Statistical properties of flip-flop processes associated to the chaotic behavior of systems with strange attractors

1987 ◽  
Vol 55 (6) ◽  
pp. 377-385 ◽  
Author(s):  
F. Aicardi ◽  
A. Borsellino
Keyword(s):  
Chaotic Behavior ◽  
Statistical Properties ◽  
Strange Attractors ◽  
Flip Flop

New developments in the ergodic theory of nonlinear dynamical systems

1994 ◽  
Vol 346 (1679) ◽  
pp. 145-157 ◽  
Keyword(s):  
Dynamical Systems ◽  
Ergodic Theory ◽  
Nonlinear Dynamical Systems ◽  
Statistical Properties ◽  
Strange Attractors ◽  
New Developments ◽  
Nonlinear Dynamical ◽  
Henon Maps ◽  
Open Questions ◽  
Hyperbolic Dynamical Systems

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.

scholarly journals Some statistical properties of strange attractors: engineering view

2008 ◽  
Vol 96 ◽  
pp. 012147 ◽  
Author(s):  
M Mijangos ◽  
V Kontorovich ◽  
J Aguilar-Torrentera
Keyword(s):  
Statistical Properties ◽  
Strange Attractors

scholarly journals Statistical Properties of Lorenz-like Flows, Recent Developments and Perspectives

2014 ◽  
Vol 24 (10) ◽  
pp. 1430028 ◽  
Author(s):  
Vitor Araujo ◽  
Stefano Galatolo ◽  
Maria José Pacifico
Keyword(s):  
Exponential Decay ◽  
Chaotic Behavior ◽  
Decay Of Correlations ◽  
Statistical Properties ◽  
Computer Assisted ◽  
Lorenz Attractor ◽  
Lorenz Equations ◽  
Physical Measure ◽  
Exponential Decay Of Correlations ◽  
Statistical Behavior

We comment on the mathematical results about the statistical behavior of Lorenz equations and its attractor, and more generally on the class of singular hyperbolic systems. The mathematical theory of such kind of systems turned out to be surprisingly difficult. It is remarkable that a rigorous proof of the existence of the Lorenz attractor was presented only around the year 2000 with a computer-assisted proof together with an extension of the hyperbolic theory developed to encompass attractors robustly containing equilibria. We present some of the main results on the statistical behavior of such systems. We show that for attractors of three-dimensional flows, robust chaotic behavior is equivalent to the existence of certain hyperbolic structures, known as singular-hyperbolicity. These structures, in turn, are associated with the existence of physical measures: in low dimensions, robust chaotic behavior for flows ensures the existence of a physical measure. We then give more details on recent results on the dynamics of singular-hyperbolic (Lorenz-like) attractors: (1) there exists an invariant foliation whose leaves are forward contracted by the flow (and further properties which are useful to understand the statistical properties of the dynamics); (2) there exists a positive Lyapunov exponent at every orbit; (3) there is a unique physical measure whose support is the whole attractor and which is the equilibrium state with respect to the center-unstable Jacobian; (4) this measure is exact dimensional; (5) the induced measure on a suitable family of cross-sections has exponential decay of correlations for Lipschitz observables with respect to a suitable Poincaré return time map; (6) the hitting time associated to Lorenz-like attractors satisfy a logarithm law; (7) the geometric Lorenz flow satisfies the Almost Sure Invariance Principle (ASIP) and the Central Limit Theorem (CLT); (8) the rate of decay of large deviations for the volume measure on the ergodic basin of a geometric Lorenz attractor is exponential; (9) a class of geometric Lorenz flows exhibits robust exponential decay of correlations; (10) all geometric Lorenz flows are rapidly mixing and their time-1 map satisfies both ASIP and CLT.

scholarly journals Chaotic Behavior and Strange Attractors in Dynamical Systems

2012 ◽  
Vol 2 (5) ◽  
pp. 25-31
Author(s):  
Md. Shariful Islam Khan
Keyword(s):  
Dynamical Systems ◽  
Chaotic Behavior ◽  
Strange Attractors

scholarly journals Calculation of the Statistical Properties of Strange Attractors

1981 ◽  
Vol 46 (24) ◽  
pp. 1547-1550 ◽  
Author(s):  
R. V. Jensen ◽  
C. R. Oberman
Keyword(s):  
Statistical Properties ◽  
Strange Attractors
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