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Correlation Dimension
In chaos theory, the correlation dimension (denoted by ''ν'') is a measure of the dimensionality of the space occupied by a set of random points, often referred to as a type of fractal dimension. For example, if we have a set of random points on the real number line between 0 and 1, the correlation dimension will be ''ν'' = 1, while if they are distributed on say, a triangle embedded in three-dimensional space (or ''m''-dimensional space), the correlation dimension will be ''ν'' = 2. This is what we would intuitively expect from a measure of dimension. The real utility of the correlation dimension is in determining the (possibly fractional) dimensions of fractal objects. There are other methods of measuring dimension (e.g. the Hausdorff dimension, the box-counting dimension, and the information dimension) but the correlation dimension has the advantage of being straightforwardly and quickly calculated, of being less noisy when only a small number of points is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chaos Theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas. Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamical Systems
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it. At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chaos Theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas. Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Approximate Entropy
In statistics, an approximate entropy (ApEn) is a technique used to quantify the amount of regularity and the unpredictability of fluctuations over time-series data. For example, consider two series of data: : Series A: (0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, ...), which alternates 0 and 1. : Series B: (0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, ...), which has either a value of 0 or 1, chosen randomly, each with probability 1/2. Moment statistics, such as mean and variance, will not distinguish between these two series. Nor will rank order statistics distinguish between these series. Yet series A is perfectly regular: knowing a term has the value of 1 enables one to predict with certainty that the next term will have the value of 0. In contrast, series B is randomly valued: knowing a term has the value of 1 gives no insight into what value the next term will have. Regularity was originally measured by exact regularity statistics, which has mainly centered on various ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recurrence Quantification Analysis
Recurrence quantification analysis (RQA) is a method of nonlinear data analysis (cf. chaos theory) for the investigation of dynamical systems. It quantifies the number and duration of recurrences of a dynamical system presented by its phase space trajectory. Background The recurrence quantification analysis (RQA) was developed in order to quantify differently appearing recurrence plots (RPs), based on the small-scale structures therein. Recurrence plots are tools which visualise the recurrence behaviour of the phase space trajectory \vec(i) of dynamical systems: :(i,j) = \Theta(\varepsilon - \, \vec(i) - \vec(j)\, ), where \Theta: \mathbf \rightarrow \ is the Heaviside function and \varepsilon a predefined tolerance. Recurrence plots mostly contain single dots and lines which are parallel to the mean diagonal (''line of identity'', LOI) or which are vertical/horizontal. Lines parallel to the LOI are referred to as ''diagonal lines'' and the vertical structures as ''vertical lines ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Correlation Integral
In chaos theory, the correlation integral is the mean probability that the states at two different times are close: :C(\varepsilon) = \lim_ \frac \sum_^N \Theta(\varepsilon - \, \vec(i) - \vec(j)\, ), \quad \vec(i) \in \mathbb^m, where N is the number of considered states \vec(i), \varepsilon is a threshold distance, \, \cdot \, a norm (e.g. Euclidean norm) and \Theta( \cdot ) the Heaviside step function. If only a time series is available, the phase space can be reconstructed by using a time delay embedding (see Takens' theorem): :\vec(i) = (u(i), u(i+\tau), \ldots, u(i+\tau(m-1))), where u(i) is the time series, m the embedding dimension and \tau the time delay. The correlation integral is used to estimate the correlation dimension. An estimator of the correlation integral is the correlation sum: :C(\varepsilon) = \frac \sum_^N \Theta(\varepsilon - \, \vec(i) - \vec(j)\, ), \quad \vec(i) \in \mathbb^m. See also *Recurrence quantification analysis Recurrence quantificati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Takens' Theorem
In the study of dynamical systems, a delay embedding theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence of observations of the state of a dynamical system. The reconstruction preserves the properties of the dynamical system that do not change under smooth coordinate changes (i.e., diffeomorphisms), but it does not preserve the geometric shape of structures in phase space. Takens' theorem is the 1981 delay embedding theorem of Floris Takens. It provides the conditions under which a smooth attractor can be reconstructed from the observations made with a generic function. Later results replaced the smooth attractor with a set of arbitrary box counting dimension and the class of generic functions with other classes of functions. Delay embedding theorems are simpler to state for discrete-time dynamical systems. The state space of the dynamical system is a \nu-dimensional manifold M. The dynamics is given by a smooth map :f: M \to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sunspot
Sunspots are phenomena on the Sun's photosphere that appear as temporary spots that are darker than the surrounding areas. They are regions of reduced surface temperature caused by concentrations of magnetic flux that inhibit convection. Sunspots appear within active regions, usually in pairs of opposite magnetic polarity. Their number varies according to the approximately 11-year solar cycle. Individual sunspots or groups of sunspots may last anywhere from a few days to a few months, but eventually decay. Sunspots expand and contract as they move across the surface of the Sun, with diameters ranging from to . Larger sunspots can be visible from Earth without the aid of a telescope. They may travel at relative speeds, or proper motions, of a few hundred meters per second when they first emerge. Indicating intense magnetic activity, sunspots accompany other active region phenomena such as coronal loops, prominences, and reconnection events. Most solar flares and coronal mas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
University Of Arizona Press
The University of Arizona Press, a publishing house founded in 1959 as a department of the University of Arizona, is a nonprofit publisher of scholarly and regional books. As a delegate of the University of Arizona to the larger world, the Press publishes the work of scholars wherever they may be, concentrating upon scholarship that reflects the special strengths of the University of Arizona, Arizona State University, and Northern Arizona University. The Press publishes about fifty books annually and has some 1,400 books in print. These include scholarly titles in American Indian studies, anthropology, archaeology, environmental studies, geography, Chicano studies, history, Latin American studies, and the space sciences. The UA Press has award-winning books in more than 30 subject areas. The UA Press also publishes general interest books on Arizona and the Southwest borderlands. In addition, the Press publishes books of personal essays, such as Nancy Mairs's ''Plaintext'' and tw ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Recurrence Plot
In descriptive statistics and chaos theory, a recurrence plot (RP) is a plot showing, for each moment i in time, the times at which the state of a dynamical system returns to the previous state at i, i.e., when the phase space trajectory visits roughly the same area in the phase space as at time j. In other words, it is a plot of :\vec(i)\approx \vec(j), showing i on a horizontal axis and j on a vertical axis, where \vec is the state of the system (or its phase space trajectory). Background Natural processes can have a distinct recurrent behaviour, e.g. periodicities (as seasonal or Milankovich cycles), but also irregular cyclicities (as El Niño Southern Oscillation, heart beat intervals). Moreover, the recurrence of states, in the meaning that states are again arbitrarily close after some time of divergence, is a fundamental property of deterministic dynamical systems and is typical for nonlinear or chaotic systems (cf. Poincaré recurrence theorem). The recurrence of states ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimensionality
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces. In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found neces ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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