|
Lyapunov
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 Petrovic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 trajectories. Quantitatively, two trajectories in phase space with initial separation vector \delta \mathbf_0 diverge (provided that the divergence can be treated within the linearized approximation) at a rate given by : , \delta\mathbf(t) , \approx e^ , \delta \mathbf_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 chaotic (provided some other conditions are met, e.g., phase space comp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aleksandr Lyapunov
Aleksandr Mikhailovich Lyapunov (russian: Алекса́ндр Миха́йлович Ляпуно́в, ; – 3 November 1918) was a Russian mathematician, mechanician and physicist. His surname is variously romanized as Ljapunov, Liapunov, Liapounoff or Ljapunow. He was the son of the astronomer Mikhail Lyapunov and the brother of the pianist and composer Sergei Lyapunov. Lyapunov is known for his development of the stability theory of a dynamical system, as well as for his many contributions to mathematical physics and probability theory. Biography Early life Lyapunov was born in Yaroslavl, Russian Empire. His father Mikhail Vasilyevich Lyapunov (1820–1868) was an astronomer employed by the Demidov Lyceum. His brother, Sergei Lyapunov, was a gifted composer and pianist. In 1863, M. V. Lyapunov retired from his scientific career and relocated his family to his wife's estate at Bolobonov, in the Simbirsk province (now Ulyanovsk Oblast). After the death of his father in 18 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Stability
Various types of stability may be discussed for the solutions of differential equations or difference equations describing dynamical systems. The most important type is that concerning the stability of solutions near to a point of equilibrium. This may be discussed by the theory of Aleksandr Lyapunov. In simple terms, if the solutions that start out near an equilibrium point x_e stay near x_e forever, then x_e is Lyapunov stable. More strongly, if x_e is Lyapunov stable and all solutions that start out near x_e converge to x_e, then x_e is asymptotically stable. The notion of exponential stability guarantees a minimal rate of decay, i.e., an estimate of how quickly the solutions converge. The idea of Lyapunov stability can be extended to infinite-dimensional manifolds, where it is known as structural stability, which concerns the behavior of different but "nearby" solutions to differential equations. Input-to-state stability (ISS) applies Lyapunov notions to systems with inputs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Vector
In applied mathematics and dynamical system theory, Lyapunov vectors, named after Aleksandr Lyapunov, describe characteristic expanding and contracting directions of a dynamical system. They have been used in predictability analysis and as initial perturbations for ensemble forecasting in numerical weather prediction. In modern practice they are often replaced by bred vectors for this purpose. Mathematical description Lyapunov vectors are defined along the trajectories of a dynamical system. If the system can be described by a d-dimensional state vector x\in\mathbb^d the Lyapunov vectors v^(x), (k=1\dots d) point in the directions in which an infinitesimal perturbation will grow asymptotically, exponentially at an average rate given by the 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 trajectories. Quantitatively, two trajectories i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prokopy Lyapunov
Prokopy Petrovich Lyapunov () (Isady, Grand Duchy of Moscow; Grand Duchy of Ryazan became a part of Grand Duchy of Moscow in 1521 and Moscow shouldn't be confused as a birth place which is located to the east of Old Ryazan, in a village that survived to this day b. ? — July 22, 1611;Most sources agree that he died no later than August 1, 1611 Tsardom of Russia) was a prominent 17th century Russian nobleman (dvoryanin), voivode (military chieftain) of, allegedly, a Rurikid origin who practically became a head of Pereyaslavl-Ryazansky lands nobility in the end 1590s; he took part in wars during power vacuum in succession crisis that happened in early 1598 as result of confusion about legitimate heir apparent following death of Feodor I, nobility infighting, war declared by Polish–Lithuanian Commonwealth (PLC) in 1605, and exhaustive Tatar raids; most famously he is remembered for organizing and leading the first unsuccessful uprising against occupation of Moscow of 1610 by ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Function
In the theory of ordinary differential equations (ODEs), Lyapunov functions, named after Aleksandr Lyapunov, are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Lyapunov functions (also called Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory. A similar concept appears in the theory of general state space Markov chains, usually under the name Foster–Lyapunov functions. For certain classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability. Whereas there is no general technique for constructing Lyapunov functions for ODEs, in many specific cases the construction of Lyapunov functions is known. For instance, quadratic functions suffice for systems with one state; the solution of a particular linear matrix inequality provides Lyapunov functions for linear systems; and conservation laws can often be used to construct Lyapunov funct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sergei Lyapunov
Sergei Mikhailovich Lyapunov (or Liapunov; russian: Серге́й Миха́йлович Ляпуно́в, ; 8 November 1924) was a Russian composer, pianist and conductor. Life Lyapunov was born in Yaroslavl in 1859. After the death of his father, Mikhail Lyapunov, when he was about eight, Sergei, his mother, and his two brothers (one of them was Aleksandr Lyapunov, later a notable mathematician) went to live in the larger town of Nizhny Novgorod. There he attended the grammar school along with classes of the newly formed local branch of the Russian Musical Society. On the recommendation of Nikolai Rubinstein, the Director of the Moscow Conservatory of Music, he enrolled in that institution in 1878. His main teachers were Karl Klindworth (piano; a former pupil of Franz Liszt), and Sergei Taneyev (composition; a former pupil of Pyotr Ilyich Tchaikovsky and his successor at the Conservatory). He graduated in 1883, more attracted by the nationalist elements in music of the New Russ ... [...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 differen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Family
The Lyapunov family (russian: Ляпуно́в) is a Russian noble family claiming descent from the Galich Rurikids, who lost their princely title in the 15th century.Their descent from Rurik is disputed by some historians, such as Sergey Soloviev. The family later served the archbishop of Veliky Novgorod, and subsequently integrated into the Ryazanian nobility. History Origin and name It's known that Lyapunov brothers were descendants of a family of of Galich and one of his sons, dukes of the both of whom itself belonged to Rurik Dynasty. Their ancestors ruled in the Principality of Galich (an appanage of the Vladimir-Suzdal Duchy), until Duke Dmitry Donskoy, Grand Duke (Prince) of Moscow annexed their domain in 1362 and exiled Prince Dmitry Ivanovich of Galich, who fled to Veliky Novgorod where he entered the service to the local archbishop. Around this time Lyapunov family has lost their Duke title (the exact date is unknown though) and were considered boyars of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lyapunov Fractal
In mathematics, Lyapunov fractals (also known as Markus–Lyapunov fractals) are bifurcational fractals derived from an extension of the logistic map in which the degree of the growth of the population, ''r'', periodically switches between two values ''A'' and ''B''. A Lyapunov fractal is constructed by mapping the regions of stability and chaotic behaviour (measured using the Lyapunov exponent \lambda) in the ''a''−''b'' plane for given periodic sequences of ''a'' and ''b''. In the images, yellow corresponds to \lambda 0 (chaos). Lyapunov fractals were discovered in the late 1980s by the Germano-Chilean physicist from the Max Planck Institute of Molecular Physiology. They were introduced to a large public by a science popularization article on recreational mathematics published in Scientific American in 1991. Properties Lyapunov fractals are generally drawn for values of ''A'' and ''B'' in the interval ,4/math>. For larger values, the interval ,1is no longe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexey Lyapunov
Alexey Andreevich Lyapunov (russian: Алексе́й Андре́евич Ляпуно́в; 25 September 1911 – 23 June 1973) was a Soviet mathematician and an early pioneer of computer science. One of the founders of Soviet cybernetics, Lyapunov was member of the Soviet Academy of Sciences and a specialist in the fields of real function theory, mathematical problems of cybernetics, set theory, programming theory, mathematical linguistics, and mathematical biology. Biography Composer Sergei Lyapunov, mathematician Aleksandr Lyapunov, and philologist Boris Lyapunov were close relatives of Alexey Lyapunov. In 1928, Lyapunov enrolled at Moscow State University to study mathematics, and in 1932 he became a student of Nikolai Luzin. Under his mentorship, Lyapunov began his research in descriptive set theory. He became world-wide known for his theorem on the range of an atomless vector-measure in finite dimensions, now called the Lyapunov Convexity Theorem. From 1934 un ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mikhail Nikolaevich Lyapunov
Mikhail Nikolaevich Lyapunov (russian: Михаил Николаевич Ляпунов; 1 March 1848 — 4 March 1909) was Russian statesman and military leader and a participant in the Russo-Japanese War. Biography Lyapunov was born into a family of hereditary nobles of Saint Petersburg Governorate. He received his secondary education at the Saint Petersburg military gymnasium and at the age of 16 he was enrolled as a cadet at the Pavel Military School . At the age of 20 he was promoted to the rank of lieutenant. The command noticed Lyapunov's legal abilities and he was sent to Odessa to the Military Law Academy. Lyapunov was promoted to staff-captain on 6 November 1872, captain from 21 October 1875, major on 20 February 1876, captain of the military judicial department from 1 April 1877, assistant military prosecutor and lieutenant colonel from 17 April 1879 and finally promoted to colonel on 30 August 1882. In December 1884 he received the post of military judge. From 27 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
