Dynamic chaos

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... 

The results of the previous section have already established that classical chaos and quantum mechanics are not incompatible in the macroscopic limit. The question then naturally arises whether observed quantum mechanical systems can be chaotic far from the classical limit This question is particularly significant as closed quantum mechanical systems are not chaotic, at least in the conventional sense of dynamical systems theory (R. Kosloff et.al., 1981 1989). In the case of observed systems it has recently been shown, by defining and computing a maximal Lyapunov exponent applicable to quantum trajectories, that the answer is in the affirmative (S. Habib et.al., 1998). Thus, realistic quantum dynamical systems are chaotic in the conventional sense and there is no fundamental conflict between quantum mechanics and the existence of dynamical chaos. 

Keywords Dynamical chaos, kicked rotor, relativistic systems... 

For the past three decades deterministic classical systems with chaotic dynamics have been the subject of extensive study (Chirikov, 1979)-(Sagdeev et. al., 1988). Dynamical chaos is a phenomenon peculiar to the deterministic systems, i.e. the systems whose motion in some state space is completely determined by a given interaction and the initial conditions. Under certain initial conditions the behaviour of these systems is unpredictable. 

Many realistic systems and their models have been considered to study dynamical chaos phenomenon. Such systems as, kicked rotor and various billiard geometries allow one to treat chaotic behavior of deterministic systems successfully. 

It should be noted that there is a limited number of works on classical relativistic dynamical chaos (Chernikov et.al., 1989 Drake and et.al., 1996 Matrasulov, 2001). However, the study of the relativistic systems is important both from fundamental as well as from practical viewpoints. Such systems as electrons accelerating in laser-plasma accelerators (Mora, 1993), heavy and superheavy atoms (Matrasulov, 2001) and many other systems in nuclear and particle physics are essentially relativistic systems which can exhibit chaotic dynamics and need to be treated by taking into account relativistic dynamics. Besides that interaction with magnetic field can also strengthen the role of the relativistic effects since the electron gains additional velocity in a magnetic field. 

Dynamical chaos in periodically driven systems has become attractive topic in many areas of contemporary physics such as atomic, molecular, nuclear and particle physics. Dynamical systems which can exhibit chaotic dynamics can be divided into two classes time independent and time-dependent systems. Billiards, atoms in a constant magnetic field, celestial systems with chaotic dynamics are time independent systems, whose dynamics can be chaotic. 

We have already mentioned the view of Daniels on the confusion in chemical kinetics. Horiuti emphasized this view by using the word "chaos . The experiment discovered a complex kinetic behaviour which is likely to have supported this point of view. But the situation has changed drastically in the last 10 years. New concepts of mathematical physics connected with the study of non-linear systems make us understand complex dynamics ("chaos ) is the result of a certain law. In kinetic Chaos, we can notice Harmony and Hope to see it more clearly. 

Chirikov, B.V. and Shepelyansky, D.L. (1984). Correlation properties of dynamical chaos in Hamiltonian systems, Physica D13, 395-400. 

May, R.M. (1987). Chaos and the dynamics of biological populations, in Dynamical Chaos, eds. M.V. Berry, I.C. Percival and N.O. Weiss (Princeton University Press, Princeton). 

Another characteristic of turbulent flows is unpredictability, that is the high sensitivity of the solution to very small perturbations that are always present in real physical systems or numerical simulations. This unpredictability, also known as dynamical chaos, is a well known feature of much simpler low-dimensional nonlinear dynamical systems. Although in a strict mathematical sense a unique solution of the Navier-Stokes equation always exists for well-posed initial conditions (at least for large finite times), in practice the details of the forcing and boundary conditions are only known within some approximations and thus the solution in the turbulent regime repre-... 

Henri Poincare seems to have been the first to recognize the existence of dynamical chaos, its intrinsic connections with the field of topology, and its importance to physics. However, the importance of his three-volume work on the subject and its implications for planetary motion, Les Methodes Nouvelles... 

A great deal of attention has been focused in recent years by workers in classical dynamics on the geometric properties of phase space structures and their manifestation on Poincare maps (also referred to as surfaces of section). The result has been the blossoming of a huge literature on the subject of nonlinear dynamics (quasiperiodicity and dynamical chaos), which is discussed in a number of recent textbooks and articles. - ... 

As alluded to previously, the reason we discuss dynamics for two-dimensional systems arises from the fact that all one-dimensional conservative Hamiltonians are integrable and therefore do not admit chaotic motion. However, two-dimensional systems are in general anharmonic and nonintegrable, except in certain special cases (a particle in a central field in two dimensions, a two-dimensional normal-mode oscillator, etc.). Thus, Hamiltonians of the type given previously represent the simplest conservative systems that can exhibit dynamical chaos. Note that any function of four variables that is equal to a constant must correspond to a three-dimensional surface embedded in the fourdimensional space. 

Farjoun, M., and Levin, M. 2004. Industry dynamism, chaos theory and the fractal dimension measure. Working paper. New York University Stem School of Business. 

Kuz min, M. V., and Stuchebrukhov, A. A. (1989). Dynamical chaos and intramolecular vibrational relaxation in polyatomic molecules. In Laser spectroscopy of highly vihrationally excited molecules (ed. V. S. Letokhov), pp. 178-264. Adam Hilger,... 

Volume 28 Applied Nonlinear Dynamics Chaos of Mechanical Systems with Discontinuities... 

The multi-dimensional extension of two-dimensional rough systems is the Morse-Smale systems discussed in Sec. 7.4. The list of limit sets of such a system includes equilibrium states and periodic orbits only furthermore, such systems may only have a finite number of them. Morse-Smale systems do not admit homoclinic trajectories. Homoclinic loops to equilibrium states may not exist here because they are non-rough — the intersection of the stable and unstable invariant manifolds of an equilibrium state along a homoclinic loop cannot be transverse. Rough Poincare homoclinic orbits (homoclinics to periodic orbits) may not exist either because they imply the existence of infinitely many periodic orbits. The Morse-Smale systems have properties similar to two-dimensional ones, and it was presumed (before and in the early sixties) that they are dense in the space of all smooth dynamical systems. The discovery of dynamical chaos destroyed this idealistic picture. 

---