications. 

Damgov, V.N. and P.G. Georgiev Amplitude Spectrum of the Possible Pendulum Motions in the Field of an External. Nonlinear to the Coordinate Periodic Force. Dynamical Systems and Chaos, World Scientific, London, Vol. 2, P. 304 (1995)... 

WIGGINS, S. Introduction to Applied Nonlinear Dynamics and Chaos. Springer-Verlag, New York, Berlin, 1989... 

Sagdeev, R. Z., D. A. Usikov, G. M. Zaslavsky. Nonlinear Physics From the Pendulum to Turbulence and Chaos. Harwood Academic Publishers, New York,... 

Abstract. Quantum chaos at finite-temperature is studied using a simple paradigm, two-dimensional coupled nonlinear oscillator. As an approach for the treatment of the finite-temperature a real-time finite-temperature field theory, thermofield dynamics, is used. It is found that increasing the temperature leads to a smooth transition from Poissonian to Gaussian distribution in nearest neighbor level spacing distribution. 

In this work we give a simple presciption for the treatment of finite-temperature effects in quantum chaos using a well-known paradigm of nonlinear dynamics, nonlinear oscillator. 

Since the first report of oscillation in 1965 (159), a variety of other nonlinear kinetic phenomena have been observed in this reaction, such as bi-stability, bi-rhythmicity, complex oscillations, quasi-periodicity, stochastic resonance, period-adding and period-doubling to chaos. Recently, the details and sub-systems of the PO reaction were surveyed and a critical assessment of earlier experiments was given by Scheeline and co-workers (160). This reaction is beyond the scope of this chapter and therefore, the mechanistic details will not be discussed here. Nevertheless, it is worthwhile to mention that many studies were designed to explore non-linear autoxidation phenomena in less complicated systems with an ultimate goal of understanding the PO reaction better. 

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