Experiments on Chaotic Motions of a Forced Nonlinear Oscillator: Strange Attractors

1980 ◽  
Vol 47 (3) ◽  
pp. 638-644 ◽  
Author(s):  
F. C. Moon
Keyword(s):  
Strange Attractor ◽  
Phase Plane ◽  
Chaotic Behavior ◽  
Multiple Equilibrium ◽  
Chaotic Motions ◽  
Magnetic Forces ◽  
Forcing Function ◽  
Melnikov Criterion ◽  
Forced Nonlinear Oscillator ◽  
Experimental Criterion

The forced vibrations of a buckled beam show nonperiodic, chaotic behavior for forced deterministic excitations. Using magnetic forces to buckle the beam, two and three stable equilibrium positions for the postbuckling state of the beam are found. The deflection of the beam under nonlinear magnetic forces behaves statically as a butterfly catastrophe and dynamically as a strange attractor. The forced nonperiodic vibrations about these multiple equilibrium positions are studied experimentally using Poincare plots in the phase plane. The apparent chaotic motions are shown to possess an intricate but well-defined structure in the Poincare plane for moderate damping. The structure of the strange attractor is unravelled experimentally by looking at different Poincare projections around the toroidal product space of the phase plane and phase angle of the forcing function. An experimental criterion on the forcing amplitude and frequency for strange attractor motions is obtained and compared with the Holmes-Melnikov criterion and a heuristic formula developed by the author.

Characteristics in Response Integration and Bifurcation of a Forced Duffing Oscillator

2007 ◽  
Author(s):  
Jang-Der Jeng ◽  
Yuan Kang ◽  
Yeon-Pun Chang ◽  
Shyh-Shyong Shyr
Keyword(s):  
Applied Sciences ◽  
Phase Plane ◽  
Duffing Oscillator ◽  
High Order ◽  
Chaotic Motions ◽  
Integration Algorithm ◽  
High Order Harmonic ◽  
Stiffness And Damping

The Duffing oscillator is well-known models of nonlinear system, with applications in many fields of applied sciences and engineering. In this paper, a response integration algorithm is proposed to analyze high-order harmonic and chaotic motions in this oscillator for modeling rotor excitations. This method numerically integrates the distance between state trajectory and the origin in the phase plane during a specific period and predicted intervals with excitation periods. It provides a quantitative characterization of system responses and can replace the role of the traditional stroboscopic technique (Poincare´ section method) to observe bifurcations and chaos of the nonlinear oscillators. Due to the signal response contamination of system, thus it is difficult to identify the high-order responses of the subharmonic motion because of the sampling points on Poincare´ map too near each other. Even the system responses will be made misjudgments. Combining the capability of precisely identifying period and constructing bifurcation diagrams, the advantages of the proposed response integration method are shown by case studies. Applying this method, the effects of the change in the stiffness and the damping coefficients on the vibration features of a Duffing oscillator are investigated in this paper. From simulation results, it is concluded that the stiffness and damping of the system can effectively suppress chaotic vibration and reduce vibration amplitude.

Radial basis function neural network chaos control of a piezomagnetoelastic energy harvesting system

2019 ◽  
Vol 25 (16) ◽  
pp. 2191-2203 ◽  
Author(s):  
R. Dehghani ◽  
H. M. Khanlo
Keyword(s):  
Energy Harvesting ◽  
Radial Basis Function ◽  
Basis Function ◽  
Output Voltage ◽  
Chaos Control ◽  
Chaotic Behavior ◽  
Rbf Network ◽  
Magnetic Forces ◽  
Radial Basis ◽  
Energy Harvesting System

In this paper, an adaptive chaos control is proposed for a typical vibratory piezomagnetoelastic energy harvesting system to return the chaotic behavior to a periodic one. Piezomagnetoelastic energy harvesting systems show chaotic behaviors in spite of harmonic input. Although, the chaotic behavior of the system gives higher output voltage than the periodic motion, it is preferred to the output voltage as this is periodic for charging a battery or a capacitor efficiently. Therefore, the chaos control is important in this system. The physical model is composed of the upper and lower piezoelectric layers on a cantilever taper beam, one attached tip magnet, and two external magnets (EM). Position of the EM is controlled by inputs. Firstly, chaotic and periodic regions are detected by utilizing the bifurcation diagrams, phase plan portrait, and Poincaré maps. Then an adaptive controller is proposed for controlling of the chaotic behaviors in the presence of uncertainty due to magnetic forces. The control law is derived based on the inverse dynamic method and the uncertainty elements of the controller are estimated using radial basis function (RBF) network. The weights of the RBF network are obtained using an adaptation law. The adaptation laws are derived based on Lyapunov stability theory and a projection operator. The distance of the tip magnet and the EM as well as the gap distance of two EM are used to control the chaotic behavior. Simulation results show that the proposed controller can return the chaotic motion to a periodic one in spite of the uncertainties in the magnetic forces.

Complicated Regular and Chaotic Motions of the Parametrically Excited Pendulum

2005 ◽  
Author(s):  
Eugene I. Butikov
Keyword(s):  
Computer Simulations ◽  
Parametric Resonance ◽  
Inverted Pendulum ◽  
Chaotic Behavior ◽  
Spectral Composition ◽  
Physical Explanation ◽  
Chaotic Motions ◽  
Upper Limit ◽  
Close Relationship ◽  
Subharmonic Resonances

Several new types of regular and chaotic behavior of the parametrically driven pendulum are discovered with the help of computer simulations. A simple physical explanation is suggested to the phenomenon of subharmonic resonances. The boundaries of these resonances in the parameter space and the spectral composition of corresponding stationary oscillations are determined theoretically and verified experimentally. A close relationship between the upper limit of stability of the dynamically stabilized inverted pendulum and parametric resonance of the non-inverted pendulum is established. Most of the newly discovered modes are still waiting a plausible physical explanation.

TRANSITION TO CHAOS IN A CURVED CARBON NANOTUBE UNDER HARMONIC EXCITATION

2012 ◽  
Vol 26 (32) ◽  
pp. 1250210 ◽  
Author(s):  
YIXIANG GENG ◽  
LIXIANG ZHANG
Keyword(s):  
Carbon Nanotube ◽  
Chaotic Behavior ◽  
Numerical Study ◽  
Melnikov Method ◽  
Equation Of Motion ◽  
Harmonic Excitation ◽  
Periodic Attractor ◽  
Runge Kutta Method ◽  
Transition To Chaos ◽  
Melnikov Criterion

The chaotic behavior of a carbon nanotube with waviness along its axis is investigated. The equation of motion involves a quadratic and cubic terms due to the curved geometry and the mid-plane stretching. Melnikov method is applied for the system, and Melnikov criterion for global homoclinic bifurcations is obtained analytically. The numerical solution of the system using a fourth-order-Runge–Kutta method confirms the analytical predictions and shows that the transition from regular to chaotic motion is often associated with increasing the energy of an oscillator. Moreover, a detailed numerical study of the periodic attractor in the period window is also carried out.

Chaotic Motion of a Nonhomogeneous Torsional Pendulum

1997 ◽  
Vol 07 (03) ◽  
pp. 733-740 ◽  
Author(s):  
Jiin-Po Yeh
Keyword(s):  
Total Energy ◽  
Lyapunov Exponents ◽  
Chaotic Motion ◽  
Local Minimum ◽  
Phase Plane ◽  
Nonlinear Oscillations ◽  
Initial Conditions ◽  
Chaotic Motions ◽  
Torsional Pendulum ◽  
Damped Systems

In this paper, the nonlinear oscillations of a nonhomogeneous torsional pendulum are investigated. Chaotic motions are shown to exist in both damped systems with two-well potential and undamped systems with one-well or two-well potential. Autocorrelations of the Poincaré mappings of the motion are presented and shown to be another useful tool to judge whether the system is chaotic. The total energy of the torsional pendulum is explored as well and it is conjectured that the irregularity of the total energy is probably one of the important factors which cause chaos. Lyapunov exponents are used as an indication of chaos in this paper. For systems with two-well potential, the phase-plane trajectories are found to stay in one well if the motion is regular, but jump from one well to another if the motion is chaotic. Making the initial conditions near the local minimum of the two-well potential is proved to be successful in preventing chaos from happening in the undamped systems.

scholarly journals Traveling Wave Solutions and Chaotic Motions for a Perturbed Nonlinear Schrödinger Equation with Power-Law Nonlinearity and Higher-Order Dispersions

2021 ◽  
Author(s):  
Mati Youssoufa ◽  
Ousmanou Dafounansou ◽  
Camus Gaston Latchio Tiofack ◽  
Alidou Mohamadou
Keyword(s):  
Power Law ◽  
Traveling Wave ◽  
Bifurcation Theory ◽  
Chaotic Behavior ◽  
Traveling Wave Solutions ◽  
Chaotic Motions ◽  
Auxiliary Equation Method ◽  
Nonlinear Schrödinger ◽  
Wave Solutions ◽  
Quasiperiodic Route To Chaos

This chapter aims to study and solve the perturbed nonlinear Schrödinger (NLS) equation with the power-law nonlinearity in a nano-optical fiber, based upon different methods such as the auxiliary equation method, the Stuart and DiPrima’s stability analysis method, and the bifurcation theory. The existence of the traveling wave solutions is discussed, and their stability properties are investigated through the modulational stability gain spectra. Moreover, the development of the chaotic motions for the systems is pointed out via the bifurcation theory. Taking into account an external periodic perturbation, we have analyzed the chaotic behavior of traveling waves through quasiperiodic route to chaos.

Chaotic Behavior of a Two Mass Bouncing System

1992 ◽  
Author(s):  
Chi-Wook Lee ◽  
Ali Seireg ◽  
Joseph Duffy
Keyword(s):  
Power Spectrum ◽  
Spectrum Analysis ◽  
Phase Plane ◽  
Chaotic Behavior ◽  
Power Spectrum Analysis ◽  
Ground Contact ◽  
Hopping Robots ◽  
Plane Technique ◽  
The Stability ◽  
Spring Constants

Abstract This study investigates the behavior of simple two mass bouncing systems which are released from a certain height. A nonlinearity exists in the discontinuity of the flight and the ground modes, although the behavior of the systems is linear in each mode. Such oscillators provide models for mechanical systems such as legged systems for hopping robots. The phase plane technique and the power spectrum analysis are used to investigate the stability of bouncing systems and the chaos that may occur. The effects of the spring constants and the damping coefficient at the ground contact on the bouncing behavior is also investigated.

Dynamics of a Fluid-Conveying Cantilevered Pipe With Intermediate Spring Support

2010 ◽  
Author(s):  
Mergen H. Ghayesh ◽  
Michael P. Pai¨doussis
Keyword(s):  
Phase Plane ◽  
Nonlinear Differential Equations ◽  
Three Dimensional ◽  
Quantitative Agreement ◽  
Chaotic Motions ◽  
Qualitative And Quantitative ◽  
Time Histories ◽  
Cantilevered Pipe ◽  
Raphson Method ◽  
Spring Support

The aim of this study is to investigate the three-dimensional (3-D) nonlinear dynamics of a fluid-conveying cantilevered pipe, additionally supported by an array of four springs attached at a point along its length. In the theoretical analysis, the 3-D equations are discretized via Galerkin’s technique, yielding a set of coupled nonlinear differential equations. These equations are solved numerically using a finite difference technique along with the Newton-Raphson method. The dynamic behaviour of the system is presented in the form of bifurcation diagrams, along with phase-plane plots, time-histories, PSD plots, and Poincare´ maps for two different spring locations and inter-spring configurations. Interesting dynamical phenomena, such as planar or circular flutter, divergence, quasiperiodic and chaotic motions, have been observed with increasing flow velocity. Experiments were conducted for the cases studied theoretically, and good qualitative and quantitative agreement was observed.

scholarly journals Harmonic Balance Method for Chaotic Dynamics in Fractional-Order Rössler Toroidal System

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Huijian Zhu
Keyword(s):  
Theoretical Analysis ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Harmonic Balance ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Chaotic Motions ◽  
Behavior Based

This paper deals with the problem of determining the conditions under which fractional order Rössler toroidal system can give rise to chaotic behavior. Based on the harmonic balance method, four detailed steps are presented for predicting the existence and the location of chaotic motions. Numerical simulations are performed to verify the theoretical analysis by straightforward computations.

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