V Arnold 2008 Nonlinearity 21 T109 doi:10.1088/0951-7715/21/7/T02
Orbits' statistics in chaotic dynamical systems
V Arnold
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This paper shows how the measurement of the stochasticity degree of a finite sequence of real numbers, published by Kolmogorov in Italian in a journal of insurances' statistics, can be usefully applied to measure the objective stochasticity degree of sequences, originating from dynamical systems theory and from number theory.
Namely, whenever the value of Kolmogorov's stochasticity parameter of a given sequence of numbers is too small (or too big), one may conclude that the conjecture describing this sequence as a sample of independent values of a random variables is highly improbable.
Kolmogorov used this strategy fighting (in a paper in 'Doklady', 1940) against Lysenko, who had tried to disprove the classical genetics' law of Mendel experimentally.
Calculating his stochasticity parameter value for the numbers from Lysenko's experiment reports, Kolmogorov deduced, that, while these numbers were different from the exact fulfilment of Mendel's 3 : 1 law, any smaller deviation would be a manifestation of the report's number falsification.
The calculation of the values of the stochasticity parameter would be useful for many other generators of pseudorandom numbers and for many other chaotically looking statistics, including even the prime numbers distribution (discussed in this paper as an example).
- PACS
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05.45.-a Nonlinear dynamics and nonlinear dynamical systems
05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
- MSC
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70K55 Transition to stochasticity (chaotic behavior) (See also 37D45)
- Subjects
- Dates
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Issue 7 (July 2008)
Received 10 January 2008
Published 21 May 2008
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Orbits' statistics in chaotic dynamical systems
V Arnold 2008 Nonlinearity 21 T109
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