Homoclinic Orbits of a Buckled Beam Subjected to Transverse Uniform Harmonic Excitation

2019 ◽  
Vol 29 (05) ◽  
pp. 1950061
Author(s):  
D. M. Zhang ◽  
Y. L. Jin ◽  
F. Li
Keyword(s):  
Melnikov Method ◽  
Homoclinic Orbits ◽  
Harmonic Excitation ◽  
Slow Manifold ◽  
Singular Perturbation Method ◽  
Smale Horseshoe ◽  
Buckled Beam ◽  
Resonance Band ◽  
The One ◽  
Geometric Singular Perturbation Method

Homoclinic orbits of a buckled beam subjected to transverse uniform harmonic excitation are investigated in the case of 1:1 internal resonance. The geometric singular perturbation method and Melnikov method are employed to show the existence of the one-bump and multi-bump homoclinic orbits that connect the equilibria in a resonance band of the slow manifold. Each bump is a fast excursion away from the resonance band, and the bumps are interspersed with slow segments near the resonance band. The results obtained imply the existence of the amplitude modulated chaos for the Smale horseshoe sense in the class of buckled beam systems.

Oscillation Suppression in a Delay-Coupled Flexible-Joint System

2013 ◽  
Vol 477-478 ◽  
pp. 123-127
Author(s):  
Shan Ying Jiang
Keyword(s):  
Time Delay ◽  
Periodic Oscillation ◽  
Slow Manifold ◽  
Fast Subsystem ◽  
Singular Perturbation Method ◽  
Joint System ◽  
Flexible Joint ◽  
The Stability ◽  
Slow Subsystem ◽  
Geometric Singular Perturbation Method

A delay-coupled flexible-joint system is investigated in this paper. Because of the different time scales, the flexible-joint system could be transformed into a fast subsystem and a slow subsystem. The geometric singular perturbation method is used to obtain the slow manifold defining as the equilibrium of the fast subsystem. The eigenvalue analysis of the fast subsystem reveals a relation between the stability of the slow manifold and the time delay. The analysis results provide an idea of suppressing the small amplitude periodic oscillation via adjusting the time delay. Numerical simulations are performed to display the effectiveness of this method.

Delay-Induced Bogdanov–Takens Bifurcation and Dynamical Classifications in a Slow-Fast Flexible Joint System

2015 ◽  
Vol 25 (09) ◽  
pp. 1550121 ◽  
Author(s):  
Jian Xu ◽  
Shanying Jiang
Keyword(s):  
Time Delay ◽  
Homoclinic Orbits ◽  
Singular Perturbation Method ◽  
Joint System ◽  
Fast System ◽  
Flexible Joint ◽  
Melnikov Theory ◽  
The Stability ◽  
Delay Differential ◽  
Geometric Singular Perturbation Method

A slow-fast delay-coupled flexible joint system is investigated in this paper. To understand the effects of time delay on the stability and oscillation of the manipulator, the geometric singular perturbation method is extended in dealing with delay differential equations. Bogdanov–Takens (BT) bifurcation of the fast subsystem is obtained, which leads to the existence of homoclinic orbits and is proved to be related to the formation of spiking. After the break of homoclinic orbits, Melnikov theory is introduced to predict the threshold curve indicating the occurrence of chaos. Numerical results show that with the increase of time delay, the stability of the system gets worse, and complicated oscillations including bursting, chaotic-bursting and complete chaos turn up. Besides, it is briefly summarized that the effect of the small parameter in the slow-fast system is to influence the convergence rate of solution trajectories, which is widely neglected in previous works.

scholarly journals A Melnikov Method for Homoclinic Orbits with Many Pulses

1998 ◽  
Vol 143 (2) ◽  
pp. 105-193 ◽  
Author(s):  
Roberto Camassa ◽  
Gregor Kovačič ◽  
Siu-Kei Tin
Keyword(s):  
Melnikov Method ◽  
Homoclinic Orbits

scholarly journals SYNCHRONIZATION AS A PROCESS OF SHARING AND TRANSFERRING INFORMATION

2012 ◽  
Vol 22 (11) ◽  
pp. 1250261 ◽  
Author(s):  
ERIK M. BOLLT
Keyword(s):  
Symbolic Dynamics ◽  
Slow Manifold ◽  
Transfer Entropy ◽  
The Other ◽  
Chaotic Oscillators ◽  
Smale Horseshoe ◽  
Global System ◽  
Identity Function ◽  
Transfer Operators ◽  
First Time

Synchronization of chaotic oscillators has become well characterized by errors which shrink relative to a synchronization manifold. This manifold is the identity function in the case of identical systems, or some other slow manifold in the case of generalized synchronizaton in the case of nonidentical components. On the other hand, since many decades beginning with the Smale horseshoe, chaotic oscillators can be well understood in terms of symbolic dynamics as information producing processes. We study here the synchronization of a pair of chaotic oscillators as a process for sharing information bearing bits transferred between each other, by measuring the transfer entropy tracked as the global system transitions to the synchronization state. Further, we present for the first time the notion of transfer entropy in the measure theoretic setting of transfer operators.

Stability of Subharmonic Oscillations in a Pipe Conveying Fluid under Harmonic Excitation

2013 ◽  
Vol 444-445 ◽  
pp. 796-800
Author(s):  
Yi Xiang Geng ◽  
Han Ze Liu
Keyword(s):  
Averaging Method ◽  
Melnikov Method ◽  
Equation Of Motion ◽  
Harmonic Excitation ◽  
Angle Variable ◽  
Galerkin Approach ◽  
Existence And Stability ◽  
The Stability ◽  
Action Angle Variable ◽  
Conveying Fluid

The existence and stability of subharmonic oscillations in a two end-fixed fluid conveying pipe whose base is subjected to a harmonic excitation are investigated. A Galerkin approach is utilized to reduce the equation of motion to a second order nonlinear differential equation. The conditions for the existence of subharmonic oscillations are given by using Melnikov method. The stability of subharmonic oscillations is discussed in detail by using action-angle variable and averaging method. It is shown that the velocity of fluid plays an important role in the stability of subharmonic oscillations.

ESTIMATION OF CHAOTIC THRESHOLDS FOR THE RECENTLY PROPOSED ROTATING PENDULUM

2013 ◽  
Vol 23 (04) ◽  
pp. 1350074 ◽  
Author(s):  
N. HAN ◽  
Q. J. CAO ◽  
M. WIERCIGROCH
Keyword(s):  
Nonlinear System ◽  
Nonlinear Behavior ◽  
Melnikov Method ◽  
Original System ◽  
Homoclinic Orbits ◽  
Heteroclinic Orbits ◽  
Chaotic Attractors ◽  
Viscous Damping ◽  
Approximate System ◽  
Smoothness Parameter

In this paper, we investigate the nonlinear behavior of the recently proposed rotating pendulum which is a cylindrically nonlinear system with irrational type having smooth and discontinuous characteristics depending on the value of a smoothness parameter. We introduce a cylindrical approximate system whose analytical solutions can be obtained successfully to reflect the nature of the original system without the barrier of irrationalities. Furthermore, Melnikov method is employed to detect the chaotic thresholds for the homoclinic orbits of the second-type, a pair of homoclinic orbits of the first and second-type and the double heteroclinic orbits under the perturbation of viscous damping and external harmonic forcing within the smooth regime. Numerical simulations show the efficiency of the proposed method and the results presented herein this paper demonstrate the predicated chaotic attractors of pendulum-type, SD-type and their mixture depending on the coupling of the nonlinearities.

scholarly journals DIFFERENTIAL GEOMETRY AND MECHANICS: APPLICATIONS TO CHAOTIC DYNAMICAL SYSTEMS

2006 ◽  
Vol 16 (04) ◽  
pp. 887-910 ◽  
Author(s):  
JEAN-MARC GINOUX ◽  
BRUNO ROSSETTO
Keyword(s):  
Differential Geometry ◽  
Dynamical Systems ◽  
Slow Manifold ◽  
Van Der Pol ◽  
New Approach ◽  
Metric Properties ◽  
Chaotic Dynamical Systems ◽  
Mechanics Concepts ◽  
The One ◽  
Curvature And Torsion

The aim of this article is to highlight the interest to apply Differential Geometry and Mechanics concepts to chaotic dynamical systems study. Thus, the local metric properties of curvature and torsion will directly provide the analytical expression of the slow manifold equation of slow-fast autonomous dynamical systems starting from kinematics variables (velocity, acceleration and over-acceleration or jerk). The attractivity of the slow manifold will be characterized thanks to a criterion proposed by Henri Poincaré. Moreover, the specific use of acceleration will make it possible on the one hand to define slow and fast domains of the phase space and on the other hand, to provide an analytical equation of the slow manifold towards which all the trajectories converge. The attractive slow manifold constitutes a part of these dynamical systems attractor. So, in order to propose a description of the geometrical structure of attractor, a new manifold called singular manifold will be introduced. Various applications of this new approach to the models of Van der Pol, cubic-Chua, Lorenz, and Volterra–Gause are proposed.

Application of the Random Melnikov Method for Single-Degree-of-Freedom Vessel Rolls Motion

2010 ◽  
Author(s):  
Kaiye Hu ◽  
Yong Ding ◽  
Hongwei Wang ◽  
Jide Li
Keyword(s):  
Global Stability ◽  
Global Bifurcation ◽  
Melnikov Method ◽  
Threshold Value ◽  
Homoclinic Orbits ◽  
Nonlinear Damping ◽  
Degree Of Freedom ◽  
Bounded Noise ◽  
Single Degree Of Freedom ◽  
Single Degree

Basing on the nonlinear dynamics theory, the global stability of ship in stochastic beam sea is researched by the global bifurcation method. In this paper, bounded noise is first briefly introduced. Bounded noise is a harmonic function with constant random frequency and phase. It has finite power and its spectral shape can be made to fit a target spectrum, such as Pierson-Moskowitz spectrum, by adjusting its parameters. This paper considered the stochastic excitation term as bounded noise and the influence of nonlinear damping and nonlinear righting moment, setup the random single degree of freedom nonlinear rolling equation. Then the random Melnikov process for the nonlinear system with homoclinic orbits under both dissipative and bounded noise perturbations is derived. The random Melnikov mean-square criterion is used to analysis the global stability of this system. The research indicates that the bounded noise can approximately simulate the wave excitation and if the noise exceeds the threshold value, the ship will undergo stochastic chaotic motion. That will lead ships to instability and even to capsizing.

Subharmonic Bifurcations and Transition to Chaos in a Pipe Conveying Fluid under Harmonic Excitation

2013 ◽  
Vol 444-445 ◽  
pp. 791-795
Author(s):  
Yi Xiang Geng ◽  
Han Ze Liu
Keyword(s):  
Chaotic Behavior ◽  
Melnikov Method ◽  
Harmonic Excitation ◽  
Phase Portraits ◽  
Pipe Conveying Fluid ◽  
Bifurcation Diagrams ◽  
Dynamical Behaviors ◽  
Transition To Chaos ◽  
Subharmonic Bifurcations ◽  
Conveying Fluid

The subharmonic and chaotic behavior of a two end-fixed fluid conveying pipe whose base is subjected to a harmonic excitation are investigated. Melnikov method is applied for the system, and Melnikov criterions for subharmonic and homoclinic bifurcations are obtained analytically. The numerical simulations (including bifurcation diagrams, maximal Lyapunov exponents, phase portraits and Poincare map) confirm the analytical predictions and exhibit the complicated dynamical behaviors.

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