Bifurcations and Chaotic Motions of a Class of Mechanical System With Parametric Excitations

2015 ◽  
Vol 10 (5) ◽  
Author(s):  
Liangqiang Zhou ◽  
Fangqi Chen ◽  
Yushu Chen
Keyword(s):  
Mechanical System ◽  
Parametric Excitation ◽  
Chaotic Dynamics ◽  
Stable And Unstable Manifolds ◽  
Chaotic Motions ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Critical Curves ◽  
Unstable Manifolds

Bifurcations and chaotic motions of a class of mechanical system subjected to a superharmonic parametric excitation or a nonlinear periodic parametric excitation are studied, respectively, in this paper. Chaos arising from the transverse intersections of the stable and unstable manifolds of the homoclinic and heteroclincic orbits is analyzed by Melnikov's method. The critical curves separating the chaotic and nonchaotic regions are plotted. Chaotic dynamics are compared for these systems with a periodic parametric excitation or a superharmonic parametric excitation, or a nonlinear periodic parametric excitation. Especially, some new dynamical phenomena are presented for the system with a nonlinear periodic parametric excitation.

Chaos and Three-Dimensional Horseshoes in Slowly Varying Oscillators

1988 ◽  
Vol 55 (4) ◽  
pp. 959-968 ◽  
Author(s):  
Stephen Wiggins ◽  
Steven W. Shaw
Keyword(s):  
Chaotic Dynamics ◽  
Three Dimensional ◽  
Equation Of Motion ◽  
Necessary Condition ◽  
Stable And Unstable Manifolds ◽  
Chaotic Motions ◽  
Single Degree Of Freedom ◽  
Single Degree ◽  
Unstable Manifolds ◽  
Additional Equation

We present general results pertaining to chaotic motions in a class of systems termed slowly varying oscillators which consist of weakly perturbed single-degree-of-freedom systems in which parameters vary slowly in time according to an additional equation of motion. Our results include an analytical method for detecting transversal intersections of stable and unstable manifolds (typically a necessary condition for chaotic motions to exist) and a detailed description of the chaotic dynamics that occur when this situation exists.

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

Chaotic dynamics for a class of single-machine-infinite bus power system

2016 ◽  
Vol 24 (3) ◽  
pp. 582-587 ◽  
Author(s):  
Liangqiang Zhou ◽  
Fangqi Chen
Keyword(s):  
Power Systems ◽  
Power System ◽  
Single Machine ◽  
Chaotic Dynamics ◽  
Numerical Results ◽  
Chaotic Motions ◽  
System Parameters ◽  
Critical Curves ◽  
Single Machine Infinite Bus

The chaotic motions are investigated both analytically and numerically for a class of single-machine-infinite bus power systems. The mechanism and parametric conditions for chaotic motions of this system are obtained rigorously. The critical curves separating the chaotic and non-chaotic regions are presented. The chaotic feature of the system parameters is discussed in detail. It is shown that there exist chaotic bands for this system, and the bands vary with the system parameters. Some new dynamical phenomena are presented. Numerical results are given, which verify the analytical ones.

Heteroclinic Bifurcations in Rigid Bodies Containing Internally Moving Parts and a Viscous Damper

1999 ◽  
Vol 66 (3) ◽  
pp. 720-728 ◽  
Author(s):  
G. L. Gray ◽  
D. C. Kammer ◽  
I. Dobson ◽  
A. J. Miller
Keyword(s):  
Chaotic Dynamics ◽  
Equations Of Motion ◽  
Major Axis ◽  
Rigid Bodies ◽  
Necessary Condition ◽  
Viscous Damping ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Melnikov Criterion ◽  
Analytical Criterion

Melnikov’s method is used to analytically study chaotic dynamics in an attitude transition maneuver of a torque-free rigid body in going from minor axis to major axis spin under the influence of viscous damping and nonautonomous perturbations. The equations of motion are presented, their phase space is discussed, and then they are transformed into a form suitable for the application of Melnikov’s method. Melnikov’s method yields an analytical criterion for homoclinic chaos in the form of an inequality that gives a necessary condition for chaotic dynamics in terms of the system parameters. The criterion is evaluated for its physical significance and for its application to the design of spacecraft. In addition, the Melnikov criterion is compared with numerical simulations of the system. The dependence of the onset of chaos on quantities such as body shape and magnitude of damping are investigated. In particular, it is found that for certain ranges of viscous damping values, the rate of kinetic energy dissipation goes down when damping is increased. This has a profound effect on the criterion for chaos.

On Melnikov's method in the study of chaotic motions of gyrostat

2002 ◽  
Vol 75 (5) ◽  
pp. 328-351 ◽  
Author(s):  
Jinlu Kuang ◽  
Soonhie Tan ◽  
A. Y. T. Leung
Keyword(s):  
Chaotic Motions ◽  
Melnikov's Method ◽  
Melnikov’S Method

Chaos in the Quasiperiodically Forced Helmholtz-Duffing Oscillator With Two-Well Potential

2005 ◽  
Author(s):  
Jing-Jun Lou ◽  
Shi-Jian Zhu ◽  
Qi-Wei He
Keyword(s):  
Hamiltonian System ◽  
Parameter Space ◽  
Homoclinic Orbit ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Duffing Oscillator ◽  
Restoring Force ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
The Asymmetry

The chaotic dynamics of the quasiperiodically excited Helmholtz-Duffing oscillator with two-well potential was investigated. The condition of the existence of homoclinic orbit in the corresponding Hamiltonian system was presented which is asymmetrical resulting from the asymmetry restoring force. It was found that the mechanism for chaos is transverse homoclinic tori and it is illustrated how transverse homoclinic tori give rise to chaos for the Helmholtz-Duffing oscillator with multi-frequency periodic forces. The criterion for the existence of chaos was given utilizing a generalization of the Melnikov’s method. The region in parameter space where chaotic dynamics may occur was given. It was also demonstrated that increasing the number of forcing frequencies increases the area in parameter space where chaotic behavior can occur.

scholarly journals Shooting methods and topological transversality

1999 ◽  
Vol 129 (6) ◽  
pp. 1137-1155 ◽  
Author(s):  
B. Buffoni
Keyword(s):  
Dynamical Systems ◽  
Chaotic Dynamics ◽  
Shooting Method ◽  
Heteroclinic Orbits ◽  
Kolmogorov Equation ◽  
Shooting Methods ◽  
Stable And Unstable Manifolds ◽  
Heteroclinic Solutions ◽  
Unstable Manifolds

We show that shooting methods for homoclinic or heteroclinic orbits in dynamical systems may automatically guarantee the topological transversality of the stable and unstable manifolds. The interest of such results is twofold. First, these orbits persist under perturbations which destroy the structure allowing the shooting method and, second, topological transversality is often sufficient when some kind of transversality is required to obtain chaotic dynamics. We shall focus on heteroclinic solutions in the extended Fisher–Kolmogorov equation.

MELNIKOV'S METHOD AND STICK–SLIP CHAOTIC OSCILLATIONS IN VERY WEAKLY FORCED MECHANICAL SYSTEMS

1999 ◽  
Vol 09 (03) ◽  
pp. 505-518 ◽  
Author(s):  
J. AWREJCEWICZ ◽  
M. M. HOLICKE
Keyword(s):  
Numerical Simulation ◽  
Chaotic Dynamics ◽  
Mechanical Systems ◽  
Degree Of Freedom ◽  
Chaotic Oscillations ◽  
Stick Slip ◽  
Melnikov's Method ◽  
Melnikov’S Method

In this paper we predict stick–slip chaotic dynamics in a one-degree-of-freedom very weakly forced (quasiautonomous) oscillator using the Melnikov's technique. Numerical simulation confirms the validity of our approach.

Chaos in a Spacecraft Attitude Maneuver Due to Time-Periodic Perturbations

1996 ◽  
Vol 63 (2) ◽  
pp. 501-508 ◽  
Author(s):  
G. L. Gray ◽  
I. Dobson ◽  
D. C. Kammer
Keyword(s):  
Chaotic Motion ◽  
Chaotic Dynamics ◽  
Equations Of Motion ◽  
Major Axis ◽  
Artificial Satellites ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Attitude Maneuver ◽  
Analytical Criterion ◽  
Time Periodic

We use Melnikov’s method to study the chaotic dynamics of an attitude transition maneuver of a torque-free rigid body in going from minor axis spin to major axis spin under the influence of small damping. The chaotic motion is due to the formation of Smale horseshoes which are caused by the oscillation of small subbodies inside the satellite. The equations of motion are derived and then transformed into a form suitable for the application of Melnikov’s method. An analytical criterion for chaotic motion is derived in terms of the system parameters. This criterion is evaluated for its significance to the design of artificial satellites.

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