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34 relations: Anosov diffeomorphism, Automorphism, Chaos theory, Dense set, Determinant, Differential equation, Eigenvalues and eigenvectors, Ergodicity, Hyperbolic equilibrium point, Inverse function, List of chaotic maps, Map projection, Mathematics, Matrix (mathematics), Measure-preserving dynamical system, Mixing (mathematics), Mixing (physics), Periodic point, Phase space, Poincaré recurrence theorem, Quotient space (linear algebra), Randomness, Rational dependence, Rational number, Recurrence plot, Shear mapping, Stable manifold, Structural stability, Symmetric matrix, Torus, Unimodular matrix, Vertex (geometry), Vladimir Arnold, Wolfram Demonstrations Project.
Anosov diffeomorphism
In mathematics, more particularly in the fields of dynamical systems and geometric topology, an Anosov map on a manifold M is a certain type of mapping, from M to itself, with rather clearly marked local directions of "expansion" and "contraction".
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Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself.
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Chaos theory
Chaos theory is a branch of mathematics focusing on the behavior of dynamical systems that are highly sensitive to initial conditions.
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Dense set
In topology and related areas of mathematics, a subset A of a topological space X is called dense (in X) if every point x in X either belongs to A or is a limit point of A, that is the closure of A is constituting the whole set X. Informally, for every point in X, the point is either in A or arbitrarily "close" to a member of A — for instance, every real number either is a rational number or has a rational number arbitrarily close to it (see Diophantine approximation).
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Determinant
In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.
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Differential equation
A differential equation is a mathematical equation that relates some function with its derivatives.
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Eigenvalues and eigenvectors
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.
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Ergodicity
In probability theory, an ergodic dynamical system is one that, broadly speaking, has the same behavior averaged over time as averaged over the space of all the system's states in its phase space.
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Hyperbolic equilibrium point
In the study of dynamical systems, a hyperbolic equilibrium point or hyperbolic fixed point is a fixed point that does not have any center manifolds.
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Inverse function
In mathematics, an inverse function (or anti-function) is a function that "reverses" another function: if the function applied to an input gives a result of, then applying its inverse function to gives the result, and vice versa.
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List of chaotic maps
In mathematics, a chaotic map is a map (.
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Map projection
A map projection is a systematic transformation of the latitudes and longitudes of locations from the surface of a sphere or an ellipsoid into locations on a plane.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Matrix (mathematics)
In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.
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Measure-preserving dynamical system
In mathematics, a measure-preserving dynamical system is an object of study in the abstract formulation of dynamical systems, and ergodic theory in particular.
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Mixing (mathematics)
In mathematics, mixing is an abstract concept originating from physics: the attempt to describe the irreversible thermodynamic process of mixing in the everyday world: mixing paint, mixing drinks, etc.
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Mixing (physics)
In physics, a dynamical system is said to be mixing if the phase space of the system becomes strongly intertwined, according to at least one of several mathematical definitions.
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Periodic point
In mathematics, in the study of iterated functions and dynamical systems, a periodic point of a function is a point which the system returns to after a certain number of function iterations or a certain amount of time.
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Phase space
In dynamical system theory, a phase space is a space in which all possible states of a system are represented, with each possible state corresponding to one unique point in the phase space.
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Poincaré recurrence theorem
In physics, the Poincaré recurrence theorem states that certain systems will, after a sufficiently long but finite time, return to a state very close to, if not exactly the same as, the initial state.
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Quotient space (linear algebra)
In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by "collapsing" N to zero.
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Randomness
Randomness is the lack of pattern or predictability in events.
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Rational dependence
In mathematics, a collection of real numbers is rationally independent if none of them can be written as a linear combination of the other numbers in the collection with rational coefficients.
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Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator.
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Recurrence plot
In descriptive statistics and chaos theory, a recurrence plot (RP) is a plot showing, for a given moment in time, the times at which a phase space trajectory visits roughly the same area in the phase space.
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Shear mapping
In plane geometry, a shear mapping is a linear map that displaces each point in fixed direction, by an amount proportional to its signed distance from a line that is parallel to that direction.
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Stable manifold
In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions embodied in the idea of an attractor or repellor.
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Structural stability
In mathematics, structural stability is a fundamental property of a dynamical system which means that the qualitative behavior of the trajectories is unaffected by small perturbations (to be exact ''C''1-small perturbations).
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Symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose.
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Torus
In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.
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Unimodular matrix
In mathematics, a unimodular matrix M is a square integer matrix having determinant +1 or −1.
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Vertex (geometry)
In geometry, a vertex (plural: vertices or vertexes) is a point where two or more curves, lines, or edges meet.
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Vladimir Arnold
Vladimir Igorevich Arnold (alternative spelling Arnol'd, Влади́мир И́горевич Арно́льд, 12 June 1937 – 3 June 2010) was a Soviet and Russian mathematician.
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Wolfram Demonstrations Project
The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields.
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Redirects here:
Arnold Cat map, Arnold cat map, Arnold's Cat map, Cat map, Cat mapping.
References
[1] https://en.wikipedia.org/wiki/Arnold's_cat_map
