Numerical study of a random dynamical system with two degrees of freedom

1975 ◽  
Vol 37 (1) ◽  
pp. 87-100 ◽  
Author(s):  
Claude Froeschl�
Keyword(s):  
Dynamical System ◽  
Degrees Of Freedom ◽  
Numerical Study ◽  
Random Dynamical System ◽  
Two Degrees Of Freedom

Non-Stationary Responses in Externally Excited Two-Degrees-of-Freedom Nonlinear Systems with 1: 2 Internal Resonance

2004 ◽  
Vol 10 (11) ◽  
pp. 1663-1697 ◽  
Author(s):  
Anil K. Bajaj ◽  
Patricia Davies ◽  
Bappaditya Banerjee
Keyword(s):  
Internal Resonance ◽  
Degrees Of Freedom ◽  
Numerical Study ◽  
Excitation Frequency ◽  
Primary Resonance ◽  
Single Mode ◽  
Two Degrees Of Freedom ◽  
Analytical Technique ◽  
Bifurcation Points ◽  
Coupled Mode

The dynamics of two-degrees-of-freedom dynamical systems with weak quadratic nonlinearities is analyzed in the neighborhood of bifurcation points when the excitation frequency varies slowly through the region of primary resonance. The two modes of vibration are in 1: 2 subharmonic internal resonance. The slowly evolving averaged equations are numerically studied for motions initiated in the vicinity of stationary responses, and observations are made about the nature of responses of the system near the transition from single-mode to coupled-mode solutions (pitchfork points), and near jump and Hopf bifurcations in the coupled-mode solutions. An analytical technique based on the dynamic bifurcation theory is developed to explain the numerical observations for passage through the bifurcations. A numerical study is carried out to determine the effects of system parameters on the dynamics near the pitchfork bifurcation points and results are compared with analytical and numerical descriptions of dynamics.

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.

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