Chaos Theory Tamed

Garnett P. Williams, Joseph Henry Press

ISBN:0309063515, Edition: 1, 1997-10-08

Price: $34.95

A
accuracy (a) A numerical measure of how close an approximation is to the truth; (b) correctness, in the
sense of lack of bias.
adaptive histogramming Any histogramming procedure in which bin width is allowed to vary (usually
according to local densities in the data).
adaptive kernel density estimator A probability-estimation technique consisting of the kernel density
estimator but with the variation that bin width changes according to local density of data points.
affine Produced by a transformation in which the variables or axes haven't all been scaled by the same
factor.
algorithm A recipe, plan, sequence of steps, set of instructions, list of rules, or set of mathematical
equations used to solve a problem, usually with the aid of a computer. Some of the many possible forms of
an algorithm include a loosely phrased group of text statements, a diagram called a flowchart, or a
computer program.
almost-periodic orbit An orbit that comes closer and closer to repeating itself with time.
amplitude (a) The extreme range of a fluctuating quantity, such as a pendulum, tidal cycle, etc.; (b) the
maximum difference between the value of a periodic oscillation and its mean. In ''next-amplitude" plots, an
amplitude is simply a variable's local maximum or peak value of each oscillation, within a sequence of
oscillations.
aperiodic (a) Not repeating with time, that is, lacking periodicity or quasiperiodicity (tantamount to having
an infinite period); (b) occurring as a transient or pulse only once over infinite time.
arithmetic mean Same as mean (the sum of all data values divided by the number of such values).
attractor The phase space point or set of points representing the various possible steady-state conditions
of a system; in other words, an equilibrium state or group of states to which a dynamical system converges.
attractor reconstruction See reconstruction of attractor.
autocorrelation Correlation of a variable at one time with itself at another time. Also called serial
correlation.
autocorrelation coefficient A dimensionless numerical indicator (calculated as autocovariance divided by
variance) of the extent to which the measurements of a time series are mutually related.
autocorrelation function The spectrum or entire series of autocorrelation coefficients for a time series.
autocorrelation time The time required for the autocorrelation function to drop to 1/e (= 1/2.718 = 0.37).
autocovariance A measure of the degree to which the observations of a time series are related to
themselves, numerically calculated as:
autonomous (a) Self-governing, propagating from within; (b) independent of time.
axis (a) One of a group of mutually perpendicular reference lines passing through the origin of a graph; (b)
a line about which a body or group of bodies can rotate.
axis vector Principal axis.

B
band-pass filter A filter that passes only those frequencies that fall within a desired range or band.
basic wave Fundamental wave.
basin of attraction The group of all possible phase space points that can evolve onto a given attractor.
basis A set of nonparallel and linearly independent vectors.
bifurcation (a) In general, a branching into parts or into connected segments (usually from one segment to
two); (b) any abrupt change in the qualitative form of an attractor or in a system's steady-state behavior, as
one or more parameters are changed.
bifurcation-rate scaling law An equation that estimates the critical parameter-value at which a particular
bifurcation occurs, within a series of bifurcations in period-doubling.
bifurcation diagram A graph showing all possible solutions (excluding transients) to an equation that
relates a variable to a control parameter. Such a graph usually is drawn specifically to show bifurcations
(period-doublings) of the variable.
bifurcation point A critical parameter value at which two or more branches of system-behavior emerge.
binary Having two equally likely, mutually exclusive possibilities.
binary digit Either of the two digits, conventionally 0 and 1, used in a binary system of reading numbers
or of naming numbers; as such, it's the smallest amount of information that can be stored in a computer.
binary system A system that is set up or operates on binary principles.
bit (a) Contraction of "binary digit;" (b) either of the digits 0 or 1; (c) the unit of information when logs are
taken to the base 2, that is, a measure of information equal to the decision content of one of two mutually
exclusive and equally likely values or states (sometimes also called a shannon).
box-counting dimension Similarity dimension as obtained from a grid of boxes.
box dimension Same as box-counting dimension.
broadband spectrum A frequency-domain (Fourier-analysis) plot revealing no outstanding periodicity.
butterfly effect Sensitive dependence on initial conditions. The "butterfly" name stems from the theoretical
possibility that a very slight change in the state of a system (such as a butterfly flapping its wings) might
create slightly different "initial" conditions and thereby influence the long-term resulting pattern (the
weather), even in some other part of the world.
byte (a) The amount of memory space needed to store one character on a computer (normally eight bits);
(b) a string of binary elements operated on or treated as a unit.

C
Cantor set A fractal obtained by dividing a line into equal subparts, deleting one or more of those parts,
and repeating that process indefinitely.
capacity A dimension Dc defined only at the limit where the scaling object approaches length zero; written
in equation form as:
cellular automaton A mathematical construction consisting of a system of entities, called cells, whose
temporal evolution is governed by a set of rules and whose behavior over time becomes, or at least may
appear, highly complex.
chain rule A rule that tells how to differentiate a function of a function, that is, how to find the derivative
of a composite function. The name stems from the fact that the various functions fit together like a chain.
For example, if y is a function of t, and t is a function of x, then the chain rule says that (dy/dx)=(dy/dt)
(dt/dx).
chaologist A person who studies chaos.
chaology The study of chaos.
chaos (a) Sustained and random-like long-term evolution that satisfies certain special mathematical criteria
and that happens in deterministic, nonlinear, dynamical systems: (b) largely unpredictable long-term
evolution occurring in a deterministic, nonlinear dynamical system because of sensitivity to initial
conditions.
chaos theory The principles and operations underlying chaos.
chaotic attractor An attractor that shows sensitivity to initial conditions (exponential divergence of
neighboring trajectories) and that, therefore, occurs only in the chaotic domain.
characteristic exponent Lyapunov exponent.
circle map Any of a class of nonlinear difference equations that map a point on a circle to another point on
the circle (Middleton 1995).
classical decomposition The breaking down of a time series into its component constituents of trend,
seasonality, cyclicity, and noise.
coarse graining Categorizing or partitioning, such as when reducing a set of observations of a continuous
variable to a stream of discrete symbols.
coefficient (a) A constant number or symbol prefixed as a multiplier to a variable or unknown quantity
(and hence a constant as opposed to a variable); (b) a dimensionless, numerical measure of a set of data, for
example, an autocorrelation coefficient.
complex number Any number of the form a+jb, where a and b are real numbers and j is an imaginary
number (the square root of -1).
complexity A type of dynamical behavior in which many independent agents continually interact in novel
ways, spontaneously organizing and reorganizing themselves into larger and more complicated patterns
over time.
conditional entropy The uncertainty in a measurement of variable y from system Y, given a value of
variable x from coupled system X.
conditional probability The likelihood that a particular state or event will happen, given that one or more
other specified states or events have taken place previously.
conservative Not losing energy (keeping instead a constant area or volume in phase space), with time.
constant A quantity that doesn't vary under specified conditions.
continuous Defined at all values of the given (independent) variable.
continuous random variable A random variable that can take on any value over a prescribed continuous
range.
control parameter A controllable constant or quantity, different values of which can produce different
cases or outcomes of an experiment or equation.
converge To approach a finite limiting value.
coordinate (noun) (a) One of a set of numbers that locate a point in space; (b) the axis of a graph, as in
"coordinate axes" or "coordinate system."
coordinate vector A vector whose starting point is at the origin (0,0) of the graph. The word "coordinate"
for such vectors is often dropped for convenience.
correlation dimension An exponent in a power law (i.e. slope of straight line) on a logarithmic plot of
correlation sum (as the dependent variable) versus the radius e of an encompassing circle, sphere, or
hypersphere.
correlation exponent Correlation dimension.
correlation integral Correlation sum.
correlation sum A normalized total number of pairs of points within a circle or sphere of radius e,
obtained by dividing that total sum by the total number of points on the attractor.
correlogram A plot of autocorrelation coefficient (on the ordinate) versus lag, on arithmetic scales.
crisis An abrupt, discontinuous change in a chaotic attractor as a parameter is varied, characterized by
either destruction of the chaotic attractor or its expansion to a much larger interval of x.
cubic polynomial A polynomial in which the highest power to which any variable is raised is 3.
cubic spline A smoothly varying, cubic-polynomial curve fitted between two data points, often used to
interpolate "data" values between the two measured points.
cycle A series of events or observations that occur in a fixed sequence and return to the original state and
which then repeat themselves in a regular pattern.
cyclicity Same as periodicity, that is, the tendency for any pattern to repeat itself over fixed intervals of
time.

D
data-adaptive Said of methods of analysis in which the operations are applied to classes or subgroups that
are defined differently from one to another, according to specified peculiarities of the data (e.g. number of
observations in a class).
decomposition (a) The numerical expression of a quantity in terms of its simpler components; (b) see
classical decomposition.
degrees of freedom The number of independent quantities (variables or parameters) or pieces of
information that must be specified to define the state of a system completely. The field of statistics also has
several other less common meanings.
delay-coordinate method Same as time-delay method.
delay method Same as time-delay method.
delay time (a) A delay, in time units, usually accompanied by an integer parameter to account for the
units; (b) in some authors' usage, same as lag time.
delay vector Same as lag vector.
density Same as probability density function.
density estimation The estimation of probabilities.
density function Same as probability density function.
dependent variable The output variable of a function, as determined by the values of the independent
variables.
derivative (a) Generally, a value that derives from, comes from, or is traceable to a particular source, such
as a point on a curve; (b) in mathematics, the slope (rate of change) of the line that is tangent to a point on a
curve.
deterministic (a) Completely and exactly specified (at least to within measuring accuracy) by one or more
mathematical equations and a given initial condition; (b) said of a system whose past and future are
completely determined by its present state and any new input.
deterministic fractal A fractal that looks exactly the same at all levels of detail.
detrend To transform trended data in such a way that they have an approximately constant mean and
variance with time, so that the trend is removed.
diffeomorphic Smooth and invertible.
diffeomorphism A differentiable mapping that has a differentiable inverse.
difference equation A recurrence equation based on changes that occur at discrete times and solved by
iteration. An example is the logistic equation. In mathematical jargon, a difference equation is an equation
in which a difference operator is applied to a dependent variable one or more times. (A difference operator
is a twofold mathematical operation in which (1) a small increment is added to the independent variable,
and then (2) the original value of the independent variable is subtracted.)
difference plot A graph on which the coordinates of each plotted point are successive differences. Each
successive difference is the difference between an observation xt and some later observation (on the
abscissa) and the difference between xt+m and a similarly lagged observation (on the ordinate).
differencing The filtering of a time series by subtracting each value, in turn, from a subsequent (lagged)
value. See also first-differencing, second-differencing, and difference plot.
differential equation An equation expressing a relationship between a function and one or more of its
derivatives and that therefore is based on changes that occur continuously.
digit A code character, such as a number from 0 to 9 or a letter of the alphabet.
digital In numerical form, as in (a) calculation by numerical methods or by discrete units or (b)
representation of data by numerical digits or discrete units.
digitize (a) To approximate by discrete samplings; (b) to put data into digital (numerical) notation (as for
use in a digital computer).
dimension Generally, a magnitude measured in a particular direction, as on the axes of a graph. In chaos
theory, "dimension" is used in any of several variations of that general meaning, such as (a) each axis of a
set of mutually perpendicular axes in Euclidean space; (b) the number of coordinates needed to locate a
point in space; (c) the maximum number of variables of a system; (d) any of various quantitative,
topological measures of an object's complexity, and other variations.
discrete Defined or occurring only at specified values.
discrete Fourier transform (DFT) A mathematical operation that transforms a series of discrete, equally
spaced observations measured over a finite range of time into a discrete, frequency-domain spectrum. Also
called a finite Fourier transform.
discrete random variable A random variable that only takes on certain specified outcomes or values, with
no possible outcomes or values in between.
discretize To extract equally spaced (in time) discrete values from a continuous time series.
dissipative system A system that loses energy with time. Evidence of energy loss with time includes an
irreversible evolution toward an asymptotic or limiting condition over time and a decrease of phase space
area or volume, with time.
distance formula The simple equation, based on the Pythagorean theorem, for finding the straight-line
distance c between two points in Nd-dimensional space:
c = ([x2-x1]2 + [y2-y1]2 + [z2-z1]2 + . . . + [w2-w1]2)0.5
where w is the Ndth variable of the group of Nd variables.
distribution function A mathematical operation that gives the proportion of members in a sample or
population having values less than or equal to any given value. See also frequency distribution and
probability distribution.
dot product A scalar quantity for two vectors that have a common origin, obtained either by (a)
multiplying their x coordinates (x1x2), their y coordinates (y1y2), etc. for any other dimensions involved, and
then summing those coordinate products (x1x2+y1y2+etc.), or (b) multiplying the vectors' lengths by the
cosine of their included angle.
dynamical Changing with time.
dynamical system (a) Anything that moves or that evolves in time; (b) any process or model in which each
successive state is a function of the preceding state.
dynamical-systems theory (a) The study of phenomena that vary with time; (b) a language that describes
the behavior of moving or evolving systems, especially as affected by external control parameters.
dynamics (a) That branch of physics (mechanics) that deals with forces and their relation to the motion and
sometimes the equilibrium of bodies; (b) the pattern of change or growth of an object or phenomenon.

E
embedding The preparation of a pseudo phase space graph to reconstruct a system's dynamics (attractor),
using successively lagged values of a single variable.
embedding dimension The total number of separate time series (consisting of the original series and
subgroups obtained by lagging that series) used in a pseudo phase space plot or in a more rigorous
mathematical analysis.
emergent Said of phenomena or systems in which new, increasingly complex levels of order appear over
time.
entropy (a) A measure of unavailable energy (thermodynamics), degree of disorder or disorganization,
probability (in inverse proportion), uncertainty, randomness, variety of choice, surprise, or information; (b)
a quantity,
computed for a discrete random variable whose ith outcome has probability P.
equation of motion An equation in which time is the independent variable.
equilibrium point Same as fixed point.
ergodic (a) The property whereby statistical measures of an ensemble don't change with time and, in
addition, all statistics are invariant from one time series to another within the ensemble; (b) said of a
system for which spatial or ensemble averages are equal to time averages (meaning that time averages are
independent of starting time and that most points visit every region of phase space with about equal
probability); (c) said of a trajectory if it comes back arbitrarily close to itself after some time; (d) the
property whereby averages computed from a data sample converge over time to ensemble averages (i.e.
statistics of all initial states ultimately lead to the same set of statistics). Eubank & Farmer (1990) mention
additional usages of the word and state that "there is no universally accepted definition of the word
'ergodic'."
ergodic hypothesis (a) The dynamical theory that says that, in the limit as the number of observations goes
to infinity, a time average equals a space average (i.e. the theory says that the point that represents the state
of the system spends, in each phase space compartment, an amount of time proportional to the volume of
that compartment); (b) the dynamical theory that says that, for a system in statistical equilibrium, all
accessible states are equally probable, so that the system passes rapidly through all of them.
ergodic theory (a) The mathematical study of the long-term average behavior of dynamical systems; (b)
the study of measure-preserving transformations; (c) ergodic hypothesis.
error The difference between a quantity and an estimate of the quantity.
Euclidean dimension The common or standard notion of "dimension" whereby a point has dimension
zero, a line dimension one, a surface dimension two, and a volume dimension three.
exponential (a) In a general sense, relating to powers (exponents); (b) referring to a rate of change that's
proportional to a constant raised to a power, where the power includes the independent variable (see
exponential divergence and exponential equation); (c) referring to a specific mathematical series known as
an "exponential series."
exponential divergence Temporal separation of two adjacent trajectories according to an exponential law,
that is, by a straight-line relation between the log of separation distance (as the dependent variable) and
time.
exponential equation (exponential function) An equation relating a dependent variable to some constant
raised to a power, where the power includes the independent (given) variable. Examples are y=cx and
y=acbx, where x and y are variables and a, b, and c are constants. An exponential equation plots as a straight
line on semilog paper, with the log scale being used for y (or y/a) and the arithmetic scale for x.

F
false nearest neighbor A point in lagged space that is close to another point only because the embedding
dimension is too low.
fast Fourier transform (FFT) Any member of a family of computer algorithms for calculating the various
real and imaginary parts of the discrete Fourier transform (DFT) efficiently and quickly in about Nlog2N
operations.
feedback That part of the output that returns to serve as input again, in a temporal process.
Feigenbaum constant Same as Feigenbaum number.
Feigenbaum number A universal constant (4.6692 . . .), discovered in the mid-1970s by Mitchell
Feigenbaum, that represents the rate (in the limit where number of periods, n, becomes infinite) at which
new periods appear during the period-doubling route to chaos. The Feigenbaum number (also known as the
Feigenbaum constant) is defined as:
fidelity A measure of how closely linked one measurement is to its predecessor.
fiducial Referring to something used as a standard of reference for measurement or calculation. Examples:
"fiducial point," fiducial trajectory.
fiducial trajectory A trajectory used as a reference trajectory from which to compute orbital gaps and the
Lyapunov exponent.
filter (linear filter) Any mathematical technique or operator that systematically changes an input series,
such as a time series or power spectrum, into a new (output) series that has certain desired qualities. Some
purposes of filters are to (a) eliminate periodicity or trend; (b) reduce or eliminate noise; (c) suppress high
or low frequencies; and (d) remove autocorrelation.
final state Point attractor.
first derivative The initial derivative of a function.
first-difference plot A graph on which the coordinates of each plotted point are first differenced data, in
the form of the difference between xt+m and xt on the abscissa and the difference between xt+2m and xt+m on the
ordinate, in which m is lag.
first-difference transformation A transformation performed by first-differencing the data.
first-differencing Calculation of the difference between a measured value (xt) and a lagged measurement
(xt+m), for all values in a time series. This is the most common form of differencing.
first harmonic A wave that has the same frequency as the fundamental wave, in Fourier analysis.
first principal axis The principal axis that is stretched the most (or reduced the least) in the phase space
evolution of an arbitrarily designated element.
first-return map A (pseudo) phase space model of Poincaré-section data, giving the value of some
variable as related to its preceding value at that section.
fixed point (a) In discrete processes, a single phase space point that is its own iterate (a point for which
xt+1=xt); (b) in continuous processes, a constant, time-independent solution to a differential equation. Also
called equilibrium point or steady state. See also stable fixed point and unstable fixed point.
fixed-point attractor Point attractor.
flip bifurcation Period-doubling.
flow (a) A set of differential equations; (b) a phase space trajectory or bundle of trajectories obtained by
solving a set of differential equations.
folding A topologist's explanation of how a particular range of output values results from (a) two different
ranges of input values during iteration; or (b) continued phase space evolution of a chaotic trajectory as it
reaches the limiting value of the variable(s) and then rebounds or deflects back onto the attractor.
forced oscillator A device to which extra energy is periodically added, by some external means.
fork-width scaling law An equation that estimates the width between the two parts of any particular
bifurcation (fork), within a series of bifurcations in period-doubling.
Fourier analysis A mathematical technique for uniquely describing a time series in terms of the
frequency-domain characteristics of its periodic constituents.
Fourier coefficient (a) A numerical value reflecting the strength of a particular constituent wave relative
to that of other constituent waves; (b) one of the coefficients needed to express a function formally in terms
of its Fourier series.
Fourier cosine series An equation for getting wave height for a particular harmonic by adding the cosines
of angles associated with constituent wavelengths, in Fourier analysis.
Fourier integral A mathematical expression that extends the Fourier series to the more general situation
of an infinitely long period by decomposing a continuous time series into sinusoidal constituents at all
frequencies and merging the variances into a continuous distribution of variances.
Fourier series An equation that describes a periodic time series in terms of the cosines and/or sines of
constituent harmonics and their associated coefficients.
Fourier sine series An equation for getting wave height for a particular harmonic by adding the sines of
angles associated with constituent wavelengths, in Fourier analysis.
Fourier transform A mathematical frequency-domain characterization of a time series, consisting of
constituent amplitudes and phases at each frequency.
fractal (a) A pattern that repeats the same design and detail or definition over a broad range of scale; (b) a
set of points whose dimension is not a whole number (Lorenz 1993).
fractal dimension A generic term for any dimension (e.g. similarity dimension, capacity, Hausdorff-
Besicovich dimension, correlation dimension, etc.) that can take on a non-integer value.
fractional dimension Fractal dimension.
frequency (a) In physics, the number of repeating wavelengths, periods, or cycles in a unit of time; (b) in
statistics, the number of observations or individuals in a class; (c) also in statistics, a probability or
proportion (relative frequency).
frequency analysis Same as Fourier analysis.
frequency distribution A list of class intervals or values and their associated number of observations
(frequencies). The frequencies or number of observations in each class are often normalized to range from
0 to 1 and are often assumed equal to probabilities.
frequency domain The representation of time series data in terms of their frequencies (or of some other
wave characteristic) and respective variances.
frequency locking Frequency adjustment by an oscillator in response to some periodic stimulus.
frequency spectrum A plot of the distribution of calculated powers as a function of frequency, as obtained
in a Fourier analysis. Also known as power-density spectrum, power spectrum, or spectral density.
function (a) An output variable or dependent variable whose value is uniquely determined by one or more
input (independent) variables; (b) an equation or relation between two groups A and B such that at least
one member of group A is matched with one member of group B (a "single-valued" function) or is matched
with two or more members of group B (a "multivalued" function). A single-valued function, for instance, is
y=3x; for any value of x, there is only one value of y, and vice versa. A multivalued function is y=x2; for
y=4, x can be+2 or -2, that is, one value of y is matched with two values of x. Many authors use "function''
to mean single-valued function.
fundamental frequency The frequency of the selected basic wave in a Fourier analysis.
fundamental wave A wave (usually the longest available, or the length of the time series) chosen as a
standard or reference wave in Fourier analysis.
fundamental wavelength The wavelength of the selected basic wave in a Fourier analysis.

G
geometric progression A sequence of terms whose successive members differ by a constant ratio or
multiplier. (Also known as a geometric series.)
geometric series A series of numbers in which the ratio of each member to its predecessor is the same
throughout the sequence. Also known as a geometric progression. Example: 1, 2, 4, 8, 16, 32, etc., in
which the ratio of any number to its predecessor is 2.
golden mean The unique value obtained by sectioning (a) a straight line such that the ratio of the shorter
segment to the longer segment equals the ratio of the longer segment to the total length, or (b) the sides of a
rectangle such that the ratio of their difference to the smaller equals the ratio of the shorter to the longer.
Gram-Schmidt orthogonalization Same as orthogonalization.

H
Hamiltonian system A system with no friction.
harmonic (a) As an adjective, expressible in terms of sine and cosine functions; (b) as a noun, any
component of a periodic quantity having a frequency that's an integer multiple of a given fundamental
frequency. In a Fourier analysis, for example, the first harmonic is the fundamental wave; the second
harmonic is the constituent wave having a frequency twice that of the fundamental wave; the third
harmonic is the constituent wave that has a frequency three times that of the fundamental wave; etc.
harmonic analysis The frequency-domain description of a time series, especially of a periodic function, by
summing sine and cosine functions (its constituent harmonics).
harmonic number The number corresponding to a particular harmonic. For example, the harmonic
number of the second harmonic is two, that of the third harmonic is three, etc.
Hausdorff dimension That critical dimension DH at which the computed value of the measure Me changes
abruptly from zero to infinity (or from infinity to zero), using the relation .
Hausdorff-Besicovich dimension Same as Hausdorff dimension.
Heaviside function A simple mathematical function or number that is zero if a specified expression is
negative (less than zero) and 1 if it is positive (zero or greater).
Hénon map A phase space plot of iterates of the equations xt+1=yt+1-axt2, and yt+1=bxt where a and b are
constants.
high-pass filter A filter that lets high frequencies pass and blocks low frequencies.
histogram A bar diagram showing the frequency distribution of a variable.
homoclinic orbit An orbit that is asymptotic to a fixed point or periodic orbit and that emanates from the
same point or orbit (Lorenz 1993).
homoclinic point The fixed point from which a homoclinic orbit emanates and which it subsequently
approaches (Lorenz 1993).
Hopf bifurcation An abrupt increase (by one) in the number of fundamental frequencies of a system,
caused by the increase of a control parameter past a critical value. Commonly applied in the quasiperiodic
route to chaos. Named for the work done on the subject by mathematician Eberhard Hopf in 1942.
hypercube (a) An imaginary phase space zone or subspace of four or more dimensions and characterizable
by a length; (b) the multidimensional analog of a cube.
hyperspace Space of more than three dimensions.
hypersphere (a) An imaginary phase space zone or subspace of four or more dimensions and
characterizable by a length (e.g. radius); (b) the multidimensional analog of a sphere.

I
identity line (or identity map) A 45° straight line on an arithmetic-scale two-coordinate graph,
representing the relation y=x.
imaginary number Any number consisting of a real number times the square root of minus one.
incremental redundancy The amount of change in the redundancy of a vector when we increase its
dimension by one, at a constant lag.
independent variable (a) An input number to a function; (b) a variable unaffected by the value taken on
by other variables; (c) a variable that an experimenter deliberately manipulates, to find its effect on some
other quantity (the latter being the dependent variable).
information A numerical measure of (a) knowledge or content of any statement, (b) how much is learned
when the contents of a message are revealed or (c) the uncertainty in the outcome of an experiment to be
done.
information dimension The slope of a straight line on a semilog plot of information, here defined as Ie
(dependent variable, on arithmetic scale) versus 1/e (log scale), where e is characteristic size of measuring
device and
information entropy A measure devised by Shannon (Shannon & Weaver 1949; called by him simply
"entropy") for the amount of information in a message, and identical in equation form to thermodynamic
entropy.
information theory The formal, standard treatment of information as a mathematically definable and
measurable quantity.
initial conditions Values of variables at the beginning of any specified time period.
inner product Same as dot product.
integer A whole number with no decimal or fractional part.
integral (a) A sum obtained by adding a group of separate elements; (b) the result of mathematically
integrating a function or an equation.
intermittency A complex steady-state behavior (often a route to chaos) consisting of orderly periodic
motion (regular oscillations, with no period-doubling) interrupted by occasional bursts of chaos or noise at
uneven intervals.
interpolation The process of estimating one or more intermediate values between two known values.
interval (a) The length of time between successive events; (b) a set of real numbers that fall between two
end-member real numbers.
invariant (a) Independent of the particular coordinates, that is. unaffected by a change in coordinates or by
a particular transformation, such as a change from an original phase space to a time-delay reconstruction;
(b) remaining forever in (never escaping from) a particular region of phase space; (c) unchanged by the
system's dynamics over time.
invariant manifold A collection of phase space trajectories, none of which ever leave the collection.
invariant measure (a) A measurable property that's unchanged by transformations; (b) a measurable
feature that doesn't change with time, that is, doesn't change under the action of the dynamics; (c) a
probability-distribution function describing the long-time likelihood of finding a system in a particular zone
of phase space.
invariant probability distribution The frequency distribution that is approached as time goes to infinity.
Also known as (an attractor's) probability distribution, probability density, probability density distribution,
invariant measure, and other combinations of these terms.
invertible (a) Having an inverse; (b) having a unique successor or predecessor, or in other words, capable
of being solved uniquely either forwards or backwards in time. Example: capable of indicating either xt or
xt+m, given the other.
isotropic Independent of direction.
iterate (a) As a verb, to repeat an operation over and over, often with the aim of coming closer and closer
to a desired result; (b) as a noun, a value calculated by any mathematical process of successive
approximation.
iteration (a) Any process of successive approximation; (b) repeated application of a mathematical
procedure, with the outcome of one solution fed back in as input for the next; (c) each successive step of an
iterative process.

J
joint entropy The average amount of information obtained (or average amount of uncertainty reduced) by
individual measurements of two or more systems.
joint probability The probability that two specified events (usually independent events) will happen
together. "Together" doesn't necessarily mean "simultaneously."
joint probability distribution A list, table, or function for two or more random variables, giving the
probability of the joint occurrence of each possible combination of values of those variables.

K
K-S entropy Kolmogorov-Sinai entropy.
kernel density estimator A probability-estimation technique in which a fixed, local probability
distribution (kernel) is applied to the neighborhood centered on each datum point.
Koch snowflake A geometric pattern formed from a straight line by applying a particular, constant,
repeated generating procedure. The most common generating procedure is to replace the middle third of the
line with an equilateral triangle (also known as "von Koch snowflake").
Kolmogorov-Sinai (K-S) entropy Entropy (based on sequence probabilities) per unit time in the limits
where time goes to infinity and bin size goes to zero. Also known as source entropy, entropy of the source,
measure-theoretic entropy, metric-invariant entropy, and metric entropy.

L
lag The basic time interval or amount of offset between any two values being compared, within a time
series.
lag-m autocorrelation coefficient Autocorrelation coefficient.
lag-m serial correlation coefficient Autocorrelation coefficient.
lag space A special but very common type of pseudo phase space in which the axes or dimensions
represent successive values of the same feature (x) separated by a constant time interval.
lag vector A plotted point as defined by successively lagged values of some variable in reconstructed
(pseudo) phase space.
lagged phase space See lag space.
limit cycle A self-sustaining phase space loop that represents periodic motion. Hence, it's a periodic
attractor.
line spectrum Periodogram.
linear (a) Pertaining to lines, usually straight ones; (b) having no variable raised to a power other than one
(see linear equation).
linear equation An equation having a straight line for its graph, that is, an equation in which the variables
are raised to the first power only (also called a polynomial equation of the first degree). Examples of linear
equations are ax+b=0, y=c+bx, and ax+by+cz=0, where x, y, and z are variables and a, b, and c are
parameters.
linear filter See filter.
linear function A mathematical relationship in which the variables appear only in the first degree,
multiplied by constants, and combined only by addition and subtraction.
linear interpolation Interpolation based on straight-line relations between observations.
linear system A system in which all observations of a given variable plot as a straight line (arithmetic
scales) against observations of a second (or lagged) variable. In lag space, for example, all observations
then are said to be "linearly related" to observations at a later time.
local Lyapunov exponent The exponential rate of trajectory convergence or divergence in a local region
of an attractor.
logarithm An exponent that is the power to which a base number (usually 10, 2, or e) is raised to give
another number.
logarithmic equation An equation of the form ay=cxd, in which a, c, and d are constants. Such an equation
plots as a straight line on semilog paper (with x on the log scale and y on the arithmetic scale).
logistic (a) In mathematics and statistics, referring to a special type of so-called growth curve (an
expression that specifies how the size of a population varies with time) (however, that curve isn't the
"logistic equation" of chaos theory; see logistic equation); (b) in a general sense, skilled in computation;
(c) in military usage: referring to the provision of personnel and material.
logistic equation (a) Historically and generally, the relation xt+1=k/(1+ea+bx) where the constant b is less than
zero; (b) in chaos theory, the relation xt+1=kxt(1-xt). Both equations are iterative types that are popular in
biology as models for population growth.
low-pass filter A filter that lets low frequencies pass and blocks high frequencies.
Lyapunov characteristic exponent Lyapunov exponent.
Lyapunov characteristic number Lyapunov number.
Lyapunov exponent The average of many local exponential rates of convergence or divergence of
adjacent trajectories, expressed in logarithms and measured over the entire attractor. As such, it reflects the
average rate of expansion or contraction of neighboring trajectories with time.
Lyapunov number The number whose logarithm, to a given base, equals the Lyapunov exponent.

M
manifold (a) In general, an object consisting of many diverse elements; (b) in mathematics, a class with
subclasses (e.g. a plane is a two-dimensional manifold of points because it is the class of all its points); (c)
any smooth geometric object (point, curve, surface, volume, or multidimensional counterpart); (d) in the
nonchaotic domain, an attractor; (e) the basic space of a dynamical system.
map (a) A function, mathematical model, or rule specifying how a dynamical system evolves; (b) a
synonym for correspondence, function, or transformation; (c) the mathematical process of taking one point
to another. In the latter sense, a map tells how x will go, by a discrete step, to a new value of x. More than
one variable can be involved. Common forms of a map are an iterative equation and a graph; in graphical
form, a map shows a historical sequence of values.
mapping (a) A series of iterations of a map; (b) a dynamical system whose variables are defined only at
discrete times; (c) a function, correspondence, or transformation.
marginal distribution The complete distribution (summing to 1.0) of probabilities for any chosen variable
within a joint distribution.
marginal probability The probability of getting a particular value of one variable within a joint
distribution, regardless of the value of the other variable.
marginal probability distribution Same as marginal distribution.
marginal redundancy Same as incremental redundancy.
marginally stable Tending to keep a perturbation at about its original magnitude, over time.
mathematical fractal An object having the property that any suitably magnified part looks exactly like the
whole.
mean Arithmetic average (sum of values divided by the number of values).
measure (noun) (a) The size or quantity of something expressed in terms of a standard unit; (b) a scalar
associated with a vector and indicating its magnitude and sense but not its orientation; (c) the probability of
finding a value of a variable within a particular domain.
measure-preserving transformation A one-to-one transformation made such that the measure of every
set and of its inverse image agree.
measure-theoretic entropy Same as Kolmogorov-Sinai (K-S) entropy.
metric (noun) (a) In general, a standard of measurement (e.g. "there is no metric for joy"); (b) in
mathematics, a way of specifying values of a variable or positions of a point (e.g. "a Euclidean metric");
(c) also in mathematics, a differential expression of distance in a generalized vector space; (d) (adjective)
referring to measurement or to the meter (100 centimeters).
metric entropy Same as Kolmogorov-Sinai (K-S) entropy.
metrical Quantitative.
microstate A phase space compartment, possible outcome, or solution.
model A simplified representation of a real phenomenon, in other words, a stripped-down or
uncomplicated description or version of a real-world process. Models can be classified into physical
(scale), mathematical, analog, and conceptual models.
monotonic Always increasing or always decreasing, that is, pertaining to a continuous line along which the
slope keeps the same sign at all points.
moving average A transformed time series in which each value in the original time series is replaced by
the arithmetic mean of that value and a specified, constant number of adjacent values that come before and
after it.
multifractal Having different fractal scalings (dimension values) at different times or places, on an
attractor.
multitaper spectral analysis A type of Fourier analysis that optimally combines information from
orthogonal tapered estimates to minimize leakage and bias in the spectral estimate at each frequency.
mutual information (a) The amount by which a measurement of one variable reduces the uncertainty in
another; (b) the quantity of information one system or variable contains about another; (c) a measure of the
degree to which two processes or random variables are mutually related; (d) the amount of surprise or
predictability associated with a new measurement.
mutually exclusive A statistical term meaning that only one of various possible outcomes can occur at a
time.

N
nat The unit of measurement for the separation (or convergence) rate of two neighboring trajectories when
logs are taken to base 2.
natural fractal An object having the property that any suitably magnified part looks approximately like
the whole, the differences being minor, negligible and ascribable to chance.
natural measure A measure or observation that is a convergent time-average value.
natural probability measure Invariant measure.
nearest neighbor A pseudo phase space datum point that plots close to another point, for a particular
embedding dimension.
next-amplitude map A one-dimensional first-return map that uses only the high values of successive
oscillations (the successive maxima) of the time series.
next-amplitude plot Next-amplitude map.
noise (a) Any unwanted disturbance superimposed on useful data and tending to obscure their information
content; (b) unexplainable variability or fluctuation in a quantity with time; (c) in a general sense, anything
that impedes communication. Such disturbance or variability may be random fluctuations, reading errors,
analytical errors, sampling errors, and other factors.
nonautonomous Time-dependent.
noninvertible Not capable of being uniquely solved backwards in time, for example, not able to indicate xt
from a given value of xt+m.
nonlinear Not having a straight-line relationship, that is, referring to a response that isn't directly (or
inversely) proportional to a given variable.
nonlinear dynamics The study of motion that doesn't follow a straight-line relation, that is, the study of
nonlinear movement or evolution. As such, nonlinear dynamics is a broad field that includes chaos theory
and many mathematical tools used in analyzing complex temporal phenomena.
nonlinear equation An equation in variables x and y which cannot be put in the form y=c+bx, where b and
c are coefficients.
nonlinear system A system in which the observations of a given variable do not plot as a straight line (on
arithmetic scales) against observations of a second (or lagged) variable.
nonmonotonic Pertaining to a continuous line along which the slope changes sign.
nonparametric Not involving assumptions about specific values of parameters or about the form of a
distribution. ("Parameter" in this case usually refers to the statistical definition, namely numerical
characteristics of a population.)
nonperiodic Same as aperiodic.
nonstationary Said of a time series for which (a) a moving average isn't approximately constant with time
or (b) the mean and variance aren't approximately constant with time.
non-uniformity factor (NUF) The standard deviation of the local rates of convergence or divergence of
neighboring trajectories, sampled over the entire attractor.
norm The magnitude of a vector.
normal distribution A special type of symmetrical (bell-shaped) and continuous frequency distribution,
the graph or curve of which is given by a particular general equation (not reproduced here).
normalization The process of adjusting or converting one or more values to some standard scale. The
standard scale for a group of values usually is from 0 to 1. The conversion then consists of dividing each
value of the original dataset by some maximum reference quantity, such as the greatest value in the dataset
or a theoretical maximum value. A vector is normalized by dividing it by its magnitude, yielding a socalled
unit vector. Probability distributions are normalized by changing the variable so that the distribution
has a mean of 0 and a variance of 1.
null hypothesis A hypothesis that supposes no significant difference between a statistic for one group and
the same statistic for another.

O
observable (noun) A physical quantity that can be measured.
one-dimensional map An equation (in the form of either a written statement or a graphical plot) that gives
the value of a variable as a function of its value at one or more earlier times. The typical expression is xt+1
as some function of xt.
orbit The path through space taken by a moving body or point. Examples: (a) a trajectory that completes a
circuit (as, "the Moon orbits the Earth"); (b) a trajectory or chronological sequence of states as represented
in phase space.
ordinary probability A nonstandard term used in this book to mean the likelihood of getting a specified
state at a particular time.
origin A reference point in ordinary space or phase space. Most often, it's the point at which all variables
have a value of zero.
orthogonal (a) Perpendicular or normal (having to do with right angles); (b) unrelated or independent; (c)
said of elements having the property that the product of any pair of them is zero.
orthogonalization A procedure for realigning two or more nonorthogonal vectors into a set of an equal
number of mutually orthogonal vectors, all of which have the same origin.
orthonormal Said of axes or vectors that are mutually perpendicular and of normalized (unit) length.
orthonormalization The process of reducing mutually orthogonal vectors to unit length.

P
parabola The curve or path of a moving point that remains equidistant from a fixed point (the focus) and
from a fixed straight line (the directrix).
parallelogram law A graphical method for adding two vectors, whereby their starting points are placed
together to form two sides of a parallelogram, the two opposite sides then are drawn in, and the sum or
resultant is given by the diagonal drawn from the two starting points to the opposite corner.
parameter (a) In physics, a controllable quantity kept constant in an experiment as a measure of some
influential or driving environmental condition; (b) in mathematics, an arbitrary constant in a mathematical
expression and that can be changed to provide different cases of the phenomenon represented; (c) in
mathematics, a special variable in terms of which two or more other variables can be written; (d) in
statistics, a numerical characteristic of a population (such as, for example, the arithmetic mean).
parametric Involving parameters.
parametric equations In math, a set of equations that express quantities in terms of the same set of
independent variables (called parameters).
partition (a) As a noun, the collection of possible outcomes of an experiment; (b) as a noun, a
compartment (bin, cell, etc.) or group of compartments within phase space; (c) as a verb, to subdivide a
phase space or the range of values of a variable into a set of discrete, connected subintervals.
partitioning The process of dividing a dataset into classes or groups such that any one observation is a
member of only one class.
percentage-of-moving-average method Same as ratio-to-moving-average method.
period The amount of time needed for a system to return to its original state, that is, the time required for a
regularly repeating phenomenon to repeat itself once.
period-doubling A process whereby increases in a control parameter in certain iterated equations produce
trajectories made up of a successively doubled number of attractor values (e.g. 2, 4, 8, 16, etc. attractor
values), eventually leading to chaos.
periodic Regularly repeating. The repetition can be exact (in a pure mathematical sense) or approximate
(as with virtually all measured data).
periodic attractor An attractor consisting of a finite set of points that form a closed loop or cycle in phase
space.
periodic toroidal attractor A torus on which the composite trajectory's motion repeats itself regularly,
i.e., on an exact or integer basis.
periodic points Points (values) that are members of a cycle (and hence that recur periodically).
periodicity (a) The number of measured observations or iterations to a cycle or period; (b) the quality of
recurring at a definite interval (in other words, repetition of a given pattern over fixed intervals).
periodogram A graph showing the relative strengths (usually variances) of constituent waves at their
discrete frequencies. Discrete wave periods can be used in place of frequencies and are probably the origin
of the name.
perturbation (a) A difference between two neighboring observations, at a given time: (b) an intentional
displacement of an observation, at any given time; (c) a deliberate change (usually slight) in one or more
parameters of an equation.
perturbation vector Principal axis.
phase (a) The stage that a dynamical system is in at any particular time; (b) the fraction of a cycle through
which a wave has passed at any instant.
phase angle The starting angle or reference point within a wave cycle, from which a process begins.
phase diagram Same as phase portrait.
phase locking The beating in harmony or resonating together of many individual oscillators.
phase portrait A phase space plot of a system's various possible conditions, as shown by one or more
trajectories.
phase space An abstract mathematical space in which coordinates represent the variables needed to
specify the phase (or state) of a dynamical system at any time.
phase space portrait Same as phase portrait.
phase space reconstruction See reconstruction of phase space.
pitchfork bifurcation The spawning of two equilibrium points from one equilibrium point, as in perioddoubling.
Poincaré map Same as return map.
Poincaré return map Same as return map.
Poincaré section A slice or cross section in phase space, cutting through an attractor transverse to the
flow. The section shows dots that represent the trajectory's intersections with the plane or slice.
point A state of a system (values of its variables at a given time).
point attractor A single fixed point in phase space, representing the final or equilibrium state of a system
that comes to rest with the passage of time.
polynomial An algebraic expression involving two or more summed terms, each term consisting of a
constant multiplier and one or more variables raised to nonnegative integer powers. (In some mathematics
texts, a polynomial can have one term.)
population Any group of items that is well defined (has a common characteristic).
power (a) The number of times, as indicated by an exponent, in which a given number (say, x) is used
when being repeatedly multiplied by itself (e.g. x3=x·x·x, and we say x is raised to the third power); (b) a
term used in Fourier analysis as a synonym for variance.
power equation An equation of the form y=axc, where x and y are variables and a and c are constants. A
power equation plots as a straight line on log paper. See also power law.
power law Any equation in which the dependent variable varies in proportion to a specified power of the
independent variable. Synonymous with power equation.
power spectrum The ensemble of variances (powers) calculated in a Fourier analysis and plotted against
their respective wave characteristic (frequency, wavelength, period, or harmonic number).
precision (a) In general, the accuracy with which a calculation is made; (b) reproducibility, as with
repeated samplings.
principal axis Any of a body's two or three axes of structural symmetry that were mutually perpendicular
before deformation.
probability (a) Limiting relative frequency; (b) the ratio of the number of times an event occurs to the
number of trials; (c) the encoding of all that we know about the likelihood that a particular event will
happen, with the encoding expressed as a number between zero (no chance that the event will take place)
and 1 (certainty that the event will happen); (d) the formal study of chance occurrences.
probability density Same as probability density function.
probability density curve Same as probability density function.
probability density distribution Same as probability density function.
probability density function The limiting relative frequency density as sample size becomes infinitely
large and bin width goes to zero, for a continuous random variable.
probability distribution A list of the possible outcomes of an experiment and their associated
probabilities, that is, a table or function that assigns to each possible value of a random variable the
probability of that value's occurrence. Also known (possibly depending on whether the variable is discrete
or continuous) as distribution function, statistical distribution, probability density function, density
function, or frequency function.
projection The image of a geometric object or vector superimposed on some other vector. As such, the
projection is a new vector and is called the projection of the first vector onto the second.
pseudo phase space An imaginary graphical space in which the first coordinate represents a physical
feature and the other coordinates represent lagged values of that feature.
Pythagorean theorem The theorem that, in a right triangle, the square of the length of the hypotenuse
equals the sum of the squared lengths of the other two sides.

Q
quadratic Of the second degree, that is, involving one or more unknowns (variables) that are squared but
none that are raised to a higher power.
quadratic map An iterated equation (and sometimes also the phase space plot of those iterates) in which
xt+1 is given as some function of xt2.
quasiperiodic toroidal attractor A torus on which the composite trajectory's motion almost but not
exactly repeats itself regularly.
quasiperiodicity (a) A dynamical state characterized by the superposition of two or more periodic motions
(characteristic frequencies); (b) a route to chaos caused by the simultaneous occurrence of two periodicities
whose frequencies are out of phase (not commensurate) with one another.

R
random (a) Based strictly on a chance mechanism (the luck of the draw), with negligible deterministic
effects (the definition used in this book); (b) disorganized or haphazard; (c) providing every member an
equal chance of selection; (d) unlikely repeatability of any observation; (e) unpredictability of individual
events to within any reasonable degree of certainty; (f) difficult to compute.
random process A process based on random selection, as in a process in which new observations depend
strictly on chance, or in which every possible observation has an equal chance of selection.
random variable (a) A variable whose value cannot be foretold, except statistically; (b) a variable that
takes on different values at different random events (experiments or trials).
ratio-to-moving-average method A classical decomposition technique for isolating the value of
[seasonality times random deviations] by dividing the value of the variable by the moving-average value,
at each sequential time.
reconstruction dimension The embedding dimension in which an attractor is reconstructed.
reconstruction of attractor The graphical or analytical recreation of the topology or es sence of a
multidimensional attractor by analyzing lagged values of one selected variable.
reconstruction of phase space A pseudo (lagged) phase space plot in two or three dimensions, made with
the hope of seeing an attractor (if there is one). Also known as phase space reconstruction, state space
reconstruction, phase portrait reconstruction, trajectory reconstruction, and similar expressions.
recursive Referring to any process or function in which each new value is generated from the preceding
value. (Hence, iterative.) Examples: (a) the logistic equation of chaos theory; (b) forecasting techniques in
which each successive forecast incorporates the preceding forecast.
reductionism The notion that the world is an assemblage of parts.
redundancy The multidimensional generalization of mutual information.
regression A mathematical model stating how a dependent variable varies with one or more independent
variables.
relative entropy A ratio of two entropies (typically an actual entropy to a reference entropy), reflecting the
magnitude of one relative to the other.
relative frequency The ratio of number of occurrences of an outcome to total number of possible
occurrences.
relative frequency density The ratio of relative frequency to class width.
relative frequency distribution A list of relative frequencies, for a particular process or system.
renormalization A mathematical scaling technique consisting of rescaling a physical variable and
transforming a control parameter such that the properties of an equation at one scale can be related to
those at another scale, and the properties at the limiting scale (infinity) can be determined.
reorthonormalization A procedure for again realigning all vectors to be mutually perpendicular and then
making each vector of unit length.
residual The difference between an observed value and the value predicted by a model fitted to the general
dataset.
resolution (a) The act of breaking something up into its constituent elements; (b) the act of determining or
rendering distinguishable the individual parts of something; (c) the size of the biggest element in a
partitioning, such as the length of a box or radius of a sphere.
resultant A vector that is the sum of two or more other vectors.
return map A rule, function, or model that tells where the trajectory will next cross a particular onedimensional
Poincaré section, given the location of a previous crossing at that section. ''Locations" along
the line are described in terms of distance (L) along the line from some arbitrary origin on the line.
Commonly, a first-return (or first-order-return) map is a model of Ln+1 as a function of Ln, for the Poincaré
section; a second-return (or second-order-return) map is a model of Ln+2 versus Ln; and so on.
Richardson plot A logarithmic graph of a coastline's measured length (on the ordinate) versus the
associated ruler length.
robust Resistant to, or steady under, perturbation.

S
saddle (saddle point) An unstable steady-state phase space condition (fixed point, limit cycle, etc.), the
flow near which is stable in some directions (converges toward the fixed point) and unstable in others
(moves away from the fixed point).
scalar A number representing a magnitude only, as might be indicated on a simple scale.
scalar product Same as dot product.
scalar time series An ordinary time series, in which each successive measurement is recorded along with
its time or order of measurement (and, hence, synonymous with the general meaning of time series).
scale (a) A sequence of collinear marks, usually representing equal steps or increases, used as a reference
in measuring; (b) the proportion that a model bears to whatever it represents (e.g. a scale of one centimeter
to a kilometer); (c) to reduce or enlarge according to a fixed ratio or proportion; (d) to determine a quantity
at some order of magnitude by using data or relationships known to be valid at other orders of magnitude.
scale invariance The property whereby an object's size cannot be determined or estimated without
additional information because the item looks like any of a family of objects of different size but similar
appearance.
scale length Length of tool or device used (usually by successive increments) to estimate an object's size.
scaling ratio A proportion (and hence a value less than 1.0), used in scaling, and equal to that fraction of
the original object which the new (smaller) object represents as measured in the direction of any
topological dimension.
scaling region A straight-line relation (the slope of which is the correlation dimension) on a plot of log of
correlation sum versus log of measuring radius.
Schmidt orthogonalization Same as orthogonalization.
seasonality Same as periodicity (the tendency for any pattern to repeat itself over fixed intervals of time).
In this book, the fixed intervals of time for seasonality are arbitrarily taken as one year or less, whereas
periodicity is used as a generic or general term and can have any fixed interval.
second derivative The derivative of the first derivative of a function.
second-differencing The computation of differences of differenced data.
second harmonic A wave that has twice the frequency as the fundamental wave, in Fourier analysis.
second-order return map Second-return map.
second-return map A pseudo phase space plot of Poincaré-section data on which each observation at the
section is plotted against its second return to that section.
self-affine That property whereby an object is reproduced by a transformation involving different scaling
factors for different dimensions or variables.
self-entropy (a) The entropy for one system or variable; (b) the average amount of information gained by
making a measurement of(or provided by) a random process about itself.
self-information Same as self-entropy.
self-organization The spontaneous, self-generated occurrence of some kind of pattern or structure from an
(usually) orderless dynamical system.
self-organized criticality The property whereby a dynamical system naturally evolves toward a critical
state (a condition where the system undergoes a sudden change). Various perturbations of the system at
that critical state provoke different responses that follow a power law.
self-similarity The property whereby any part of an object, suitably magnified, looks mostly like the
whole.
self-similarity dimension Similarity dimension.
sensitive dependence on initial conditions (a) The quality whereby two slightly different values of an
input variable evolve to two vastly different trajectories; (b) the quality whereby two initially nearby
trajectories diverge exponentially with time.
sensitivity to initial conditions Sensitive dependence on initial conditions.
separatrix A boundary between regions of phase space, such as a boundary between two basins of
attraction in a dissipative system.
sequence probability A nonstandard term used in this book to mean the joint probability of certain
successive events in time and representing the probability that a given sequence of events will take place.
serial correlation Same as autocorrelation.
set A group of points or values that have a common characteristic or rule of formation.
similarity dimension A dimension D defined as D=logN/log(1/r), used in transforming figures into similar
figures of different sizes.
sine wave A wave corresponding to the equation y=sinx, or more generally, y=A sin (x+f), where A is wave
amplitude and f is phase angle.
singular spectral analysis An alternative to Fourier analysis that uses empirical basis functions formed by
a principal component analysis of the embedded phase space data matrix instead of the sine/cosine basis
functions.
singular system analysis A phase space reconstruction technique in which orthonormal reconstruction
axes near each point xt are a maximal set of linearly independent vectors that are derived from the local
distribution of points near xt by singular value decomposition.
singular value decomposition Singular system analysis.
sink A phase space point toward which all nearby points flow.
sinusoid (a) The sine curve (y=sin x); (b) any curve derivable from the sine curve by multiplying by a
constant or by adding a constant; (c) any curve shaped like a sine curve but with different amplitude,
period, or intercepts with the axis.
sinusoidal (a) Relating to, and shaped like, a sine curve; (b) describable by a sine or cosine equation.
smooth Said of a frequency distribution that is continuous or, in mathematical terms, is differentiable at
every point.
smoothing (a) In general, reduction or removal of discontinuities or fluctuations in data; (b) more
specifically, application of a low-pass filter, that is, any mathematical technique that combines two or more
observations to reduce variability and to highlight general trends.
smoothing parameter Bin width, usually as required in methods for estimating probability densities.
soliton Solitary wave.
source A phase space point away from which all nearby points flow.
space-filling curve A curve passing through every point in a space of two or more dimensions.
spectral analysis The calculation and interpretation of a spectrum, especially in terms of the frequencydomain
characteristics of a time series. Generally used synonymously with Fourier analysis.
spectrum A collection of all the frequencies, wavelengths, or similar components of a complex process,
showing the distribution of those components according to their magnitudes.
spline (a) A flexible strip used in drafting to form"and to help draw"a curve between two fixed points;
(b) a smoothly varying curve between two data points, often fitted by a cubic polynomial and used to
interpolate "data" values between the two measured points.
stable Tending to dampen perturbations or initial differences, over time.
stable fixed point A type of attractor in the form of a phase space point that attracts all neighboring
points, that is, a point to which all iterates beginning from some other point converge.
stable manifold The set of points that asymptotically approach a given point, over time.
standard deviation A descriptive statistic that reflects the spread of a group of values about their mean,
usually computed as:
As such, it's the square root of the average squared deviation, or the square root of the variance.
standard phase space A term used in this book for the usual phase space, the coordinates for which
represent the measured and different physical features that are to be plotted on the associated graph or
phase portrait.
standardization A transformation done by subtracting the mean from each observation of a dataset and
then dividing each such difference by the standard deviation. Such a transformation, for example, removes
seasonality. It converts data of different units to a common or standard scale, namely a scale for which the
mean is zero and the standard deviation is one. The new scale measures the transformed variable in terms
of the deviation from its mean, in units of standard deviation, and is dimensionless.
state (a) The condition of a system or values of its variables, at any one time; (b) an arbitrarily defined
subrange (usually with specified numerical boundaries) that one or more variables of a system can be in at
one particular time.
state space Same as phase space.
state space reconstruction See reconstruction of phase space.
state space vector Same as lag vector.
stationary Time-invariant, that is, (a) lacking a trend; or (b) (more rigidly), keeping a constant mean and
variance with time; or (c) (more formally still), said of a randomlike process whose statistical properties
are independent of time. ("Independence of time" is often a matter of the particular time scale being used.)
statistical self-similarity The property whereby any suitably magnified part of an object looks
approximately like the whole, the differences being minor, negligible and ascribable to chance.
steady state (a) A condition that doesn't change with time; (b) the state toward which the system's behavior
becomes asymptotic as time goes to infinity. The associated equation gives a constant solution.
stochastic (a) Having no regularities in behavior; (b) characterizable only by statistical properties, thus
involving randomness and probability as opposed to being mainly deterministic; (c) random; (d) developed
according to a probabilistic model.
strange attractor (a) Same as chaotic attractor; (b) an attractor having such geometrical features as
fractal dimension, Cantor-set structure, and so on.
stretching (a) A topologist's interpretation of either (1) the amplification of a certain range of input values
to a larger range of output values during iteration or (2) the phase space exponential divergence of two
nearby chaotic trajectories; (b) a transformation of the form x'=ax and y'=ay, where a>1.
stroboscopic map A lag-time phase space model in which data are taken at equal time intervals.
subharmonic bifurcation Period-doubling.
successive-maxima plot Same as next-amplitude map.
surface of section Poincaré section.
surrogate data Artificially generated data that mimic certain selected features of an observed time series
but that are otherwise stochastic.
system (a) An arrangement of interacting parts, units, components, variables, etc. into a whole; (b) a
group, series, or sequence of elements, often in the form of a chronological dataset.

T
tangent bifurcation That special situation on a one-dimensional map whereby the function is tangent to
the identity line.
theoretic (a) Restricted to theory; (b) lacking verification.
thermodynamic entropy A measure conceived by Clausius for describing the unavailable energy of a
closed system such as a heat-producing engine, and subsequently modified for application to other
thermodynamic systems.
time-delay method A lag-time analytical technique that uses data of just one measured physical feature (x)
(regardless of any other features that may have been measured), whereby xt is compared to one or more
lagged subseries, often with the aim of reconstructing an attractor.
time domain The representation of time series data in their raw or unaltered form.
time series (a) A chronological list or plot of the values of any variable or variables and their time or order
of measurement; (b) in a narrow and more rare sense, a set of values that vary randomly with time.
topological dimension An integer that reflects the complexity of a geometric continuum, equal to 1+the
Euclidean dimension of the simplest geometric object that can subdivide that continuum. The topological
dimension usually has the same value as the Euclidean dimension.
topology (a) That branch of geometry that deals with the properties of figures that are unaltered by
imposed deformations or cumulative transformations; (b) the branch of mathematics that studies the
qualitative properties of spaces, as opposed to the more delicate and refined geometric or analytic
properties.
toroidal attractor Torus.
torus (a) A three-dimensional ring- or doughnut-shaped surface or solid generated by rotating a circle
about any axis that doesn't intersect that circle; (b) an attractor that represents two or more limit cycles.
Also called a toroid.
trajectory (a) A path taken by a moving body or point (and hence an orbit); (b) a sequence of measured
values or list of successive states of a dynamical system; (c) a solution to a differential equation; (d)
graphically, a line on a phase space plot, connecting points in chronological order.
transformation (a) A change in the numerical description or scale of measurement of a variable (in other
words, a filter); (b) a change in position or direction of the axes of a coordinate system, without altering
their relative angles; (c) a mapping between two spaces.
transient An early, atypical observation or temporary behavior in a system when first activated that dies
out with time.
translation A shifting (transformation) of the axes of a coordinate system while keeping the same
orientation of those axes. For example, say a is the amount that the x axis is shifted and b is the amount that
the y axis is shifted; then the origin of new axis x' is at x+a and the origin of new axis y' is at y+b; any point
plotted on the new coordinates is located at x'=x-a and y'=y-b.
trend (a) A systematic change, prevailing tendency, or general drift (in chaos, usually a prevailing
direction of plotted points over some period of time); (b) a nonconstant mean.
triangle law A graphical method of adding two vectors whereby the starting point of the second vector is
placed at the terminal point of the first vector, the sum or resultant being given by the new vector drawn
from the starting point of the first vector to the terminal point of the second.
truncate (a) To shorten a number by keeping only the first few (significant) digits and discarding all
others; (b) to approximate an infinite series by a finite number of terms; (c) to exclude sample values that
are greater (or less) than a specified constant value.
two-dimensional map A pair of equations, in each of which xt and yt together are used to yield xt+m (with
the first equation) and yt+m (with the second equation).

U
unit vector A vector having a magnitude of 1. It's usually obtained by dividing a vector by its length
(magnitude).
universal Typical of entire classes of systems.
universality The property whereby many seemingly unrelated systems or equations behave alike in a
particular respect, so that they can be grouped into a class by their generic behavior.
unstable Tending to amplify perturbations or initial differences, over time.
unstable fixed point A fixed point from which successive iterates move farther and farther away.
unstable manifold The set of points that exponentially diverge from a given point, over time.
unstable orbit (unstable trajectory) A trajectory for which, arbitrarily close to any input value, there's
another possible input which gives rise to a vastly different trajectory.

V
variable A characteristic or property that can have different numerical values.
variable-partition histogramming Same as adaptive histogramming.
variance A measure of the spread or dispersion of a group of values about their mean, specifically the
average squared deviation from the mean, or
vector A directed straight line, that is, a straight line representing a quantity that has both magnitude and
direction, drawn from its starting point (point of origin) to its terminal point.
vector array The list of paired values that make up a vector time series.
vector time series A listing or series of measurements of a variable in which each value is associated not
with its time or order of measurement explicitly but rather with its value at some constant lag time later.
von Koch snowflake Koch snowflake.

W
wave amplitude A measure of the maximum height of a wave, taken either as the vertical height from
trough to crest or as half the vertical height from trough to crest.
wave frequency See frequency.
wavelength The distance from any point on a wave to the equivalent point on the next wave.
wave period The time needed for one full wavelength or cycle.
weight A factor by which some quantity is multiplied and that reflects that quantity's relative importance.
weighted sum A sum obtained by adding weighted quantities.
weighting The multiplication of each item of data by some number that reflects the item's relative
importance.
white noise Data that are mutually unrelated. For instance, an observation made at one time has no relation
to observations made at earlier times. Such noise might, for example, be generated from independent,
identically distributed observations of a variable. The label "white" comes from the analogy with white
light and means that all possible periodic oscillations or frequencies are present with equal power or
variance.
window A selected subrange or interval of values.