Chaos of liquid surface waves in a vessel under vertical excitation with slowly modulated amplitude

Free surface waves in a cylinder of liquid under vertical excitation with slowly modulated amplitude are investigated in the current paper. It is shown by both theoretical analysis and numerical simulation that chaos may occur even for a single mode with modulation which can be used to explain Gollub and Meyer's experiment. The implied resonant mechanism accounting for this phenomenon is further elucidated.

Direct dynamical test for deterministic chaos and optimal embedding of a chaotic time-series

We propose here a local exponential divergence plot which is capable of providing an alternative means of characterizing a complex time series. The suggested plot defines a time-dependent exponent and a ''plus'' exponent. Based on their changes with the embedding dimension and delay time, a criterion for estimating simultaneously the minimal acceptable embedding dimension, the proper delay time, and the largest Lyapunov exponent has been obtained. When redefining the time-dependent exponent LAMBDA(k) curves on a series of shells, we have found that whether a linear envelope to the LAMBDA(k) curves exists can serve as a direct dynamical method of distinguishing chaos from noise.

Direct dynamical test for deterministic chaos

We present a direct and dynamical method to distinguish low-dimensional deterministic chaos from noise. We define a series of time-dependent curves which are closely related to the largest Lyapunov exponent. For a chaotic time series, there exists an envelope to the time-dependent curves, while for a white noise or a noise with the same power spectrum as that of a chaotic time series, the envelope cannot be defined. When a noise is added to a chaotic time series, the envelope is eventually destroyed with the increasing of the amplitude of the noise.

Prediction, Chaos and Ecological Perspective.

As defined, the modeling procedure is quite broad. For example, the chosen compartments may contain a single organism, a population of organisms, or an ensemble of populations. A population compartment, in turn, could be homogeneous or possess structure in size or age. Likewise, the mathematical statements may be deterministic or probabilistic in nature, linear or nonlinear, autonomous or able to possess memory. Examples of all types appear in the literature. In practice, however, ecosystem modelers have focused upon particular types of model constructions. Most analyses seem to treat compartments which are nonsegregated (populations or trophic levels) and homogeneous. The accompanying mathematics is, for the most part, deterministic and autonomous. Despite the enormous effort which has gone into such ecosystem modeling, there remains a paucity of models which meets the rigorous &! validation criteria which might be applied to a model of a mechanical system. Most ecosystem models are short on prediction ability. Even some classical examples, such as the Lotka-Volterra predator-prey scheme, have not spawned validated examples.

Quantifying chaos: practical estimation of the correlation dimension

Be it a physical object or a mathematical model, a nonlinear dynamical system can display complicated aperiodic behavior, or "chaos." In many cases, this chaos is associated with motion on a strange attractor in the system's phase space. And the dimension of the strange attractor indicates the effective number of degrees of freedom in the dynamical system.

In this thesis, we investigate numerical issues involved with estimating the dimension of a strange attractor from a finite time series of measurements on the dynamical system.

Of the various definitions of dimension, we argue that the correlation dimension is the most efficiently calculable and we remark further that it is the most commonly calculated. We are concerned with the practical problems that arise in attempting to compute the correlation dimension. We deal with geometrical effects (due to the inexact self-similarity of the attractor), dynamical effects (due to the nonindependence of points generated by the dynamical system that defines the attractor), and statistical effects (due to the finite number of points that sample the attractor). We propose a modification of the standard algorithm, which eliminates a specific effect due to autocorrelation, and a new implementation of the correlation algorithm, which is computationally efficient.

Finally, we apply the algorithm to chaotic data from the Caltech tokamak and the Texas tokamak (TEXT); we conclude that plasma turbulence is not a low- dimensional phenomenon.

Few-cycle spatiotemporal soliton wave excited by filamentation of a femtosecond laser pulse in materials with anomalous dispersion

The nonlinear dynamics of 1.6-mu m fs laser pulses propagating in fused silica is investigated by employing a full-order dispersion model. Different from the x-wave generation in normally dispersive media, a few-cycle spatiotemporally compressed soliton wave is generated with the contrary contributions of anomalous group velocity dispersion (GVD) and self-phase-modulation. However, at the tailing edge of the pulse forms a shock wave which generates separate and strong supercontinuum peaked at 670 nm. It is also the origin of conical emission formed both in time and frequency domain with the contribution of normal GVD at visible light.

Ghosts of order on the frontier of chaos

What kinds of motion can occur in classical mechanics? We address this question by looking at the structures traced out by trajectories in phase space; the most orderly, completely integrable systems are characterized by phase trajectories confined to low-dimensional, invariant tori. The KAM theory examines what happens to the tori when an integrable system is subjected to a small perturbation and finds that, for small enough perturbations, most of them survive.

The KAM theory is mute about the disrupted tori, but, for two-dimensional systems, Aubry and Mather discovered an astonishing picture: the broken tori are replaced by "cantori," tattered, Cantor-set remnants of the original invariant curves. We seek to extend Aubry and Mather's picture to higher dimensional systems and report two kinds of studies; both concern perturbations of a completely integrable, four-dimensional symplectic map. In the first study we compute some numerical approximations to Birkhoff periodic orbits; sequences of such orbits should approximate any higher dimensional analogs of the cantori. In the second study we prove converse KAM theorems; that is, we use a combination of analytic arguments and rigorous, machine-assisted computations to find perturbations so large that no KAM tori survive. We are able to show that the last few of our Birkhoff orbits exist in a regime where there are no tori.

Spatiotemporal evolution and multiple self-focusing of ultrashort pulses in a resonant two-level medium

The spatiotemporal evolutions of ultrashort pulses in two dimensions are investigated numerically by solving the coupled Maxwell-Bloch equations without invoking the slowly varying envelope approximation and rotating-wave approximation. For an on-axis 2n pi sech pulse, local delay makes the temporal split 2 pi sech pulses crescent-shaped in the transverse distribution. Due to the transverse effect, the temporal split 2 pi sech pulses become unstable and experience reshaping during the propagation process. Then, interference occurs between the successive crescent-shaped pulses and multiple self-focusing can form.

Spatiotemporal instabilities in nonlinear Kerr media in the presence of arbitrary higher-order dispersions

Spatiotemporal instabilities in nonlinear Kerr media with arbitrary higher-order dispersions are studied by use of standard linear-stability analysis. A generic expression for instability growth rate that unifies and expands on previous results for temporal, spatial, and spatiotemporal instabilities is obtained. It is shown that all odd-order dispersions contribute nothing to instability, whereas all even-order dispersions not only affect the conventional instability regions but may also lead to the appearance of new instability regions. The role of fourth-order dispersion in spatiotemporal instabilities is studied exemplificatively to demonstrate the generic results. Numerical simulations confirm the obtained analytic results. (C) 2002 Optical Society of America.

Spatiotemporal behaviors and singularity of ultrashort pulsed Elegant Hermite-Gaussian beams

Analytical propagation expressions of ultrashort pulsed Elegant Hermite-Gaussian beams are derived and spatiotemporal properties of the pulses with different transverse modes are studied. Singularity of the complex amplitude envelope solution of the pulses obtained under slowly varying envelope approximation is analyzed in detail. The rigorous analytical solution of the pulse is deduced and no singularity emerges in the solution. The obtained results indicate that the transverse mode affects not only the spatiotemporal properties but also the singularity of the pulses. Time delay of the off-axis maximum intensity is more obvious and the singularity is located nearer to the z-axis for the pulse with higher transverse modes. (C) 2007 Elsevier GmbH. All rights reserved.

Spatiotemporal segmentation based on two-dimensional spatiotemporal entropic thresholding

A novel spatiotemporal segmentation technique is further developed for extracting uncovered background and moving objects from the image sequences, then the following motion estimation is performed only on the regions corresponding to moving objects. The frame difference contrast (FCON) and local variance contrast (LCON), which are related to the temporal and spatial homogeneity of the image sequence, are selected to form the 2-D spatiotemporal entropy. Then the spatial segmentation threshold is determined by maximizing the 2-D spatiotemporal entropy, and the temporal segmentation point is selected to minimize the complexity measure for image sequence coding. Since both temporal and spatial correlation of an image sequence are exploited, this proposed spatiotemporal segmentation technique can further be used to determine the positions of reference frames adaptively, hence resulting in a low bit rate. Experimental results show that this segmentation-based coding scheme is more efficient than usual fixed-size coding algorithms. (C) 1997 Society of Photo-Optical Instrumentation Engineers.