Quasi-periodic Orbits in Siegel Disks/Balls and the Babylonian Problem

We investigate numerically complex dynamical systems where a fixed point is surrounded by a disk or ball of quasi-periodic orbits, where there is a change of variables (or conjugacy) that converts the system into a linear map. We compute this “linearization” (or conjugacy) from knowledge of a single quasi-periodic trajectory. In our computations of rotation rates of the almost periodic orbits and Fourier coefficients of the conjugacy, we only use knowledge of a trajectory, and we do not assume knowledge of the explicit form of a dynamical system. This problem is called the Babylonian problem: determining the characteristics of a quasi-periodic set from a trajectory. Our computation of rotation rates and Fourier coefficients depends on the very high speed of our computational method “the weighted Birkhoff average”.     

Codimension two bifurcations of fixed points in a class of vibratory systems with symmetrical rigid stops

A vibratory system having symmetrically placed rigid stops and subjected to periodic excitation is considered. Local codimension two bifurcations of the vibratory system with symmetrical rigid stops, associated with double Hopf bifurcation and interaction of Hopf and pitchfork bifurcation, are analyzed by using the center manifold theorem technique and normal form method of maps. Dynamic behavior of the system, near the points of codimension two bifurcations, is investigated by using qualitative analysis and numerical simulation. Hopf-flip bifurcation of fixed points in the vibratory system with a single stop are briefly analyzed by comparison with unfoldings analyses of Hopf-pitchfork bifurcation of the vibratory system with symmetrical rigid stops. Near the value of double Hopf bifurcation there exist period-one double-impact symmetrical motion and quasi-periodic impact motions. The quasi-periodic impact motions are represented by the closed circle and “tire-like” attractor in projected Poincaré sections. With change of system parameters, the quasi-periodic impact motions usually lead to chaos via “tire-like” torus doubling.     

COMPARISON OF TWO MODELS AND TWO ESTIMATION METHODS FOR CATCH AND EFFORT DATA

We consider the problem of estimating the optimal steady effort level from a time series of catch and effort data, taking account of errors in the observation of the “effective effort” as well as randomness in the stock-production function. The “total least squares” method ignores the time series nature of the data, while the “approximate likelihood” method takes it into account. We compare estimation schemes based upon these two methods by applying them to artificial data for which the “correct” parameters are known. We use a similar procedure to compare the effectiveness of a “power model” for stock and production with the “Ricker model.” We apply these estimation methods to some sets of real data, and obtain an interval estimate of the optimal effort.     

An Analytical Model of Periodic Waves in Shallow Water

An explicit, analytical model is presented of finite-amplitude waves in shallow water. The waves in question have two independent spatial periods, in two independent horizontal directions. Both short-crested and long-crested waves are available from the model. Every wave pattern is an exact solution of the Kadomtsev-Petviashvili equation, and is based on a Riemann theta function of genus 2. These biperiodic waves are direct generalizations of the well-known (simply periodic) cnoidal waves. Just as cnoidal waves are often used as one-dimensional models of “typical” nonlinear, periodic waves in shallow water, these biperiodic waves may be considered to represent “typical” nonlinear, periodic waves in shallow water without the assumption of one-dimensionality.     

An efficient generalized least squares algorithm for periodic trended regression with autoregressive errors

Time series data with periodic trends like daily temperatures or sales of seasonal products can be seen in periods fluctuating between highs and lows throughout the year. Generalized least squares estimators are often computed for such time series data as these estimators have minimum variance among all linear unbiased estimators. However, the generalized least squares solution can require extremely demanding computation when the data is large. This paper studies an efficient algorithm for generalized least squares estimation in periodic trended regression with autoregressive errors. We develop an algorithm that can substantially simplify generalized least squares computation by manipulating large sets of data into smaller sets. This is accomplished by coining a structured matrix for dimension reduction. Simulations show that the new computation methods using our algorithm can drastically reduce computing time. Our algorithm can be easily adapted to big data that show periodic trends often pertinent to economics, environmental studies, and engineering practices.     

近似周期时间序列分析（英文）

本文首先给出了近似周期时间序列概念,即:具有周期特征但是周期长度变化的时间序列.比如,太阳黑子序列具有11年左右的周期,但是其周期并不是11,而是在11左右变化,这就是一个近似周期序列.然后给出了提取近似周期趋势方法,并且提出了广义差分算子,这里提出的广义差分算子不仅可以消除时间序列的长期趋势和周期性,而且还可以消除近似周期性.最后,以太阳黑子序列为例说明了广义差分算子的应用.     

Action minimizers under topological constraints in the planar equal-mass four-body problem

It is shown that in the planar equal-mass four-body problem, there exist two sets of action minimizers connecting two planar boundary configurations with fixed symmetry axes and specific order constraints: a double isosceles configuration and an isosceles trapezoid configuration, while order constraints are introduced on the boundary configurations. By applying the level estimate method, these minimizers are shown to be collision-free and they can be extended to two new sets of periodic or quasi-periodic orbits.     

带有多重休假及预防维修的冲击模型二维更换策略

研究了修理工带有多重休假且定期检测的累积冲击模型.为了延长系统的运行时间,在检测时考虑了预防维修.将事后维修和预防维修结合起来运用于可修系统,且假定预防维修能够"修复如新",而事后维修为"修复非新".以系统的检测周期和故障次数为二维决策变量,选取系统经长期运行单位时间内期望费用为目标函数.并通过数值分析,求出了最优策略.     

Canonical correlation for principal components of time series

With contemporary data collection capacity, data sets containing large numbers of different multivariate time series relating to a common entity (e.g., fMRI, financial stocks) are becoming more prevalent. One pervasive question is whether or not there are patterns or groups of series within the larger data set (e.g., disease patterns in brain scans, mining stocks may be internally similar but themselves may be distinct from banking stocks). There is a relatively large body of literature centered on clustering methods for univariate and multivariate time series, though most do not utilize the time dependencies inherent to time series. This paper develops an exploratory data methodology which in addition to the time dependencies, utilizes the dependency information between S series themselves as well as the dependency information between p variables within the series simultaneously while still retaining the distinctiveness of the two types of variables. This is achieved by combining the principles of both canonical correlation analysis and principal component analysis for time series to obtain a new type of covariance/correlation matrix for a principal component analysis to produce a so-called “principal component time series”. The results are illustrated on two data sets.     

Impurity bands and quasi-Bloch waves for a one-dimensional model of modulated crystal

This paper investigates a simple one-dimensional model of incommensurate “harmonic crystal” in terms of the spectrum of the corresponding Schrödinger equation. Two angles of attack are studied: the first exploits techniques borrowed from the theory of quasi-periodic functions while the second relies on periodicity properties in a higher-dimensional space. It is shown that both approaches lead to essentially the same results; that is, the lower spectrum is split between “Cantor-like zones” and “impurity bands” to which correspond critical and extended eigenstates, respectively. These “new bands” seem to emerge inside the band gaps of the unperturbed problem when certain conditions are met and display a parabolic nature. Numerical tests are extensively performed on both steady and time-dependent problems.     

A Diophantine duality applied to the KAM and Nekhoroshev theorems

In this paper, we use geometry of numbers to relate two dual Diophantine problems. This allows us to focus on simultaneous approximations rather than small linear forms. As a consequence, we develop a new approach to the perturbation theory for quasi-periodic solutions dealing only with periodic approximations and avoiding classical small divisors estimates. We obtain two results of stability, in the spirit of the KAM and Nekhoroshev theorems, in the model case of a perturbation of a constant vector field on the $n$ -dimensional torus. Our first result, which is a Nekhoroshev type theorem, is the construction of a “partial” normal form, that is a normal form with a small remainder whose size depends on the Diophantine properties of the vector. Then, assuming our vector satisfies the Bruno–Rüssmann condition, we construct an “inverted” normal form, recovering the classical KAM theorem of Kolmogorov, Arnold and Moser for constant vector fields on torus.     

On the spectrum of almost periodic solution of second-order differential equations with piecewise constant argument

We study the spectrum containment of almost periodic solution to second order differential equations with piecewise constant argument. Some known (periodic, quasi-periodic) results would be expanded. As a corollary, it is shown that such equations with periodic perturbations possess a quasi-periodic solution and no periodic solution. This phenomena is due to the piecewise constant argument. The results are extended to nonlinear equations.     

Dynamic analysis of a pest management SEI model with saturation incidence concerning impulsive control strategy

According to biological strategy for pest control, we investigate the dynamic behavior of a pest management SEI model with saturation incidence concerning impulsive control strategy-periodic releasing infected pests at fixed times. We prove that all solutions of the system are uniformly ultimately bounded and there exists a globally asymptotically stable pest-eradication periodic solution when the impulsive period is less than some critical value. When the impulsive period is larger than some critical value, the stability of the pest-eradication periodic solution is lost; the system is uniformly permanent. Thus, we can use the stability of the positive periodic solution and its period to control insect pests at acceptably low levels. Numerical results show that the system we consider can take on various kinds of periodic fluctuations and several types of attractor coexistence and is dominated by period-doubling cascade, symmetry-breaking pitchfork bifurcation, quasi-periodic oscillate, chaos, and non-unique dynamics.     

IS THE NILE OUTFLOW FRACTAL? HURST'S ANALYSIS REVISITED

In a now classic study, Hurst [1951, 1955] found significant long-term correlations among fluctuations in Nile River outflows and described these correlations in terms of power laws. Mandelbrot's theory of random fractals [1982] later provided an axiomatic framework for Hurst's work. More recently, Bak, Tang and Weisenfeld's [1987] theory of “self-organized criticality” predicted that the fluctuations in Nile River outflows should follow power laws such as those observed by Hurst. In reexamining Hurst's data, we found small but significant deviations from these power laws, and in particular, evidence for a natural time scale of 32–128 years in global climate dynamics, possibly driven by ocean dynamics.     

Dynamics in coupled oscillators with recurrent/random forcing, a PDE approach

The current paper is devoted to the study of coupled oscillators with recurrent/random forcing. Special attention is given to the solutions having the same recurrence/randomness as that of the forcing (recurrent/random solutions for short). By embedding coupled oscillators into coupled parabolic equations, it establishes a general theorem on the existence of recurrent/random solutions. It also finds conditions under which such solutions are unique. When the recurrent forcing is actually quasi-periodic or almost periodic, recurrent solutions are refereed to as quasi-periodic or almost periodic solutions in a weak sense and they are quasi-periodic or almost periodic in the classical sense under the uniqueness conditions. In addition, applications of the general theory to coupled Duffing type oscillators and Josephson junctions are considered and the results obtained extend several existing ones for quasi-periodic Duffing oscillators.     

Singular approximation of the method of periodic components in statistical mechanics of composite materials

A quasi-periodic model is developed for random structures of composites, when the locations of inclusions are given in terms of random deviations from nodes of an ideal periodic lattice. Solution of the stochastic boundary problem of the theory of elasticity is examined for a quasi-periodic component by the method of periodic components, which is reduced to determination of the field of deviations from the known solution for a corresponding periodic composite. The solution is presented for the tensor of effective elastic properties of a quasi-periodic composite in singular approximation of the method of periodic components in terms of familiar solutions for tensors of the effective elastic properties of composites with periodic and chaotic structures and the parameters of the quasi-periodic structure: the coefficient of periodicity and the tensor of the anisotropy of inclusion disorder. A numerical calculation is performed for the effective transversally isotropic elastic properties of unidirectional fibrous composites with different degrees of fiber disorder.Perm' State Technical University, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 33, No. 4, pp. 460–473, July–August, 1997.     

Computational study of the US stock market evolution: a rank correlation-based network model

This paper presents a computational study of global characteristics of the US stock market using a network-based model referred to as the market graph. The market graph reflects similarity patterns between stock return fluctuations via linking pairs of stocks that exhibit “coordinated” behavior over a specified period of time. We utilized Spearman rank correlation as a measure of similarity between stocks and considered the evolution of the market graph over the recent decade between 2001–2011. The observed market graph characteristics reveal interesting trends in the stock market over time, as well as allow one to use this model to identify cohesive clusters of stocks in the market.     

Temporal Taylor’s scaling of facial electromyography and electrodermal activity in the course of emotional stimulation

High frequency psychophysiological data create a challenge for quantitative modeling based on Big Data tools since they reflect the complexity of processes taking place in human body and its responses to external events. Here we present studies of fluctuations in facial electromyography (fEMG) and electrodermal activity (EDA) massive time series and changes of such signals in the course of emotional stimulation. Zygomaticus major (ZYG; “smiling” muscle) activity, corrugator supercilii (COR; “frowning” muscle) activity, and phasic skin conductance (PHSC; sweating) levels of 65 participants were recorded during experiments that involved exposure to emotional stimuli (i.e., IAPS images, reading and writing messages on an artificial online discussion board). Temporal Taylor’s fluctuations scaling were found when signals for various participants and during various types of emotional events were compared. Values of scaling exponents were close to one, suggesting an external origin of system dynamics and/or strong interactions between system’s basic elements (e.g., muscle fibres). Our statistical analysis shows that the scaling exponents enable identification of high valence and arousal levels in ZYG and COR signals.     

具有反馈控制离散Logistic系统的分支分析（英文）

The paper studies the dynamical behaviors of a discrete Logistic system with feedback control. The system undergoes Flip bifurcation and Hopf bifurcation by using the center manifold theorem and the bifurcation theory. Numerical simulations not only illustrate our results, but also exhibit the complex dynamical behaviors of the system, such as the period-doubling bifurcation in periods 2, 4, 8 and 16, and quasi-periodic orbits and chaotic sets.