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.

- A: anticontrol
- C: chaology/chaologist | chaos/chaotic | cycle
- D: datum/data | directionality/time irreversibility | discretization
- E: element
- N: noise/noisy
- R: record
- S: sequence | series | set | statistic | stream
- T: test | trace

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 ischaotic[this is now the accepted use of the wordchaos, 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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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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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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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 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 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