A new entropy formula of Ledrappier–Young type for linear toral dynamics

2012 ◽  
Vol 34 (3) ◽  
pp. 1037-1054
Author(s):  
JIAN-SHENG XIE
Keyword(s):  
Dynamical System ◽  
Ergodic Theory ◽  
Linear Dynamical System ◽  
Linear Dynamics ◽  
Entropy Formula ◽  
System A ◽  
Smooth Ergodic Theory

AbstractA detailed Ledrappier–Young theory is presented for linear toral dynamics. First, a proof is given for a simplified definition for local entropies which holds in more general settings besides the current linear dynamics. Then it is shown that the transverse dimensions can be defined directly via the Smale structure of the linear dynamical system. A new entropy formula of Ledrappier–Young type is obtained. The conjecture of Ledrappier and Xie [Vanishing transverse entropy in smooth ergodic theory. Ergod. Th. & Dynam. Sys. 31(4) (2011), 1229–1235] is also discussed for such linear dynamics.

scholarly journals Inverse-square law between time and amplitude for crossing tipping thresholds

2019 ◽  
Vol 475 (2222) ◽  
pp. 20180504 ◽  
Author(s):  
Paul Ritchie ◽  
Özkan Karabacak ◽  
Jan Sieber
Keyword(s):  
Dynamical System ◽  
Feedback Loop ◽  
Tipping Point ◽  
Dimensional System ◽  
Stochastic Forcing ◽  
Inverse Square Law ◽  
System A ◽  
Asymptotic Expressions ◽  
Higher Dimensional ◽  
Random Disturbances

A classical scenario for tipping is that a dynamical system experiences a slow parameter drift across a fold tipping point, caused by a run-away positive feedback loop. We study what happens if one turns around after one has crossed the threshold. We derive a simple criterion that relates how far the parameter exceeds the tipping threshold maximally and how long the parameter stays above the threshold to avoid tipping in an inverse-square law to observable properties of the dynamical system near the fold. For the case when the dynamical system is subject to stochastic forcing we give an approximation to the probability of tipping if a parameter changing in time reverses near the tipping point. The derived approximations are valid if the parameter change in time is sufficiently slow. We demonstrate for a higher-dimensional system, a model for the Indian summer monsoon, how numerically observed escape from the equilibrium converge to our asymptotic expressions. The inverse-square law between peak of the parameter forcing and the time the parameter spends above a given threshold is also visible in the level curves of equal probability when the system is subject to random disturbances.

A novel unified approach to invariance conditions for a linear dynamical system

2017 ◽  
Vol 298 ◽  
pp. 351-367 ◽  
Author(s):  
Zoltán Horváth ◽  
Yunfei Song ◽  
Tamás Terlaky
Keyword(s):  
Dynamical System ◽  
Linear Dynamical System ◽  
Unified Approach

scholarly journals Communicating vessels: a non-linear dynamical system

2017 ◽  
Vol 39 (3) ◽  
Author(s):  
Roberto De Luca ◽  
Orazio Faella
Keyword(s):  
Dynamical System ◽  
Fluid Flow ◽  
Ideal Fluid ◽  
Oscillatory Motion ◽  
Linear Dynamical System ◽  
Static Properties ◽  
Bernoulli’S Equation ◽  
Non Linear ◽  
Dynamical Properties

The dynamics of an ideal fluid contained in two communicating vessels is studied. Despite the fact that the static properties of this system have been known since antiquity, the knowledge of the dynamical properties of an ideal fluid flowing in two communicating vessels is not similarly widespread. By means of Bernoulli's equation for non-stationary fluid flow, we study the oscillatory motion of the fluid when dissipation can be neglected.

scholarly journals On symmetric logarithm and some old examples in smooth ergodic theory

2003 ◽  
Vol 180 (3) ◽  
pp. 241-255 ◽  
Author(s):  
K. Frączek ◽  
M. Lemańczyk
Keyword(s):  
Ergodic Theory ◽  
Smooth Ergodic Theory

Smooth Ergodic Theory for Endomorphisms

2009 ◽  
Author(s):  
Min Qian ◽  
Jian-Sheng Xie ◽  
Shu Zhu
Keyword(s):  
Ergodic Theory ◽  
Smooth Ergodic Theory
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