Elizabeth Bradley

Recurrence plots of experimental data: To embed or not to embed?

A recurrence plot is a visualization tool for analyzing experimental data. These plots often reveal correlations in the data that are not easily detected in the original time series. Existing recurrence plot analysis techniques, which are primarily application oriented and completely quantitative, require that the time-series data first be embedded in a high-dimensional space, where the embedding dimension d(E) is dictated by the dimension d of the data set, with d(E)>/=2d+1. One such set of recurrence plot analysis tools, recurrence quantification analysis, is particularly useful in finding locations in the data where the underlying dynamics change. We have found that for certain low-dimensional systems the same results can be obtained with no embedding. (c) 1998 American Institute of Physics.

Nonlinear time-series analysis revisited

In 1980 and 1981, two pioneering papers laid the foundation for what became known as nonlinear time-series analysis: the analysis of observed data-typically univariate-via dynamical systems theory. Based on the concept of state-space reconstruction, this set of methods allows us to compute characteristic quantities such as Lyapunov exponents and fractal dimensions, to predict the future course of the time series, and even to reconstruct the equations of motion in some cases. In practice, however, there are a number of issues that restrict the power of this approach: whether the signal accurately and thoroughly samples the dynamics, for instance, and whether it contains noise. Moreover, the numerical algorithms that we use to instantiate these ideas are not perfect; they involve approximations, scale parameters, and finite-precision arithmetic, among other things. Even so, nonlinear time-series analysis has been used to great advantage on thousands of real and synthetic data sets from a wide variety of systems ranging from roulette wheels to lasers to the human heart. Even in cases where the data do not meet the mathematical or algorithmic requirements to assure full topological conjugacy, the results of nonlinear time-series analysis can be helpful in understanding, characterizing, and predicting dynamical systems.

Recurrence plots and unstable periodic orbits

A recurrence plot is a two-dimensional visualization technique for sequential data. These plots are useful in that they bring out correlations at all scales in a manner that is obvious to the human eye, but their rich geometric structure can make them hard to interpret. In this paper, we suggest that the unstable periodic orbits embedded in a chaotic attractor are a useful basis set for the geometry of a recurrence plot of those data. This provides not only a simple way to locate unstable periodic orbits in chaotic time-series data, but also a potentially effective way to use a recurrence plot to identify a dynamical system. (c) 2002 American Institute of Physics.

Implications of systems dynamic models and control theory for environmental approaches to the prevention of alcohol- and other drug use-related problems

The approach described in this article is premised on the idea that drug and alcohol use-related problems are heterogeneously distributed with respect to population and geography, and therefore, are essentially local problems. More specifically, it is argued that viewing a local community as an interacting set of systems that support or buffer the occurrence of specific substance misuse outcomes, opens up to research two important prospects. The first of these involves creating adequate systems models that can capture the primary community structures and relationships that support public health problems such as alcohol and drug misuse and related outcomes. The second entails rationally testing control strategies that have the potential to moderate or reduce these problems. Understanding and controlling complex dynamic systems models nowadays pervades all scientific disciplines, and it is to research in areas such as biology, ecology, engineering, computer sciences, and mathematics that researchers in the field of addictions must turn to in order to better study the complexity that confronts them as they try to understand and prevent problems resulting from alcohol and drug use and misuse. Here we set out what such a systems-based understanding of alcohol- and drug use-related problems will require and discuss its implications for public policy and prevention programming.

Automatic construction of accurate models of physical systems

This paper describes an implemented computer program called PRET that automates the process of system identification: given hypotheses, observations, and specifications, it constructs an ordinary differential equation model of a target system with no other inputs or intervention from its user. The core of the program is a set of traditional system identification (SID) methods. A layer of artificial intelligence (AI) techniques built around this core automates the high-level stages of the identification process that are normally performed by a human expert. The AI layer accomplishes this by selecting and applying appropriate methods from the SID library and performing qualitative, symbolic, algebraic, and geometric reasoning on the users inputs. For each supported domain (e.g., mechanics), the program uses a few powerful encoded rules (e.g., σF=0) to combine hypotheses into models. A custom logic engine checks models against observations, using a set of encoded domain-independent mathematical rules to infer facts about both, modulo the resolution inherent in the specifications, and then searching for contradictions. The design of the next generation of this program is also described in this paper. In it, discrepancies between sets of facts will be used to guide the removal of unnecessary terms from a model. Power-series techniques will be exploited to synthesize new terms from scratch if the users hypotheses are inadequate, and sensors and actuators will allow the tool to take aninput-output approach to modeling real physical systems.

AUTONOMOUS EXPLORATION AND CONTROL OF CHAOTIC SYSTEMS

Control algorithms that exploit chaotic behavior can vastly improve the performance of many practical and useful systems. Phase-locked loops, for example, are normally designed using linearization. The approximations thus introduced lead to lock and capture range limits. Design techniques that are equipped to exploit the real nonlinear nature of the device loosen these limitations. The program Perfect Moment is built around a collection of such techniques. Given a differential equation and two points in the systems state space, it automatically selects and maps the region of interest, chooses a set of trajectory segments from the maps, uses them to construct a composite path between the points, and causes the system to follow that path via appropriate parameter changes at the segment junctions. Rules embodying theorems and definitions from nonlinear dynamics are used to limit complexity by focusing the mapping and search on the areas of interest. Even so, these processes are computationally intensive. Ho...

Using chaos to broaden the capture range of a phase-locked loop

Phase-space trajectories on a chaotic attractor densely cover a set of nonzero measures, making all points in that set reachable from any initial condition in its basin of attraction. Moreover, the size, shape, and position of the attractor are affected by changes in system parameters, following certain highly characteristic patterns. These properties have been used, in simulations, to broaden the capture range of the common phase locked loop circuit. An external modulating input is used to throw the unlocked loop into a chaotic regime that overlaps the original capture range. The chaos-inducing modulation is then turned off, allowing the loops original dynamics to capture the signal. This technique is not limited to this system or even to this branch of engineering; it applies, with a few constraints and limitations, to any system that exhibits chaotic behavior and that is subject to design requirements. >

Computing connectedness: disconnectedness and discreteness

We consider finite point-set approximations of a manifold or fractal with the goal of determining topological properties of the underlying set. We use the minimal spanning tree of the finite set of points to compute the number and size of its -connected components. By extrapolating the limiting behavior of these quantities as ! 0 we can say whether the underlying set appears to be connected, totally disconnected, or perfect. We demonstrate the effectiveness of our techniques for a number of examples, including a family of fractals related to the Sierpinski triangle, Cantor subsets of the plane, the Henon attractor, and cantori from four-dimensional symplectic sawtooth maps. For zero-measure Cantor sets, we conjecture that the growth rate of the number of -components as ! 0 is equivalent to the box-counting dimension.

COMPUTING CONNECTEDNESS : AN EXERCISE IN COMPUTATIONAL TOPOLOGY

We reformulate the notion of connectedness for compact metric spaces in a manner that may be implemented computationally. In particular, our techniques can distinguish between sets that are connected, have a finite number of connected components, have infinitely many connected components, or are totally disconnected. We hope that this approach will prove useful for studying structures in the phase space of dynamical systems.

Computer systems are dynamical systems.

In this paper, we propose a nonlinear dynamics-based framework for modeling and analyzing computer systems. Working with this framework, we use a custom measurement infrastructure and delay-coordinate embedding to study the dynamics of these complex nonlinear systems. We find strong indications, from multiple corroborating methods, of low-dimensional dynamics in the performance of a simple program running on a popular Intel computer-including the first experimental evidence of chaotic dynamics in real computer hardware. We also find that the dynamics change completely when we run the same program on a different type of Intel computer, or when that program is changed slightly. This not only validates our framework; it also raises important issues about computer analysis and design. These engineered systems have grown so complex as to defy the analysis tools that are typically used by their designers: tools that assume linearity and stochasticity and essentially ignore dynamics. The ideas and methods developed by the nonlinear dynamics community, applied and interpreted in the context of the framework proposed here, are a much better way to study, understand, and design modern computer systems.