|
Lyapunov Exponent
In mathematics, the Lyapunov exponent or Lyapunov characteristic exponent of a dynamical system is a quantity that characterizes the rate of separation of infinitesimally close trajectory, trajectories. Quantitatively, two trajectories in phase space with initial separation vector \boldsymbol_0 diverge (provided that the divergence can be treated within the linearized approximation) at a rate given by , \boldsymbol(t) , \approx e^ , \boldsymbol_0 , where \lambda is the Lyapunov exponent. The rate of separation can be different for different orientations of initial separation vector. Thus, there is a spectrum of Lyapunov exponents—equal in number to the dimensionality of the phase space. It is common to refer to the largest one as the maximal Lyapunov exponent (MLE), because it determines a notion of predictability for a dynamical system. A positive MLE is usually taken as an indication that the system is chaos theory, chaotic (provided some other conditions are met, e.g., ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oskar Perron
Oskar Perron (7 May 1880 – 22 February 1975) was a German mathematician. He was a professor at the University of Heidelberg from 1914 to 1922 and at the University of Munich from 1922 to 1951. He made numerous contributions to differential equations and partial differential equations, including the Perron method to solve the Dirichlet problem for elliptic partial differential equations. He wrote an encyclopedic book on continued fractions ''Die Lehre von den Kettenbrüchen''. He introduced ''Perron's paradox'' to illustrate the danger of assuming that the solution of an optimization problem exists: :''Let N be the largest positive integer. If N > 1, then N2 > N, contradicting the definition of N. Hence N = 1''. Works * ''Über die Drehung eines starren Körpers um seinen Schwerpunkt bei Wirkung äußerer Kräfte'', Diss. München 1902 * ''Grundlagen für eine Theorie der Jacobischen Kettenbruchalgorithmus'', Habilitationsschrift Leipzig 1906 * ''Die Lehre von den Ketten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Dimension
In the mathematics of dynamical systems, the concept of Lyapunov dimension was suggested by Kaplan and Yorke for estimating the Hausdorff dimension of attractors. Further the concept has been developed and rigorously justified in a number of papers, and nowadays various different approaches to the definition of Lyapunov dimension are used. Remark that the attractors with noninteger Hausdorff dimension are called strange attractors. Since the direct numerical computation of the Hausdorff dimension of attractors is often a problem of high numerical complexity, estimations via the Lyapunov dimension became widely spread. The Lyapunov dimension was named after the Russian mathematician Aleksandr Lyapunov because of the close connection with the Lyapunov exponents. Definitions Consider a dynamical system \big(\_, (U\subseteq \mathbb^n, \, \cdot\, )\big) , where \varphi^t is the shift operator along the solutions: \varphi^t(u_0) = u(t,u_0), of ODE \dot = f(), t \leq 0, or differ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Synchronization Of Chaos
Synchronization of chaos is a phenomenon that may occur when two or more dissipative chaotic systems are coupled. Because of the exponential divergence of the nearby trajectories of chaotic systems, having two chaotic systems evolving in synchrony might appear surprising. However, synchronization of coupled or driven chaotic oscillators is a phenomenon well established experimentally and reasonably well-understood theoretically. The stability of synchronization for coupled systems can be analyzed using master stability. Synchronization of chaos is a rich phenomenon and a multi-disciplinary subject with a broad range of applications. Synchronization may present a variety of forms depending on the nature of the interacting systems and the type of coupling, and the proximity between the systems. Identical synchronization This type of synchronization is also known as complete synchronization. It can be observed for identical chaotic systems. The systems are said to be completely s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobian Matrix And Determinant
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of components of function values, then its determinant is called the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian. They are named after Carl Gustav Jacob Jacobi. The Jacobian matrix is the natural generalization to vector valued functions of several variables of the derivative and the differential of a usual function. This generalization includes generalizations of the inverse function theorem and the implicit function theorem, where the non-nullity of the derivative is replaced by the non-nullity of the Jacobian determinant, and the multiplicative inverse of the derivative is replaced by the inverse of the Jacobian matrix. The Jacobian determinant is fundamentally use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalues
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a constant factor \lambda when the linear transformation is applied to it: T\mathbf v=\lambda \mathbf v. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor \lambda (possibly a negative or complex number). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or shrunk. If the eigenvalue is negative, the eigenvector's direction is reversed. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Vector
Lyapunov (, in old-Russian often written Лепунов) is a Russian surname that is sometimes also romanized In linguistics, romanization is the conversion of text from a different writing system to the Roman (Latin) script, or a system for doing so. Methods of romanization include transliteration, for representing written text, and transcription, ... as Ljapunov, Liapunov or Ljapunow. Notable people with the surname include: * Alexey Lyapunov (1911–1973), Russian mathematician * Aleksandr Lyapunov (1857–1918), son of Mikhail (1820–1868), Russian mathematician and mechanician, after whom the following are named: ** Lyapunov dimension ** Lyapunov equation ** Lyapunov exponent ** Lyapunov function ** Lyapunov fractal ** Lyapunov stability ** Lyapunov's central limit theorem ** Lyapunov time ** Lyapunov vector ** Lyapunov (crater) * Boris Lyapunov (1862–1943), son of Mikhail (1820–1868), Russian expert in Slavic studies * Mikhail Lyapunov (1820–1868) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Richard H
Richard is a male given name. It originates, via Old French, from Old Frankish and is a compound of the words descending from Proto-Germanic language">Proto-Germanic ''*rīk-'' 'ruler, leader, king' and ''*hardu-'' 'strong, brave, hardy', and it therefore means 'strong in rule'. Nicknames include " Richie", " Dick", " Dickon", " Dickie", " Rich", " Rick", "Rico (name), Rico", " Ricky", and more. Richard is a common English (the name was introduced into England by the Normans), German and French male name. It's also used in many more languages, particularly Germanic, such as Norwegian, Danish, Swedish, Icelandic, and Dutch, as well as other languages including Irish, Scottish, Welsh and Finnish. Richard is cognate with variants of the name in other European languages, such as the Swedish "Rickard", the Portuguese and Spanish "Ricardo" and the Italian "Riccardo" (see comprehensive variant list below). People named Richard Multiple people with the same name * Richard Ander ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Exponents Of The Mandelbrot Set (The Mini-Mandelbrot) - Matlab
Lyapunov (, in old-Russian often written Лепунов) is a Russian surname that is sometimes also romanized as Ljapunov, Liapunov or Ljapunow. Notable people with the surname include: * Alexey Lyapunov (1911–1973), Russian mathematician * Aleksandr Lyapunov (1857–1918), son of Mikhail (1820–1868), Russian mathematician and mechanician, after whom the following are named: ** Lyapunov dimension ** Lyapunov equation ** Lyapunov exponent ** Lyapunov function ** Lyapunov fractal ** Lyapunov stability ** Lyapunov's central limit theorem ** Lyapunov time ** Lyapunov vector ** Lyapunov (crater) * Boris Lyapunov (1862–1943), son of Mikhail (1820–1868), Russian expert in Slavic studies * Mikhail Lyapunov (1820–1868), Russian astronomer * Mikhail Nikolaevich Lyapunov (1848–1909), Russian military officer and lawyer * Prokopy Lyapunov (d. 1611), Russian statesman * Sergei Lyapunov (1859–1924), son of Mikhail (1820–1868), Russian composer * Zakhary Lyapunov Zakhary Petro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Time
In mathematics, the Lyapunov time is the characteristic timescale on which a dynamical system is chaotic. It is named after the Russian mathematician Aleksandr Lyapunov. It is defined as the inverse of a system's largest Lyapunov exponent. Use The Lyapunov time mirrors the limits of the predictability of the system. By convention, it is defined as the time for the distance between nearby trajectories of the system to increase by a factor of '' e''. However, measures in terms of 2-foldings and 10-foldings are sometimes found, since they correspond to the loss of one bit of information or one digit of precision respectively. While it is used in many applications of dynamical systems theory, it has been particularly used in celestial mechanics where it is important for the problem of the stability of the Solar System. However, empirical estimation of the Lyapunov time is often associated with computational or inherent uncertainties. Examples Typical values are:Pierre Gaspard, ''Cha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplicative Inverse
In mathematics, a multiplicative inverse or reciprocal for a number ''x'', denoted by 1/''x'' or ''x''−1, is a number which when Multiplication, multiplied by ''x'' yields the multiplicative identity, 1. The multiplicative inverse of a rational number, fraction ''a''/''b'' is ''b''/''a''. For the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or 0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the Function (mathematics), function ''f''(''x'') that maps ''x'' to 1/''x'', is one of the simplest examples of a function which is its own inverse (an Involution (mathematics), involution). Multiplying by a number is the same as Division (mathematics), dividing by its reciprocal and vice versa. For example, multiplication by 4/5 (or 0.8) will give the same result as division by 5/4 (or 1.25). Therefore, multiplication by a number followed by multiplication by its reciprocal yie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
