Volume 4, Issue 1, March 1994
Index of content:
4(1994); http://dx.doi.org/10.1063/1.166051View Description Hide Description
A detailed study of the effects of quantum fluctuations in a chaotic single mode laser is presented. It has been well established that the linear noise approximation eventually becomes invalid for the case of chaotic dynamics. A more accurate description of the laser is achieved through use of nonlinear Langevin equations. Simple expressions for the time evolution of the phases of the electric field and polarization are derived. These expressions predict that chaotic dynamics will greatly enhance phase diffusion. This prediction is verified through numerical simulations. A quantitative method, for determining the amount of amplification of quantum noise by chaos is discussed. This method makes use of a metric introduced in symbolic dynamics. The fluctuations are shown to have been amplified by over two orders of magnitude, making them macroscopically visible.
4(1994); http://dx.doi.org/10.1063/1.166052View Description Hide Description
A generalization of the Swift–Hohenberg (SH) equation is used to study several stationary patterns that appear in hydrodynamical instabilities. The corresponding amplitude equations allow one to find the stability of planforms with different symmetries. These results are compared with numerical simulations of a generalized SH equation (GSHE). The transition between different symmetries, the hysteretic effects, and the characteristics of the defects observed in experiments are well reproduced in these simulations. The existence of steady fronts between domains with different symmetries is also analyzed. Steady domain boundaries between hexagons and rolls, and between hexagons and squares are possible solutions in the amplitude equation framework and are obtained in numerical simulations for a full range of coefficients in the GSHE.
4(1994); http://dx.doi.org/10.1063/1.166053View Description Hide Description
Results are presented for experimental laser‐scanning investigations of the statistical characteristics of wind‐driven ocean waves. The method involves counting the number of specular points during scanning of the sea surface by a narrow laser beam on a moving ship. The data analyzed are the set of specular points recorded along a track traced out by the laser beam as a result of the motion of the ship and the scanning beam. A prominent feature is the large‐scale variability of the number of specular points and the self‐similar nature of the process over a rather wide range of spatial scales. A fractalanalysis of the process shows a clear power‐law interval in the spatial spectrum of the distribution of specular points.
4(1994); http://dx.doi.org/10.1063/1.166054View Description Hide Description
We present a general scheme to describe particle kinetics in the case of incomplete Hamiltonian chaos when a set of islands of stability forms a complicated fractal space‐time dynamics and when there is orbit stickiness to the islands’ boundary. This kinetics is alternative to the ‘‘normal’’ Fokker–Planck–Kolmogorov equation. A new kinetic equation describes random wandering in the fractal space‐time. Critical exponents of the anomalous kinetics are expressed through dynamical characteristics of a Hamiltonian using the renormalization group approach. Renormalization transformation has been applied simultaneously for space and time and fractional calculus has been exploited.
4(1994); http://dx.doi.org/10.1063/1.166055View Description Hide Description
We report the results of an analytical and numerical study of the contour line and surface geometry in two models of continuum percolation with quasiperiodic properties. Both the fractal dimension of long isolines and the scaling coefficient ν are determined analytically for the two‐dimensional percolation problem. The scaling characteristics of the isosurfaces of the three‐dimensional potential function with an icosahedral symmetry are obtained using computer graphic representation.
4(1994); http://dx.doi.org/10.1063/1.166056View Description Hide Description
Some representative potentials of the anharmonic‐oscillator type are constructed. Some corresponding spectra‐shift operators are also constructed. These operators are a natural generalization of Fok creation and annihilation operators. The Schrödinger problem for these potentials leads to an equidistant energy spectrum for all excited states, which are separated from the ground state by an energy gap. The general properties of the dynamic system generated by spectral‐shift operators of third degree are analyzed. Several examples of such anharmonic oscillators are discussed. The relationship between the eigenvectors of the Schrödinger problem and a certain type of nonclassical orthogonal polynomials is established.
4(1994); http://dx.doi.org/10.1063/1.166057View Description Hide Description
New characteristics, Ω‐dimension ( D(Ω)) and spectral density of dimension ( D’(Ω)), of deterministically generated irregular signals are proposed. They are the functions that are calculated employing spectral transformation such as filtering with limit transmission frequency Ω. The Ω‐dimension of the time seriesgenerated by a dynamical system with a homogeneous strange attractor does not depend on Ω and coincides with the dimension measured using a standard technique. If the time series does not possess the similarity property as the time scale changes (i.e., it is multiscaled in time), the calculation of D(Ω) gives additional information on the properties of the signal. In particular, it allows for the estimation of additional degrees of freedom in the time series on signal transmission through the communication channel and preliminary processing.
4(1994); http://dx.doi.org/10.1063/1.166058View Description Hide Description
The chaotic dynamics of sound rays in a near‐bottom waveguide channel is studied on the basis of the Hamiltonian dynamics of nonparaxial rays in inhomogeneous moving media. The bottom is assumed to have a two‐dimensional roughness. The mapping of the coordinates of the rays upon reflection from the rough bottom is derived through a solution of the corresponding ray equations in an unperturbed waveguide with a horizontal bottom. A numerical analysis of the mapping reveals that a chaotic instability of rays which start out at small angles from the horizontal develops at short distances from the source. Because of this instability, the path segments of a ray along the horizontal coordinates and the signal passage time along a ray are random functions of the angle at which the ray emerges from the source. Upon a further reflection of rays from the rough bottom, there is a diffusion of rays in a stochastic ring which forms in the plane of horizontal ray directions as a result of the overlap and intersection of resonance curves. A qualitative analysis of this effect is carried out. This effect leads to a nearly isotropic distribution of ray directions.
4(1994); http://dx.doi.org/10.1063/1.166059View Description Hide Description
Computer simulations of a nonlinear differential equation with time delay have been carried out to determine the possible steady states over a wide range of parameter values. A variety of nonlinear phenomena, including chaotic attractors and multiple coexisting attractors, are observed. Precision of the solutions is verified by means of evaluating the computational error at each time step. A number of bifurcations are observed, and the involvement of unstable periodic orbits is confirmed. The phase space of the system is infinite dimensional, but nonetheless all the bifurcation phenomena observed, including the blue sky disappearance (boundary crisis) of a chaotic attractor, show geometric structures which are consistent with familiar low‐dimensional center‐manifold descriptions.
4(1994); http://dx.doi.org/10.1063/1.166060View Description Hide Description
Channeling describes the collimated motion of energetic charged particles along the lattice plane or axis in a crystal. The energetic particles are steered through the channels formed by strings of atomic constituents in the lattice. In the case of planar channeling, the motion of a charged particle between the atomic planes can be periodic or quasiperiodic, such as a simple oscillatory motion in the transverse direction. In practice, however, the periodic motion of the channeling particles can be accompanied by an irregular, chaotic behavior. In this paper, the Moliere potential, which is considered as a good analytical approximation for the interaction of channeling particles with the rows of atoms in the lattice, is used to simulate the channeling behavior of positively charged particles in a tungsten (100) crystal plane. By appropriate selection of channeling parameters, such as the projectile energy E 0 and incident angle ψ0, the transition of channeling particles from regular to chaotic motion is demonstrated. It is argued that the fine structures that appear in the angular scan channeling experiments are due to the particles’ chaotic motion.
4(1994); http://dx.doi.org/10.1063/1.166061View Description Hide Description
Pattern selection at medium and high nonlinearity is investigated. While in the former the transient time levels off for large system sizes, in the latter it diverges exponentially giving rise to supertransients. In both cases, the final attractors are quite stable with as a consequence that even at high nonlinearity an attractor can easily be reached by means of a parameter sweep.
4(1994); http://dx.doi.org/10.1063/1.166062View Description Hide Description
The transition to turbulence via spatiotemporal intermittency is investigated for coupled maps defined on generalized Sierpinski gaskets, a class of deterministic fractal lattices. Critical exponents that characterize the onset of intermittency are computed as a function of the fractal dimension of the lattice. Windows of spatiotemporal intermittency are found as the coupling parameter is varied for lattices with a fractal dimension greater than two. This phenomenon is associated with a collective chaotic behavior of the fractal array of coupled maps.
Erratum: ‘‘ℏ expansion for the periodic orbit quantization of chaotic systems’’ [Chaos 3, 601 (1993)]4(1994); http://dx.doi.org/10.1063/1.166063View Description Hide Description