Volume 1, Issue 1, July 1991
Index of content:
1(1991); http://dx.doi.org/10.1063/1.165811View Description Hide Description
The conditions for the appearance of a stochastic web in degenerate dynamic systems and typical physical problems that lead to such a web are analyzed. Examples of webs are considered, as well as their symmetry, width, and structural changes. A description is given of a change in the diffusion dynamics along the web channels as a function of the number of the degrees of freedom and the phenomenon of stochastic percolation is discussed.
1(1991); http://dx.doi.org/10.1063/1.165810View Description Hide Description
The periodic forcing of nonlinear oscillations can often be cast as a problem involving self‐maps of the circle. Consideration of the effects of changes in the frequency and amplitude of the periodic forcing leads to a problem involving the bifurcations of circle maps in a two‐dimensional parameter space. The global bifurcations in this two‐dimensional parameter space is described for periodic forcing of several simple theoretical models of nonlinear oscillations. As was originally recognized by Arnold, one motivation for the formulation of these models is their connection with theoretical models of cardiac arrhythmias originating from the competition and interaction between two pacemakers for the control of the heart.
1(1991); http://dx.doi.org/10.1063/1.165814View Description Hide Description
The coding theory of rotations (by inspecting closely their relation to flows) and the continued fractions algorithm (by considering even two‐coloring of the integers with a given proportion of, say, blue and red) are revisited. Then, even n‐coloring of the integers is defined. This allows one to code rotations on the (n−1)‐torus by considering linear flows on the n‐torus and yields a simple geometric approach to renormalization on tori by first return maps on the coding regions.
1(1991); http://dx.doi.org/10.1063/1.165815View Description Hide Description
In this paper a review is given of experimental techniques in chaotic dynamics of solid mechanical systems based on modern ideas of nonlinear dynamics. These methods include Poincaré maps, double Poincaré sections, symbol dynamics, and fractal dimension. The physical problems discussed include nonlinear elasticbeams, forced motion of a string, flow‐induced vibration of a rod, forced motions of a magnetic pendulum, and rigid body dynamics of a magnet and high‐temperature superconductor.
1(1991); http://dx.doi.org/10.1063/1.165816View Description Hide Description
The effect of a small or slow perturbation on a Hamiltonian system with one degree of freedom is considered. It is assumed that the phase portrait (‘‘phase plane’’) of the unperturbed system is divided by separatrices into several regions and that under the action of the perturbations phase points can cross these separatrices. The probabilistic phenomena are described that arise due to these separatrix crossings, including the scattering of trajectories, random jumps in the values of adiabatic invariants, and adiabatic chaos. These phenomena occur both in idealized problems in classical mechanics and in real physical systems in planetary science and plasma physics contexts.
1(1991); http://dx.doi.org/10.1063/1.165817View Description Hide Description
When a nonlinear crystal is placed within a multimode solid‐statelaser cavity, deterministic fluctuations are induced in the output intensity. In this paper, the results of our studies of the intensity noise in a diode pumped, intracavity frequency doubled Nd:YAG(neodymiumdopedyttriumaluminumgarnet) laser will be presented. First, a novel technique to eliminate these fluctuations is described. Second, the observation of antiphase states in the laser output is discussed. These states are characterized by a cyclic periodic pulsing of the individual longitudinal mode intensities. Finally, the statistical properties of chaotic intensity fluctuations are characterized. It is be demonstrated that it is possible to accurately model the laser dynamics by a system of globally coupled, nonlinear oscillators.
1(1991); http://dx.doi.org/10.1063/1.165818View Description Hide Description
Spatially complex, temporally chaotic dynamics of N‐coupled impact oscillators connected by a string are studied experimentally using a discrete measure of the motion for each of the masses. For N=8, a binary assignment of symbols, corresponding to whether or not the masses impact an amplitude constraint, is used to code the spatial pattern as a binary number and to store its change in time in a computer. A spatial pattern return map is then used to observe the change in spatial patterns with time. Bifurcations in spatial impact patterns are observed in this experiment. An entropymeasure is also used to characterize the dynamics. Numerical simulation shows behavior similar to the experimental system.
Diffusion‐induced instability in chemically reacting systems: Steady‐state multiplicity, oscillation, and chaos1(1991); http://dx.doi.org/10.1063/1.165819View Description Hide Description
The dynamical behavior of two coupled cells or reactors is described. The cells are coupled by diffusion, e.g., through a semipermeable membrane, and the chemical reactions and initial or feed concentrations of all species are the same in the two cells. Each cell has only a single stable steady state in the absence of coupling, and the coupled system may exhibit multiple steady states, periodic oscillation, or chaos. The attractors of the coupled system may be either homogeneous (the two cells have equal concentrations) or inhomogeneous. Three two‐variable kinetic models are examined: the Brusselator, a model of the chlorine dioxide–iodine–malonic acidreaction, and the Degn–Harrison model. The dynamical behavior of the coupled system is determined by the nonlinearities in the uncoupled subsystems and by two ratios, that of the diffusion constants of the two species and that of the area of the membrane to the product of the membrane thickness and the volume of a cell.
1(1991); http://dx.doi.org/10.1063/1.165813View Description Hide Description
Nonlinear wave phenomena are often characterized by the appearance of ‘‘solitary wave coherent structures’’ traveling at speeds determined by their amplitudes and morphologies. Assuming that time intervals exist in which these structures are essentially noninteracting, a method for identifying the number of independent features and their respective speeds is proposed and developed. The method is illustrated with a variety of increasingly realistic specific applications, beginning with a simple nonlinear but analytically tractable Gaussian model, continuing with (numerically generated) data describing multisoliton solutions to the Korteweg–de Vries equation, and concluding with (numerical) data from a realistic simulation of nonlinear waveinteractions in plasma turbulence. These studies reveal both strengths and limitations of the method in its present incarnation and suggest topics for future investigations.
1(1991); http://dx.doi.org/10.1063/1.165820View Description Hide Description
The quantum localization of chaotically diffusive classical motion is reviewed, using the kicked rotator as a simple but instructive example. The specific quantum steady state, which results from statistical relaxation in the discrete spectrum, is described in some detail. A new phenomenological theory of quantum dynamical relaxation is presented and compared with the previously existing theory.
1(1991); http://dx.doi.org/10.1063/1.165807View Description Hide Description
The field of quantum chaos has recently focused attention on the quantum description of chaotic scattering processes. The new physical intuition, analytical methods, and numerical tools developed in the study of the quantum behavior of classically chaotic bound systems, like quantum billiards or atoms in strong fields, has led to exciting new predictions for the scattering of electromagnetic waves in curved waveguides, electrons in mesoscopic wires, and atoms off molecules. After a brief review of recent progress in the field of quantum chaos, this paper focuses on specific results relating to ballisticelectrontransport in small, mesoscopic devices. Several specific geometries are suggested for experimental studies of this ‘‘game’’ of quantum pinball with explicit predictions for the fluctuations in the electrical conductivity as functions of the electron Fermi energy and of an applied magnetic field.
1(1991); http://dx.doi.org/10.1063/1.165808View Description Hide Description
The ionization of the highly excited hydrogen atom in a strong external microwave field is a classically chaotic, near‐classical quantum system for microwave frequencies somewhat below the initial Kepler electron orbit frequency. The addition of microwavenoise is found to reduce the sinewave microwave field needed for ionization, modifying the near‐classical fast process responsible for the microwave energy absorption. A classical numerical calculation based upon a many‐frequency model of the noise qualitatively reproduces the observed noise enhancement.