The structure of a strange attractor

For a set S of n lines labeled from 1 to n, we say that S supports an n-vertex planar graph G if for every labeling from 1 to n of its vertices, G has a straight-line crossing-free drawing with each vertex drawn as a point on its associated line. It is known from previous work [4] that no set of n parallel lines supports all n-vertex planar graphs. We show that intersecting lines, even if they intersect at a common point, are more “powerful” than a set of parallel lines. In particular, we prove that every such set of lines supports outerpaths, lobsters, and squids, none of which are supported by any set of parallel lines. On the negative side, we prove that no set of n lines that intersect in a common point supports all n-vertex planar graphs. Finally, we show that there exists a set of n lines in general position that does not support all n-vertex planar graphs.

Catadioptric sensors refer to the combination of lens-based devices and reflective surfaces. These systems are useful because they may have a field of view which is greater than hemispherical, providing the ability to simultaneously view in any direction. Configurations which have a unique effective viewpoint are of primary interest, among these is the case where the reflective surface is a parabolic mirror and the camera is such that it induces an orthographic projection and which we call paracatadioptric. We present an algorithm for the calibration of such a device using only the images of lines in space. In fact, we show that we may obtain all of the intrinsic parameters from the images of only three lines and that this is possible without any metric information. We propose a closed-form solution for focal length, image center, and aspect ratio for skewless cameras and a polynomial root solution in the presence of skew. We also give a method for determining the orientation of a plane containing two sets of parallel lines from one uncalibrated view. Such an orientation recovery enables a rectification which is impossible to achieve in the case of a single uncalibrated view taken by a conventional camera. We study the performance of the algorithm in simulated setups and compare results on real images with an approach based on the image of the mirror's bounding circle.

Rotational splitting Δω(n,l,m) of the eigenfrequencies of a star rotating with angular velocity Ω(r, θ) about a unique axis can be represented as a weighted integral of Ω over r and θ, (r, θ, φ) being spherical polar coordinates about the axis of rotation. For high-frequency acoustic modes, Δω/m collapses essentially to a function of ω — ω /(l+1/2) and M = m/(l + 1/2) alone, and the weighting kernel K(r,θ) becomes asymptotically degenerate, each factor being of essentially Abel type. Therefore, formally, the splitting integral can be inverted, once a procedure has been found for extending Δω over the domain of (ω, M) such that the turning points (rt,ωt), given by (c(rt)/rt, sinωt) = (ω, M) where c is sound speed, span the star. Obtaining that representation is the most difficult stage of the inversion. We report on a procedure that treats the inverted two-dimensional Abel integral as a repeated double integral, representing the data successively along a set of parallel lines M =constant. The method is illustrated by an inversion of artificial data which is compared with the angular velocity from which those data were computed.

The desire to increase the amount of information that can be encoded onto a single photon has driven research into many areas of optics. One such area is optical orbital angular momentum (OAM) [1]. These beams have helical phasefronts and carry an orbital angular momentum of mbar per photon, where the integer m is unbounded, giving a large state space in which to encode information. We recently developed a telescope system comprising two bespoke refractive optical elements to transform OAM states into transverse momentum states [2]. This is achieved by mapping the azimuthal position of the input plane to the lateral position in the output [3]. A mapping of this type transforms a set of concentric rings at the input plane into a set of parallel lines in the output plane. A lens can then separate the resulting transverse momentum states into specified lateral positions, allowing for the efficient measurement of multiple OAM states simultaneously. Separating OAM states in this way presents an opportunity for this larger alphabet to improve the data capacity of a free space link and has potential application in both the classical and quantum regimes. We will present our latest design, increasing the bandwidth of measurable states to over 50 OAM modes. In such a system we study the crosstalk introduced by a thin phase turbulence, showing that turbulence similarly degrades the purity of all the modes within this range.

Synthesis, physico-chemical and DNA interaction studies of homo- and hetero-trinuclear complexes

The electrostatic potential in an ionic crystal is presented in terms of the zeroth-order MacDonald function by means of decomposition of the crystal lattice into sets of parallel lines with the periodic charge distribution with the zero net charge within a period. The method is applied to the calculation of the Madelung constants of a series of standard cubic lattices, doped fullerides and idealized Y-Ba-Cu-O crystals.

Determining object shape from local velocity measurements

With a still developing and largely unpoliced code of ethics, the Internet forces users to renegotiate continually the boundaries and registers of publicity and privacy, and thus, it is the technic that constellates the ideal conditions for gossip, quite apart from its speed and efficiency in circulating tittle-tattle. Proceeding from observations about how the Internet challenges definitional certitudes about the practice and performance of gossip, this article considers the queer performances of Stephen Karam's darkly comic play Speech & Debate, in which three high-school misfits meet and bond over shared sexual secrets that begin to circulate online. Karam's representation suggests how the new platforms or stages for gossip that develop out of networking sites and technologies may create new counter-publics, capable, in their turn, of occasioning new forms of counter-intimacy. These platforms offer not only new potentialities for intertextual citation and public discourse but also mechanisms for self-promotion and, in this way, raise questions about the distinction between confession and gossip and between intimacy and publicity: two sets of parallel lines that have begun to bend toward one another.

Given an arbitrary simple polygon with N vertices we present an algorithm for shading the interior of the polygon with a set of parallel lines where the slope and the distance between lines are prespecified. If the number of shading line segments is M, the algorithm described in the paper runs in 0(N log N + M) time. The algorithm is generalizable to shade any region or regions of an arbitrary planar subdivision.

We have seen briefly in the previous chapter how tori and so-called "strange attractors" in phase space may provide the underlying mathematical representations of the processes involved in the creation of turbulence. Stationary and temporally periodic solutions, which we have covered in depth in Chapters 2 to 14, provide the simplest examples of attractors--the phase space limits of model trajectories--and their properties are relatively easy to describe analytically. However, with the need to consider more complicated temporal flows when modeling the transition to turbulence, it becomes essential to define quantitative measures for characterizing and categorizing the various types of attractors. In this chapter we introduce the Lyapunov exponents (Oseledec, 1968), the Lyapunov dimension (Kaplan and Yorke, 1979), and the correlation dimension (Grassberger and Procaccia, 1983a,b). These quantities give information about the attractor structure, solution transitions, and predictability characteristics of nonlinear models. We investigate how these quantitative measures of attractor dynamics might be helpful in distinguishing different types of chaotic attractors, and thereby possibly distinguishing some of the different routes to chaos. Moreover, because the seventh modeling principle suggests that a model must have sufficient complexity to represent a phenomenon in adequate detail, these quantitative measures may help us evaluate the adequacy of a particular model. Specifically, we illustrate the utility of the Lyapunov and correlation dimensions in characterizing attractors in several low-order spectral models, including the Lorenz (1963) model, the seven-coefficient generalized Lorenz system of Chang and Shirer (1984), and a related eleven-coefficient model.

Is a hyperchaotic attractor superposition of two multifractals?

A single fractal number cannot fully describe the complex structure of a strange attractor embedded in phase space of dynamic systems. Hence it is essential to seek a procedure through which fractal dimension provides more information about the underlying dynamic system, in order to serve such a purpose correlation integrals are calculated at every row or off-diagonal from the distance matrix obtained from a set of points on a strange attractor. The slope of each integral is plotted versus the number of row or off-diagonal considered in the calculation. Hence, an asymptotic convergence to correlation dimension is obtained in this manner. More information is revealed for the dynamic system including its lacunarity.

This paper describes the geometric structure of a strange attractor observed from the simplest known chaotic electronic circuit which the author conceived and proposed to Professor Matsumoto from Waseda University in 1983. Computer simulation of this circuit by Professor Matsumoto in October 1983 confirmed its rich chaotic dynamics [1]. Subsquent laboratory experiments using an operational amplifier circuit further confirmed the robust nature of the chaotic attractor [2], and its rich bifurcation phenomena [3]. The strange attractor is now widely known as the double scroll [4] because it resembles, from a distance, a pair of Saturn plants attached to the opposite ends of a tube formed by rolling two long strips of paper together into a scroll whose cross section consists of two tightly wound spirals.

For three-dimensional dynamical systems with continuous time (flows), a classification of strange homoclinic attractors containing an unique saddle equilibrium state is constructed. The structure and properties of such attractors are determined by the triple of eigenvalues of the equilibrium state. The method of a saddle charts is used for the classification of homoclinic attractors. The essence of this method is in the construction of an extended bifurcation diagram for a wide class of three-dimensional flows (whose linearization matrix is written in the Frobenius form). Regions corresponding to different configurations of eigenvalues are marked in this extended bifurcation diagram. In the space of parameters defining the linear part of the considered class of three-dimensional flows bifurcation surfaces bounding 7 regions are constructed. One region corresponds to the stability of the equilibrium states while other 6 regions correspond to various homoclinic attractors of the following types: Shilnikov attractor, 2 types of spiral figure-eight attractors, Lorenz- like attractor, saddle Shilnikov attractor and attractor of Lyubimov-Zaks-Rovella. The paper also discusses questions related to the pseudohyperbolicity of homoclinic attractors of three-dimensional flows. It is proved that only homoclinic attractors of two types can be pseudohyperbolic: Lorenz-like attractors containing a saddle equilibrium with a two-dimensional stable manifold whose saddle value is positive and saddle Shilnikov attractors containing a saddle equilibrium state with a two-dimensional unstable manifold.

Consider orbitally stable attractors of those flows on the open solid torus D 2 × S 1 {D^2} \times {S^1} which have uniform velocity in the S 1 {S^1} direction (uniform flows). It is found that any such attractor is the frontier of a strictly nested sequence of positively invariant open solid tori. Necessary and sufficient conditions related to these tori are derived for an arbitrary set to be an orbitally stable attractor. When the cross-section of an orbitally stable attractor is a Cantor set, the first return map is found to be conjugate to an irrational rotation on a certain compact abelian group. New examples are constructed of orbitally stable attractors of uniform C ∞ {C^\infty } flows whose cross-sections have uncountably many components (one of these attractors has positive 3 3 -dimensional Lebesgue measure).