Dynamical systems with time scale separation: averaging, stochastic modelling, and central limit theorems

2001 ◽  
pp. 189-209 ◽  
Author(s):  
Christian Rödenbeck ◽  
Christian Beck ◽  
Holger Kantz
Keyword(s):  
Dynamical Systems ◽  
Central Limit ◽  
Time Scale ◽  
Limit Theorems ◽  
Stochastic Modelling ◽  
Central Limit Theorems ◽  
Scale Separation ◽  
Time Scale Separation

Towards Model Simplification of Uncertain Complex Dynamical Systems

2005 ◽  
Author(s):  
Thordur Runolfsson
Keyword(s):  
Dynamical Systems ◽  
Time Scale ◽  
Simulation Models ◽  
Operating Conditions ◽  
Time Dependent ◽  
Model Simplification ◽  
Physical Processes ◽  
Scale Separation ◽  
Industrial Systems ◽  
Time Scale Separation

The use of simulation models in the design and verification of complex industrial systems is becoming standard practice. All models of physical processes are uncertain either due to unknown/ignored phenomena or inherent uncertainty due to variations in operating conditions and/or physical properties. This paper considers the second class of problems where the uncertainty can be characterized through an external uncertain parameter that may be time dependent as well as random. We formulate an approach for analysis of such system based on the theory of Random Dynamical Systems and present a model reduction approach under the assumption of time scale separation between the system and parameter dynamics.

scholarly journals Large deviations and central limit theorems for sequential and random systems of intermittent maps

2020 ◽  
pp. 1-28
Author(s):  
MATTHEW NICOL ◽  
FELIPE PEREZ PEREIRA ◽  
ANDREW TÖRÖK
Keyword(s):  
Central Limit Theorem ◽  
Limit Theorem ◽  
Dynamical Systems ◽  
Central Limit ◽  
Limit Theorems ◽  
Random Systems ◽  
Central Limit Theorems ◽  
Random Dynamical Systems ◽  
Stat Phys ◽  
Quenched Central Limit Theorem

Abstract We obtain large and moderate deviation estimates for both sequential and random compositions of intermittent maps. We also address the question of whether or not centering is necessary for the quenched central limit theorems obtained by Nicol, Török and Vaienti [Central limit theorems for sequential and random intermittent dynamical systems. Ergod. Th. & Dynam. Sys.38(3) (2018), 1127–1153] for random dynamical systems comprising intermittent maps. Using recent work of Abdelkader and Aimino [On the quenched central limit theorem for random dynamical systems. J. Phys. A 49(24) (2016), 244002] and Hella and Stenlund [Quenched normal approximation for random sequences of transformations. J. Stat. Phys.178(1) (2020), 1–37] we extend the results of Nicol, Török and Vaienti on quenched central limit theorems for centered observables over random compositions of intermittent maps: first by enlarging the parameter range over which the quenched central limit theorem holds; and second by showing that the variance in the quenched central limit theorem is almost surely constant (and the same as the variance of the annealed central limit theorem) and that centering is needed to obtain this quenched central limit theorem.

scholarly journals Central limit theorems for sequential and random intermittent dynamical systems

2016 ◽  
Vol 38 (3) ◽  
pp. 1127-1153 ◽  
Author(s):  
MATTHEW NICOL ◽  
ANDREW TÖRÖK ◽  
SANDRO VAIENTI
Keyword(s):  
Time Series ◽  
Fixed Point ◽  
Dynamical Systems ◽  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Stationary Time Series ◽  
Stationary Time

We establish self-norming central limit theorems for non-stationary time series arising as observations on sequential maps possessing an indifferent fixed point. These transformations are obtained by perturbing the slope in the Pomeau–Manneville map. We also obtain quenched central limit theorems for random compositions of these maps.

scholarly journals Pressure Function and Limit Theorems for Almost Anosov Flows

2021 ◽  
Vol 382 (1) ◽  
pp. 1-47
Author(s):  
Henk Bruin ◽  
Dalia Terhesiu ◽  
Mike Todd
Keyword(s):  
Thermodynamic Properties ◽  
Central Limit ◽  
Limit Theorems ◽  
Central Limit Theorems ◽  
Stable Laws ◽  
Analytic Framework ◽  
Pressure Function ◽  
Hyperbolic Flows ◽  
Anosov Flows ◽  
Almost Anosov

AbstractWe obtain limit theorems (Stable Laws and Central Limit Theorems, both standard and non-standard) and thermodynamic properties for a class of non-uniformly hyperbolic flows: almost Anosov flows, constructed here. The link between the pressure function and limit theorems is studied in an abstract functional analytic framework, which may be applicable to other classes of non-uniformly hyperbolic flows.

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