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This glossary is intended to define terms that we use to describe our work. When possible, we include references to definitions by others in the literature. Generally, we attempt to distinguish when our definition differs from what we view to be standard definitions in the literature and in the practice of chaotic/nonlinear time-series analysis.



anticontrol is a form of chaos control. The objective in anticontrol is to cause, to enhance or to maintain chaotic behavior. This differs from the usual objective of chaos control. Both anticontrol and control captitalize on sensitive dependence on initial conditions.

Also known as "maintenance of chaos" (In 1995, In 1997)

References: In (1995), In (1997).

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There is no such thing as "Chaos Theory", except in the popular media or sources of that nature. Nonetheless, chaology has been used to refer to the science or study of chaos, and hence those who study chaos would be termed chaologists. The movie/book Jurassic Park was among the first works to use the terms chaology and chaologist, but few researchers or practicioners in chaos seem to have adopted the terms (one occasionally hears chaostician). Chaos arose as an interdisciplinary, collaborative body of knowledge, so there are very few people who study pure "chaology". Instead, they might be mathematicians, biologists, physicists, engineers, physicians, economists, ecologists, chemists, psychologists or meteorologists who study systems that exhibit characteristics of chaos.

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No-one truly knows what chaos is, much less how to define it. The worst (and best) definition is by Anonymous (1995):

Chaos is what I say it is.

Although there is no standard definition of chaos, most "chaologists" agree on certain characteristics, which we summarize according as:

Chaos is bounded, continuously unstable aperiodic fluctuations displaying sensitive dependence to initial conditions.

Note that this definition is ours, and 9 out of 10 researchers would disagree or add their own terms.

Our definition is very similar to that of Strogatz (1994):

Chaos is aperiodic long-term behavior in a deterministic system that exhibits sensitive dependence on initial conditions.

The recognized first mention of chaos (ie, the "coining" of the term) is by Li and Yorke (1975). Their definition was more to do with nonperiodicities in data.

Eckmann and Ruelle (1985) stated:

Sensitive dependence on initial conditions is also expressed by saying that the system is chaotic [this is now the accepted use of the word chaos, even though the original use by Li and Yorke (1975) was somewhat different].

Lorenz (1993) differentiates "full" and "limited" chaos: "1. The property that characterizes a dynamical system in which most orbits exhibit sensitive dependence; full chaos. 2. Limited chaos; the property that characterizes a dynamical system in which some special orbits are nonperiodic but most are periodic or almost periodic."

References: Li (1975), Eckmann and Ruelle (1985), Lorenz (1993), Strogatz (1994), Anonymous (1995)

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Generally, a repetitive process. For instance, a day represent a cycle: the sun rises, crosses the sky, and sets, a process which repeats as long as the Earth rotates and the sun shines. In four-stoke internal combustion engines, four processes constitute an engine cycle: intake of fresh charge (fuel and air), combustion initiation, combustion power release, and exhaust of combustion products. In a slugging fluidized bed, we define a cycle as the repetition of the following events: settled bed, slug formation, slug rise, slug breakup and particle collapse to a settled bed. In time series, a cycle generally refers to an orbit in phase space. We refer to a cycle time as the real time for the signal to complete a cycle.

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A datum is an item of information; its plural form is data. A datum may be quantitative (eg, "73.6" or "A") or qualitative (eg, "hot" or "dark"). When referring to "data", one writes "data are" or "data were" (ie, using a plural verb with a plural noun). The modern practice of writing "data is/was" (ie, using a singular verb with a plural noun) is incorrect and careless.

In rare instances, the plural datums may be used, specifically when datum refers to a reference or standard value.

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directionality/time irreversibility

Directionality is the property of a time series which makes it seem different (qualitatively or quantitatively) when viewed in forward (natural) time and in reverse time. Irreversibility is important because the presence of directionality in a time series excludes linear Gaussian random processes, and static transforms thereof, as possible models [Weiss (1975)]. Conventional time-series analysis has focused on linear Gaussian process models (eg, ARMA), but this modeling is insufficient for nonlinear or chaotic time series; Lawrance (1991) notes
The view taken here is that directionality is an aspect of time series analysis which deserves wider recognition; for instance, it does not make sense to forecast with a time series model which is reversible, when past data are definitely irreversible. In simulating inputs to a system based on directional historical data, directional simulated data should be used. Such obvious requirements are not met by the use of Gaussian ARMA models.

References: Weiss (1975), Lawrance (1991)

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Discretization refers to the process of partitioning a larger range of values into a smaller range of values. Specifically, for analog-to-digital conversion, the analog signal is continuously variable - in theory, its precision is infinite, as one could always increasingly magnify portions of it for finer inspection; in practice, different types of uncertainties limit precision - but the process of digitization must place these

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An element is an individual item of a collection, such as vectors or sequences.

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van Goor (1998) defines noise as:
high-dimensional dynamics - occurring either internal or external to our system - that perturb the system or our observations, but that are outside our direct scope of interest.
He adds, "In effect, we consider noise as a secondary level of dynamics which does not totally dominate the behavior we are interested in, but does compromise our ability to observe the lower-dimensional features we care about."

References: van Goor (1998)

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A record is an individual item in a series or set. Often, the word point is used interchangeably, but we reserve point to describe a location in phase space and record to describe an item in a series.

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A sequence is a short grouping of consecutive, ordered records. Generally, it is shorter in length than a series, as a sequence might decribe an event in time, where a series would describe many events in time; ie, many sequences may occur in a single series.

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A series is a finite-length grouping of data, usually in a particular order (eg, in temporal order). A time series is a temporally ordered collection of data with a distinct beginning and end (compare stream).

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A set is a grouping or collection, not necessary in any order. In time-series analysis, "data set" generally refers to a time series, but in general "data set" can refer to any collection of data. In computer applications, a "data set" is stored within a single file (but this really is not necessary). An individual item in a set is referred to as a member or a record.

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A statistic is a number which summarizes data. Examples are the data mean or standard deviation. Statistics should be used with care, as often they do not reveal information about a data set without further assumptions applied in the form of a test.

Kennel (1997) writes:

It is important to recognize the difference between a statistic and a test. A statistic is a quantity deterministically computable from a particular dataset; common examples include the sample mean or the value of a correlation integral. A test combines a statistic with some additional knowledge and assumptions concerning the distribution of that statistic expected under some hypthesis of interest.

References: Kennel (1997).

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A stream is a flow of data, and in an abstract sense, its beginning and end are of no particular concern. A stream may be sampled for a finite duration to produce a series. As in computer applications, a stream is an abstract information source.

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A test usually combines a statistic with additional information about how that statistic should be expected to behave in order to answer a specific question or to verify a hypothesis.

See Kennel's comments comparing a test and a statistic.

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A trace refers to the graphical temporal history of a variable, either a signal as displayed on an oscilloscope or, say, a time series as displayed on a computer montitor. Usually, a trace is strongly associated with a physical measurement (eg, "pressure trace").

Also known as "oscillogram", although trace implies something more fleeting and incomplete than an oscillogram.

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Anonymous (name withheld) (1995). 1995 Annual Meeting of the American Institute of Chemical Engineers (Miami Beach, Florida USA; 1995 November).

Eckmann J-P, Ruelle D (1985). Ergodic theory of chaos and strange attractors, Reviews of Modern Physics 57(3), 617-656.

Goor NA van (1998). Control of chaotic and nonlinear systems influenced by dynamic noise, Ph.D. Thesis, University of Tennessee, Knoxville.

In V, Mahan SE, Ditto WL, Spano ML (1995). Experimental maintenance of chaos, Physical Review Letters 74(22), 4420-4423.

In V, Spano ML, Neff JD, Ditto WL, Daw CS, Edwards KD, Nguyen K (1997). Maintenance of chaos in a computational model of a thermal pulse combustor, Chaos 7(4), 605-613.

Kennel MB (1997). Statistical test for dynamical nonstationarity in observed time-series data, Physical Review E 56(1), 316-321.

Lawrance AJ (1991). Directionality and reversibility in time series, International Statistical Review 59(1), 67-79.

Li TY, Yorke JA (1975). Period three implies chaos, American Mathematics Monthly 82, 985-992.

Lorenz EN (1993). The essence of chaos (University of Washington Press, ISBN 0-295-97270-X).

Strogatz SH (1994). Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering (Addison-Wesley Publishing Company, ISBN 0-201-54344-5).

Weiss G (1975). Time-reversibility of linear stochastic processes, Journal of Applied Probability 12, 831-836.

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Updated: 2001-12-27 ceaf