Statistical properties of chaotic dynamical systems which exhibit strange attractors

The purpose of this paper is to give a survey of recent results on non-uniformly hyperbolic dynamical systems. The emphasis is on the existence of strange attractors and Sinai-Ruelle-Bowen measures for Henon maps, but we also describe results about statistical properties of such dynamical systems and state some of the open questions in this area.

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Statistical properties of discretizations of a class of chaotic dynamical systems

Computers & Mathematics with Applications ◽

10.1016/0898-1221(96)00065-x ◽

1996 ◽

Vol 31 (11) ◽

pp. 83-95 ◽

Cited By ~ 6

Author(s):

P. Diamond ◽

M. Suzuki ◽

P. Kloeden ◽

P. Pokrovskii

Keyword(s):

Dynamical Systems ◽

Statistical Properties ◽

Chaotic Dynamical Systems

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On statistical properties of chaotic dynamical systems

American Mathematical Society Translations: Series 2 - Sinai’s Moscow Seminar on Dynamical Systems ◽

10.1090/trans2/171/06 ◽

1995 ◽

pp. 57-71

Author(s):

N. I. Chernov

Keyword(s):

Dynamical Systems ◽

Statistical Properties ◽

Chaotic Dynamical Systems

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Unstable Periodic Orbits Sampling and Its Applications to Climate Models

10.5194/egusphere-egu21-5457 ◽

2021 ◽

Author(s):

Chiara Cecilia Maiocchi ◽

Valerio Lucarini

Keyword(s):

Dynamical Systems ◽

Periodic Orbits ◽

Numerical Study ◽

Building Blocks ◽

Statistical Properties ◽

Mathematical Framework ◽

Multiple Time ◽

Chain Process ◽

Unstable Periodic Orbits ◽

Chaotic Dynamical Systems

Climate can be interpreted as a complex, high dimensional non-equilibrium stationary system characterised by multiple time and space scales spanning various orders of magnitude. Statistical mechanics and dynamical system theory have been key mathematical frameworks in the study of the climate system. In particular, unstable periodic orbits (UPOs) have been proven to provide relevant insight in the understanding of its statistical properties. In a recent paper Lucarini and Gritsun [1] provided an alternative approach for understanding the properties of the atmosphere.In general, UPOs decomposition plays a relevant role in the study of chaotic dynamical systems. In fact, UPOs densely populate the attractor of a chaotic system, and can therefore be thought as building blocks to construct the dynamic of the system itself. Since they are dense in the attractor, it is always possible to find a UPO arbitrarily near to a chaotic trajectory: the trajectory will remain close to the UPO, but it will never follow it indefinitely, because of its instability. Loosely speaking, a chaotic trajectory is repelled between neighbourhoods of different UPOs and can thus be approximated in terms of these periodic orbits. The statistical properties of the system can then be reconstructed from the full set of periodic orbits in this fashion.The numerical study of UPOs thus represents a relevant problem and an interesting research topic for Climate Science and chaotic dynamical systems in general. In this presentation we address the problem of sampling UPOs for the paradigmatic Lorenz-63 model. First, we present results regarding the measure of the system, thus its statistical properties, using UPOs theory, namely with the trace formulas. Second, we introduce a more innovative approach, considering UPOs as global states of the system. We approximate the exact dynamics by a suitable Markov chain process, describing how the system hops on different UPOs, and we compare the two different approaches. &#160;[1] V. Lucarini and A. Gritsun, &#8220;A new mathematical framework for atmospheric blocking events,&#8221; Climate Dynamics, vol. 54, pp. 575&#8211;598, Jan 2020.

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Pseudorandom Number Generators Based on Chaotic Dynamical Systems

Open Systems & Information Dynamics ◽

10.1023/a:1011950531970 ◽

2001 ◽

Vol 08 (02) ◽

pp. 137-146 ◽

Cited By ~ 16

Author(s):

Janusz Szczepański ◽

Zbigniew Kotulski

Keyword(s):

Dynamical Systems ◽

Communication Systems ◽

Initial Conditions ◽

Pseudorandom Number ◽

New Approach ◽

Chaotic Dynamical Systems ◽

Pseudorandom Number Generators ◽

Transmission Of Information ◽

Extreme Sensitivity ◽

Contemporary Technology

Pseudorandom number generators are used in many areas of contemporary technology such as modern communication systems and engineering applications. In recent years a new approach to secure transmission of information based on the application of the theory of chaotic dynamical systems has been developed. In this paper we present a method of generating pseudorandom numbers applying discrete chaotic dynamical systems. The idea of construction of chaotic pseudorandom number generators (CPRNG) intrinsically exploits the property of extreme sensitivity of trajectories to small changes of initial conditions, since the generated bits are associated with trajectories in an appropriate way. To ensure good statistical properties of the CPRBG (which determine its quality) we assume that the dynamical systems used are also ergodic or preferably mixing. Finally, since chaotic systems often appear in realistic physical situations, we suggest a physical model of CPRNG.

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