Spontaneous parametric transition from periodic motion to chaos in a dynamical system with two degrees of freedom

1999 ◽  
Vol 42 (2) ◽  
pp. 232-237
Author(s):  
S. N. Vladimirov
Keyword(s):  
Dynamical System ◽  
Periodic Motion ◽  
Degrees Of Freedom ◽  
Two Degrees Of Freedom

scholarly journals The Vibrations of a Particle about a Position of Equilibrium—Part 2 :The Relation between the Elliptic Function and Series Solutions

1921 ◽  
Vol 40 ◽  
pp. 34-49 ◽  
Author(s):  
Bevan B. Baker
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Elliptic Function ◽  
Degrees Of Freedom ◽  
Elliptic Functions ◽  
Series Solutions ◽  
Two Degrees Of Freedom ◽  
Function Solution

In a previous paper, entitled the “Vibrations of a Particle about a Position of Equilibrium,” by the author in collaboration with Professor E. B. Ross (Proc. Edin. Math. Soc., XXXIX, 1921, pp. 34–57), a particular dynamical system having two degrees of freedom was chosen and solutions of the corresponding differential equations were obtained in terms of periodic series and also in terms of elliptic functions. It was shown that for certain values of the frequencies of the principal vibrations, the periodic series become divergent, whereas the elliptic function solution continues to give finite results.

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 On the transition from regular to chaotic behaviors in the two degrees of freedom dynamical system

2007 ◽  
Vol 29 (3) ◽  
pp. 353-374
Author(s):  
Nguyen Van Khang ◽  
Nguyen Hoang Duong
Keyword(s):  
Dynamical System ◽  
Differential Equations ◽  
Degrees Of Freedom ◽  
Numerical Solutions ◽  
Nonlinear Differential Equations ◽  
Dynamical Behavior ◽  
Control Parameters ◽  
Lagrange Equations ◽  
Chaotic Motions ◽  
Two Degrees Of Freedom

The main objective of the present paper is to study the transition from periodic regular mot ion to chaos in a two degrees of freedom dynamical system by changing control parameters. The nonlinear differential equations governing motion of the system are derived from the Lagrange equations. By use of the Poincare map, the dynamical behavior is identified based on numerical solutions of the ordinary differential equations. The Lyapunov exponent and the frequency spectrum are calculated to identify chaos. From numerical simulations, it is indicated that the periodic, quasi-periodic and chaotic motions occur in the considered system.

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.

scholarly journals Characteristics of Periodic Orbits in Elliptical Galaxies

1983 ◽  
Vol 74 ◽  
pp. 271-274
Author(s):  
N. Caranicolas
Keyword(s):  
Dynamical System ◽  
Periodic Orbits ◽  
Degrees Of Freedom ◽  
Basic Characteristic ◽  
Elliptical Galaxies ◽  
Resonance Case ◽  
Two Degrees Of Freedom ◽  
Exact Resonance ◽  
Near Resonance ◽  
Families Of Periodic Orbits

AbstractThe properties of the characteristic curves of several families of periodic orbits, in a conservative dynamical system of two degrees of freedom, symmetric with respect to both axes, are reviewed. The two main types of families are presented. One sees that the pattern of the characteristics in the exact resonance case is similar to that of the near resonance case except for the basic characteristic . The form of the characteristics can be found theoretically by means of the second integral.

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.

The scientific bases of autocontrol with vibration phase for a resonance beating-vibration system with two degrees of freedom driven by debalances

1996 ◽  
Vol 18 (2) ◽  
pp. 43-48
Author(s):  
Tran Van Tuan ◽  
Do Sanh ◽  
Luu Duc Thach
Keyword(s):  
Degrees Of Freedom ◽  
Regulation System ◽  
Vibration System ◽  
Two Degrees Of Freedom ◽  
System Operating

In the paper it is introduced a method for studying dynamics of beating-vibrators by means of digital calculation with the help of the machine in accordance with the needs by the helps of an available auto regulation system operating with high reability.

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