Chaos transition

In the case of the kicked rotor we were able to predict the critical perturbation strength by applying the Chirikov overlap criterion. This criterion does two things for us. First, it provides us with an excellent physical picture which explains qualitatively the mechanism of the chaos transition secondly, it provides us with an analytical estimate for the critical field. 

Yorke, E. D.,and Yorke, J. A. (1979) Metastable chaos Transition to sustained chaotic... 

The model outlined here is taken further in chapter 10, which deals with the order-to-chaos transition in atoms, and a classification of the different kinds of transition which can occur for excitation near avoided crossings will be given. 

The beauty of the magnetic field problem is the degree of control over its structure which can be achieved. In particular, increasing a single parameter (the energy or n value) allows the system to be raised to the semiclassical limit. Varying the field strength, on the other hand, allows the point where the order-to-chaos transition occurs to be modified. 

J. Hoffnagle, R.V. DeVoe, L. Reyna, R.G. Brewer Order-chaos transition of two trapped ions. Phys. Rev. Lett. 61, 255 (1988)... 

Ouyang Q and Swinney FI L 1991 Transition to chemical turbulence Chaos 1 411-20... 

Stehle, P. (1994) Order, Chaos, Order The Transition from Classical to Quantum Physics (Oxford University Press, Oxford) pp. 55, 123. 

K. Kassner, C. Misbah, H. Miiller-Krumbhaar. Transition to chaos in directional solidification. Phys Rev Lett 67 1551, 1991. 

Behavior for a > aoo- What happens for a > Qoo The simple answer is that the logistic map exhibits a transition to chaos, with a variety of different attractors for Qoo < a < 4 exhibiting exponential divergence of nearby points. To leave it at that, however, would surely bo a great disservice to the extraordinarily beautiful manner in which this trairsition takes place. 

However intuitive the edge-of-chaos idea appears to be, one shoidd be aware that it has received a fair amount of criticism in recent years. It is not clear, for example, how to even define complexity in more complicated systems like coevolutionary systems, much less imagine a phase transition between diffen ent complexity regimes. Even Langton s sugge.stion that effective computation within the limited domain of cellular automata can take place only in the transition region has been challenged. ... 

Iang90] Langton, C.G., Computation at the edge of chaos phase transitions and emergent computation , Physica, 42D (1990) 12-37. 

The main objective of the Workshop was to bring together people working in areas of Fundamental physics relating to Quantum Field Theory, Finite Temperature Field theory and their applications to problems in particle physics, phase transitions and overlap regions with the areas of Quantum Chaos. The other important area is related to aspects of Non-Linear Dynamics which has been considered with the topic of chaology. The applications of such techniques are to mesoscopic systems, nanostructures, quantum information, particle physics and cosmology. All this forms a very rich area to review critically and then find aspects that still need careful consideration with possible new developments to find appropriate solutions. 

Reichl, L. E. The Transition to Chaos. (Springer-Verlag, New York, 2004). 

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. 

Joyeux, M. (1992), The Transition towards Vibrational Chaos in Triatomic Molecules. A Numerical and Analytical Approach, Chem. Phys. 167, 299. 

The experiments and the simulation of CSTR models have revealed a complex dynamic behavior that can be predicted by the classical Andronov-Poincare-Hopf theory, including limit cycles, multiple limit cycles, quasi-periodic oscillations, transitions to chaotic dynamic and chaotic behavior. Examples of self-oscillation for reacting systems can be found in [4], [17], [18], [22], [23], [29], [30], [32], [33], [36]. The paper of Mankin and Hudson [17] where a CSTR with a simple reaction A B takes place, shows that it is possible to drive the reactor to chaos by perturbing the cooling temperature. In the paper by Perez, Font and Montava [22], it has been shown that a CSTR can be driven to chaos by perturbing the coolant flow rate. It has been also deduced, by means of numerical simulation, that periodic, quasi-periodic and chaotic behaviors can appear. 

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]. 

Some of the main examples of biological rhythms of nonelectrical nature are discussed below, among which are glycolytic oscillations (Section III), oscillations and waves of cytosolic Ca + (Section IV), cAMP oscillations that underlie pulsatile intercellular communication in Dictyostelium amoebae (Section V), circadian rhythms (Section VI), and the cell cycle clock (Section VII). Section VIII is devoted to some recently discovered cellular rhythms. The transition from simple periodic behavior to complex oscillations including bursting and chaos is briefly dealt with in Section IX. Concluding remarks are presented in Section X. 

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