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of the Lyapunov spectrum it is possible to obtain the so-called Kaplan – Yorke dimension, that is defined as follows:
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Lyapunov and spectrum
Thus, there is a spectrum of Lyapunov exponents — equal in number to the dimensionality of the phase space.
For a dynamical system with evolution equation in an n – dimensional phase space, the spectrum of Lyapunov exponents
The Lyapunov spectrum can be used to give an estimate of the rate of entropy production
One of the most used and effective numerical techniques to calculate the Lyapunov spectrum for a smooth dynamical system relies on periodic
Lyapunov and is
Stability for nonlinear systems that take an input is input-to-state stability ( ISS ), which combines Lyapunov stability and a notion similar to BIBO stability.
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.
Lyapunov is a Russian surname that is sometimes romanized as Ljapunov, Liapunov or Ljapunow and may refer to:
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.
where is the Lyapunov exponent.
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.
The exponent is named after Aleksandr Lyapunov.
The largest Lyapunov exponent is as follows
A. M. Lyapunov proved that if the system of the first approximation is regular ( e. g., all systems with constant and periodic coefficients are regular ) and its largest Lyapunov exponent is negative, then the solution of the original system is asymptotically Lyapunov stable.
In 1930 O. Perron constructed an example of the second-order system, the first approximation of which has negative Lyapunov exponents along a zero solution of the original system but, at the same time, this zero solution of original nonlinear system is Lyapunov unstable.
Also it is possible to construct reverse example when first approximation has positive Lyapunov exponents along a zero solution of the original system but, at the same time, this zero solution of original nonlinear system
is Lyapunov stable.
If the system is dissipative, the sum of Lyapunov exponents is negative.
Lyapunov and possible
The billiard was introduced by Yakov G. Sinai as an example of an interacting Hamiltonian system that displays physical thermodynamic properties: all of its possible trajectories are ergodic and it has a positive Lyapunov exponent.
Lyapunov and so-called
Interest in it started suddenly during the Cold War ( 1953-1962 ) period when the so-called " Second Method of Lyapunov " was found to be applicable to the stability of aerospace guidance systems which typically contain strong nonlinearities not treatable by other methods.
Lyapunov and Kaplan
The Lyapunov exponents, λ are approximately 0. 1981, 0, − 0. 6581 and the Kaplan – Yorke dimension, D < sub > KY </ sub > ≈ 2. 3010
Lyapunov and –
* Alexander Lyapunov ( 1857 – 1918 ) in the 1890s marks the beginning of stability theory.
:* Escape-time fractals – use a formula or recurrence relation at each point in a space ( such as the complex plane ); usually quasi-self-similar ; also known as " orbit " fractals ; e. g., the Mandelbrot set, Julia set, Burning Ship fractal, Nova fractal and Lyapunov fractal.
Aleksandr Mikhailovich Lyapunov (, ; – November 3, 1918 ) was a Russian mathematician, mechanician and physicist.
His father Mikhail Vasilyevich Lyapunov ( 1820 – 1868 ) was an astronomer and a head of the Demidovski lyceum.
* Lyapunov – Malkin theorem
* Mikhail Lyapunov ( 1820 – 1868 ), Russian astronomer
* Aleksandr Lyapunov ( 1857 – 1918 ), Russian mathematician and mechanician
* Sergei Lyapunov ( 1859 – 1924 ), Russian composer
* Boris Lyapunov ( 1862 – 1943 ), Russian expert in Slavic studies
* Alexey Lyapunov ( 1911 – 1973 ), Russian mathematician
Moreover, the sum of all the positive Lyapunov exponents gives an estimate of the Kolmogorov – Sinai entropy accordingly to Pesin's theorem.
Gram – Schmidt orthonormalization of the Lyapunov vectors to avoid a misalignment of all the vectors along the direction of maximal expansion.
As a result, a variety of nomenclatures are common: applied to ordinary differential equations, it is called Floquet theory ( or occasionally the Lyapunov – Floquet theorem ).
In 1608 – 1610, Lyapunov led the army of service class people against a peasant uprising in the Ryazan region and supporters of False Dmitriy II.
Mikhail Vasilyevich Lyapunov ( 1820 – 1868 ) was a Russian astronomer and a head of the Demidov Lyceum in Yaroslavl.
Lyapunov and defined
The maximal Lyapunov exponent can be defined as follows:
) The Lyapunov exponents describe the behavior of vectors in the tangent space of the phase space and are defined from the Jacobian matrix
If are the eigenvalues of, then the Lyapunov exponents are defined by
Generally the calculation of Lyapunov exponents, as defined above, cannot be carried out analytically, and in most cases one must resort to numerical techniques.
spectrum and is
The observed intensity of the radio emission of Venus is much higher than the expected thermal intensity, although the spectrum indicated by measurements at wave lengths near 3 cm and 10 cm is like that of a black body at about 600-degrees.
This result suggests a very high temperature at the solid surface of the planet, although there is the possibility that the observed radiation may be a combination of both thermal and non-thermal components and that the observed spectrum is that of a black body merely by coincidence.
For the case of Jupiter, the radio emission spectrum is definitely not like the spectrum of a black-body radiator, and it seems very likely that the radiation reaching the earth is a combination of thermal radiation from the atmosphere and non-thermal components.
There is a broad spectrum of organisms from which selection for a specified military purpose might be made.
An artist is a person engaged in one or more of any of a broad spectrum of activities related to creating art, practicing the arts and / or demonstrating an art.
Ampicillin is a beta-lactam antibiotic that is part of the aminopenicillin family and is roughly equivalent to its successor, amoxicillin in terms of spectrum and level of activity.
Generally the visible spectrum ( vmag ) is used for the apparent magnitude, but other spectrums such as the near-infrared J-band magnitude are also used.
In the visible spectrum Sirius is the brightest star in the night sky, while in the near-infrared J-band, Betelgeuse is the brightest star in the night sky.
In many countries, over-the-air broadcast television of analog audio and analog video signals is being discontinued, to allow the re-use of the television broadcast radio spectrum for other services such as datacasting and subchannels.
Thermal noise is white noise meaning that the power spectral density is constant throughout the frequency spectrum.
Flicker noise is electronic noise with a 1 / ƒ frequency spectrum ; as f increases, the noise decreases.
In classical LS AAS, as it has been proposed by Alan Walsh, the high spectral resolution required for AAS measurements is provided by the radiation source itself that emits the spectrum of the analyte in the form of lines that are narrower than the absorption lines.
In CS AAS, in contrast, a single lamp, emitting a continuum spectrum over the entire spectral range of interest is used for all elements.
The club coaches a broad spectrum of age groups, including local swimmer Niamh Robinson who is sponsored by local company TP Properties.
Arcturus is a type K1. 5 IIIpe orange giant star — the letters " pe " stand for " peculiar emission ," which indicates that the spectrum of light given off by the star is unusual and full of emission lines.
0.184 seconds.