scholarly journals Entropy and irreversibility in dynamical systems

2013 ◽  
Vol 371 (2005) ◽  
pp. 20120349 ◽  
Author(s):  
Oliver Penrose
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Chaotic System ◽  
Degrees Of Freedom ◽  
Usual Method ◽  
Chaotic Dynamical System ◽  
Two Degrees Of Freedom ◽  
Arnold’S Cat Map ◽  
Non Equilibrium

A method of defining non-equilibrium entropy for a chaotic dynamical system is proposed which, unlike the usual method based on Boltzmann’s principle , does not involve the concept of a macroscopic state. The idea is illustrated using an example based on Arnold’s ‘cat’ map. The example also demonstrates that it is possible to have irreversible behaviour, involving a large increase of entropy, in a chaotic system with only two degrees of freedom.

scholarly journals Integrals of dynamical systems linear in the velocities

1971 ◽  
Vol 17 (3) ◽  
pp. 241-244
Author(s):  
C. D. Collinson
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Integral Conditions ◽  
Two Degrees Of Freedom ◽  
Generalized Coordinates ◽  
Image Position ◽  
Linear Integrals

Kilmister (1) has discussed the existence of linear integrals of a dynamical system specified by generalized coordinates qα(α = 1, 2, …, n) and a Lagrangianrepeated indices being summed from 1 to n. He derived covariant conditions for the existence of such an integral, conditions which do not imply the existence of an ignorable coordinate. Boyer (2) discussed the conditions and found the most general Lagrangian satisfying the conditions for the case of two degrees of freedom (n = 2).

scholarly journals VII.—On the Adelphic Integral of the Differential Equations of Dynamics

1918 ◽  
Vol 37 ◽  
pp. 95-116 ◽  
Author(s):  
E. T. Whittaker
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Differential Equations ◽  
Periodic Solutions ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Two Degrees Of Freedom ◽  
Image Position ◽  
Kinetic And Potential Energies

§ 1. Ordinary and singular periodic solutions of a dynamical system. — The present paper is concerned with the motion of dynamical systems which possess an integral of energy. To fix ideas, we shall suppose that the system has two degrees of freedom, so that the equations of motion in generalised co-ordinates may be written in Hamilton's formwhere (q1q2) are the generalised co-ordinates, (p1, p2) are the generalised momenta, and where H is a function of (q1, q2, p1, p2) which represents the sum of the kinetic and potential energies.

A Novel 4D Chaotic System Based on Two Degrees of Freedom Nonlinear Mechanical System

2018 ◽  
Vol 73 (7) ◽  
pp. 595-607 ◽  
Author(s):  
Sezgin Kacar ◽  
Zhouchao Wei ◽  
Akif Akgul ◽  
Burak Aricioglu
Keyword(s):  
Mechanical System ◽  
Chaotic System ◽  
Degrees Of Freedom ◽  
Electronic Circuit ◽  
Chaotic Behaviour ◽  
Two Degrees Of Freedom ◽  
Nonlinear Mechanical System ◽  
Linear Mechanical System ◽  
Low Amplitude ◽  
The Mathematical Model

AbstractIn this study, a non-linear mechanical system with two degrees of freedom is considered in terms of chaos phenomena and chaotic behaviour. The mathematical model of the system was moved to the state space and presented as a four dimensional (4D) chaotic system. The system’s chaotic behaviour was investigated by performing dynamic analyses of the system such as equilibria, Lyapunov exponents, bifurcation analyses, etc. Also, the electronic circuit realisation is implemented as a real-time application. This system exhibited vibration along with noise-like behaviour because of its very low amplitude values. Thus, the system is scaled to increase the amplitude values. As a result, the electronic circuit implementation of the 4D chaotic system derived from the model of a physical system is realised.

OPTIMAL NONLINEAR MODELS FROM EMPIRICAL TIME SERIES: AN APPLICATION TO CLIMATE

2004 ◽  
Vol 14 (06) ◽  
pp. 2041-2052 ◽  
Author(s):  
RAFAEL M. GUTIÉRREZ
Keyword(s):  
Dynamical System ◽  
Time Series ◽  
Surface Temperature ◽  
Degrees Of Freedom ◽  
Minimum Description Length ◽  
Lower Atmosphere ◽  
External Parameter ◽  
Optimal Size ◽  
Chaotic Dynamical System ◽  
Macroscopic System

In this work we propose a method that exploits the feedback between empirical and theoretical knowledge of a complex macroscopic system in order to build a nonlinear model. We apply the method to the monthly earth's mean surface temperature time series. The problems of contamination and stationarity are considered noting the importance of observation and modeling scales. We construct a dynamical system of ordinary differential equations where the vector field relating the relevant degrees of freedom and their variations in time is expressed in terms of a polynomial base orthonormal to the measure associated to the time series under study. The optimal size of the model and the values of its parameters are estimated with the principle of minimum description length and the Adams–Molton predictor–corrector method. This procedure is self-consistent because it does not use any external parameter or assumption. We then present a first approach to find the closest chaotic dynamical system corresponding to the earth's mean surface temperature and compare it with scale consistent theoretical or phenomenological models of the lower atmosphere. This comparison allows us to obtain an explicit functional form of the heat capacity of the earth's surface as a function of the earth's mean surface temperature.

scholarly journals Centre manifolds of forced dynamical systems

1991 ◽  
Vol 32 (4) ◽  
pp. 401-436 ◽  
Author(s):  
S. M. Cox ◽  
A. J. Roberts
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Degrees Of Freedom ◽  
Rational Approach ◽  
Centre Manifold ◽  
Formal Scheme ◽  
Dimensional Models ◽  
Centre Manifolds ◽  
Low Dimensional ◽  
Sivashinsky Equation

AbstractCentre manifolds arise in a rational approach to the problem of forming low-dimensional models of dynamical systems with many degrees of freedom. When a dynamical system with a centre manifold is subject to a small forcing, F, there are two effects: to displace the centre manifold; and to alter the evolution thereon. We propose a formal scheme for calculating the centre manifold of such a forced dynamical system. Our formalism permits the calculation of these effects, with errors of order |F|2. We find that the displacement of the manifold allows a reparameterisation of its description, and we describe two “natural” ways in which this can be carried out. We give three examples: an introductory example; a five-mode model of the atmosphere to display the quasi-geostrophic approximation; and the forced Kuramoto-Sivashinsky equation.

scholarly journals On the Number of Isolating Integrals in Systems with Three Degrees of Freedom

1971 ◽  
Vol 10 ◽  
pp. 110-117
Author(s):  
Claude Froeschle
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Weak Coupling ◽  
Degrees Of Freedom ◽  
Model Problem ◽  
Energy Integral ◽  
Dimensional Mapping ◽  
Three Degrees Of Freedom

AbstractDynamical systems with three degrees of freedom can be reduced to the study of a four-dimensional mapping. We consider here, as a model problem, the mapping given by the following equations: We have found that as soon as b ≠ 0, i.e. even for a very weak coupling, a dynamical system with three degrees of freedom has in general either two or zero isolating integrals (besides the usual energy integral).

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