Verge and Foliot Clock Escapement: A Simple Dynamical System

2010 ◽  
Vol 48 (6) ◽  
pp. 374-376
Author(s):  
Mark Denny
Keyword(s):  
Dynamical System ◽  
Simple Dynamical System

Mechanical Representation and Stability of Dynamical Systems Containing Fractional Springpot Elements

2018 ◽  
Author(s):  
Matthias Hinze ◽  
André Schmidt ◽  
Remco I. Leine
Keyword(s):  
Dynamical System ◽  
Potential Energy ◽  
Lyapunov Functional ◽  
Analogue Model ◽  
Infinite Dimensional ◽  
Mechanical Analogue ◽  
Stability Of Dynamical Systems ◽  
Functional Differential ◽  
Invariance Theorem ◽  
Simple Dynamical System

In this article we consider the Lyapunov stability of mechanical systems containing fractional springpot elements. We obtain the potential energy of a springpot by an infinite dimensional mechanical analogue model. Furthermore, we consider a simple dynamical system containing a springpot as a functional differential equation and use the potential energy of the springpot in a Lyapunov functional to prove uniform stability and discuss asymptotic stability of the equilibrium with the help of an invariance theorem.

scholarly journals A simple dynamical system that mimics open‐flow turbulence

1990 ◽  
Vol 2 (11) ◽  
pp. 1983-2001 ◽  
Author(s):  
G. S. Bhat ◽  
R. Narasimha ◽  
S. Wiggins
Keyword(s):  
Dynamical System ◽  
Open Flow ◽  
Flow Turbulence ◽  
Simple Dynamical System

scholarly journals The Formulations of Classical Mechanics with Foucault’s Pendulum

Physics ◽  
2020 ◽  
Vol 2 (4) ◽  
pp. 531-540
Author(s):  
Nicolas Boulanger ◽  
Fabien Buisseret
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Classical Mechanics ◽  
Earth’S Rotation ◽  
Canonical Transformations ◽  
Open Door ◽  
Earth's Rotation ◽  
Foucault’S Pendulum ◽  
Simple Dynamical System

Since the pioneering works of Newton (1643–1727), mechanics has been constantly reinventing itself: reformulated in particular by Lagrange (1736–1813) then Hamilton (1805–1865), it now offers powerful conceptual and mathematical tools for the exploration of dynamical systems, essentially via the action-angle variables formulation and more generally through the theory of canonical transformations. We propose to the (graduate) reader an overview of these different formulations through the well-known example of Foucault’s pendulum, a device created by Foucault (1819–1868) and first installed in the Panthéon (Paris, France) in 1851 to display the Earth’s rotation. The apparent simplicity of Foucault’s pendulum is indeed an open door to the most contemporary ramifications of classical mechanics. We stress that adopting the formalism of action-angle variables is not necessary to understand the dynamics of Foucault’s pendulum. The latter is simply taken as well-known and simple dynamical system used to exemplify and illustrate modern concepts that are crucial in order to understand more complicated dynamical systems. The Foucault’s pendulum first installed in 2005 in the collegiate church of Sainte-Waudru (Mons, Belgium) will allow us to numerically estimate the different quantities introduced.

The Nature of Chaos in a Simple Dynamical System

1979 ◽  
Vol 34 (11) ◽  
pp. 1283-1289 ◽  
Author(s):  
Akira Shibata ◽  
Toshihiro Mayuyama ◽  
Masahiro Mizutani ◽  
Nobuhiko Saitô
Keyword(s):  
Dynamical System ◽  
Distribution Function ◽  
Probability Distribution ◽  
Correlation Functions ◽  
Probability Distribution Function ◽  
Time Correlation ◽  
Time Correlation Functions ◽  
One Dimensional ◽  
Initial Distributions ◽  
Simple Dynamical System

A simple one-dimensional transformation xn = axn-1 + 2 - a (0 ≦ xn-1 ≦ 1 - 1 / a ) , xn = a( 1 - xn-1) (1 - 1 a ≦ xn-1 ≦ 1) (1 ≦ a ≦ 2) is investigated by introducing the probability distribution function Wn(x). Wn ( x ) converges when n → oo for a > V 2 , but oscillates for 1 < a ≦ V2. The final distribution of Wn(x) does not depend on the initial distributions for a > V2, but does for 1 < a ≦V2 Time-correlation functions are also calculated

scholarly journals Elementary bifurcations for a simple dynamical system under non-Gaussian Lévy noises

2012 ◽  
Vol 32 (4) ◽  
pp. 1391-1398 ◽  
Author(s):  
Chen Huiqin ◽  
Duan Jinqiao ◽  
Zhang Chengjian
Keyword(s):  
Dynamical System ◽  
Non Gaussian ◽  
Simple Dynamical System

Periodic Motions in a First-Order, Time-Delayed, Nonlinear System

2018 ◽  
Author(s):  
Siyuan Xing ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical System ◽  
Differential Equation ◽  
Nonlinear System ◽  
Polynomial Functions ◽  
Periodic Motions ◽  
Approximate Analytical Expression ◽  
Bifurcation Tree ◽  
First Order ◽  
The Stability ◽  
Simple Dynamical System

In this paper, periodic motions in a first-order, time-delayed, nonlinear system are investigated. For time-delay terms of non-polynomial functions, the traditional analytical methods have difficulty in determining periodic motions. The semi-analytical method is used for prediction of periodic motion. This method is based on implicit mappings obtained from discretization of the original differential equation. From the periodic nodes, the corresponding approximate analytical expression can be obtained through discrete finite Fourier series. The stability and the bifurcations of such periodic motions are determined by eigenvalue analysis. The bifurcation tree of period-1 to period-4 motions are obtained and the numerical results and analytical predictions are compared. The complexity of periodic motions in such a simple dynamical system is discussed.

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