Stability and Bifurcation Analysis of Periodic Motions in a Periodically Forced Duffing Oscillator

2011 ◽  
Author(s):  
Albert C. J. Luo ◽  
Jianzhe Huang
Keyword(s):  
Approximate Solution ◽  
Bifurcation Analysis ◽  
Duffing Oscillator ◽  
Approximate Solutions ◽  
Nonlinear Damping ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Stability And Bifurcation Analysis

In this paper, the analytical, approximate solutions of period-1 motions in the nonlinear damping, periodically forced, Duffing oscillator is obtained. The corresponding stability and bifurcation analysis of the HB2 approximate solution of period-1 motions in the forced Duffing oscillator is carried out. Numerical illustrations of period-1 motions are presented.

On Complex Periodic Motions in a Time-Delayed, Double-Well Duffing Oscillator With Strong Excitation

2016 ◽  
Author(s):  
Albert C. J. Luo ◽  
Siyuan Xing
Keyword(s):  
Differential Equation ◽  
Bifurcation Analysis ◽  
Analytical Method ◽  
Particle Physics ◽  
Duffing Oscillator ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Stability And Bifurcation Analysis ◽  
Strong Excitation

The time-delayed double-well Duffing oscillator is extensively applied in engineering and particle physics. Determination of periodic motions in such a system is significant. Thus, in this paper, period-1 motions in the time-delayed double-well Duffing oscillator are discussed through a semi-analytical method. The semi-analytical method is based on the implicit mappings constructed by discretization of the corresponding differential equation. Complex period-1 motions are predicted and the corresponding stability and bifurcation analysis are completed. From predictions, complex periodic motions are simulated numerically, and the harmonic amplitudes and phases are presented. Through this study, the complexity of periodic motions in the time-delayed Duffing oscillator can be better understood.

SINGULARITY, SWITCHABILITY AND BIFURCATIONS IN A 2-DOF, PERIODICALLY FORCED, FRICTIONAL OSCILLATOR

2013 ◽  
Vol 23 (03) ◽  
pp. 1330009 ◽  
Author(s):  
ALBERT C. J. LUO ◽  
MOZHDEH S. FARAJI MOSADMAN
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Analysis ◽  
Degrees Of Freedom ◽  
Eigenvalue Analysis ◽  
Analytical Dynamics ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis ◽  
The Stability

In this paper, the analytical dynamics for singularity, switchability, and bifurcations of a 2-DOF friction-induced oscillator is investigated. The analytical conditions of the domain flow switchability at the boundaries and edges are developed from the theory of discontinuous dynamical systems, and the switchability conditions of boundary flows from domain and edge flows are presented. From the singularity and switchability of flow to the boundary, grazing, sliding and edge bifurcations are obtained. For a better understanding of the motion complexity of such a frictional oscillator, switching sets and mappings are introduced, and mapping structures for periodic motions are adopted. Using an eigenvalue analysis, the stability and bifurcation analysis of periodic motions in the friction-induced system is carried out. Analytical predictions and parameter maps of periodic motions are performed. Illustrations of periodic motions and the analytical conditions are completed. The analytical conditions and methodology can be applied to the multi-degrees-of-freedom frictional oscillators in the same fashion.

Periodic Motions in the Periodically Forced Oscillator With Multiple Discontinuities

2006 ◽  
Author(s):  
Albert C. J. Luo ◽  
Mehul T. Patel
Keyword(s):  
Dynamical System ◽  
Bifurcation Analysis ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
Mapping Structures ◽  
Forced Oscillator ◽  
Stability And Bifurcation Analysis ◽  
The Stability

In this paper, the stability and bifurcation of periodic motions in periodically forced oscillator with multiple discontinuities is investigated. The generic mappings are introduced for the analytical prediction of periodic motions. Owing to the multiple discontinuous boundaries, the mapping structures for periodic motions are very complicated, which causes more difficulty to obtain periodic motions in such a dynamical system. The analytical prediction of complex periodic motions is carried out and verified numerically, and the corresponding stability and bifurcation analysis are performed. Due to page limitation, grazing and stick motions and chaos in this system will be investigated further.

Approximate solutions of periodic motions in nonlinear systems via a generalized harmonic balance

2011 ◽  
Vol 18 (11) ◽  
pp. 1661-1674 ◽  
Author(s):  
Albert CJ Luo ◽  
Jianzhe Huang
Keyword(s):  
Harmonic Balance ◽  
Nonlinear Dynamical Systems ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Approximate Solutions ◽  
Nonlinear Damping ◽  
Balance Method ◽  
Periodic Motions ◽  
Approximate Analytical Solutions ◽  
Generalized Harmonic Balance Method

In this paper, the generalized harmonic balance method is presented for approximate, analytical solutions of periodic motions in nonlinear dynamical systems. The nonlinear damping, periodically forced, Duffing oscillator is studied as a sample problem. The approximate, analytical solution of period-1 periodic motion of such an oscillator is obtained by the generalized harmonic balance method. The stability and bifurcation analysis of the HB2 approximate solution of period-1 motions in the forced Duffing oscillator is carried out, and the parameter map for such HB2 solutions is achieved. Numerical illustrations of period-1 motions are presented. Similarly, the same ideas can be extended to period- k motions in such an oscillator. The methodology presented in this paper can be applied to other nonlinear vibration systems, which are independent of small parameters.

On Periodic Motions in a Parametric Hardening Duffing Oscillator

2014 ◽  
Vol 24 (01) ◽  
pp. 1430004 ◽  
Author(s):  
Albert C. J. Luo ◽  
Dennis M. O'Connor
Keyword(s):  
Fourier Series ◽  
Bifurcation Analysis ◽  
Analytical Solutions ◽  
Duffing Oscillator ◽  
Periodic Motions ◽  
Period 2 ◽  
Approximate Analytical Solutions ◽  
Stability And Bifurcation Analysis ◽  
Series Expression ◽  
Amplitude Characteristics

In this paper, analytical solutions for periodic motions in a parametric hardening Duffing oscillator are presented using the finite Fourier series expression, and the corresponding stability and bifurcation analysis for such periodic motions are carried out. The frequency-amplitude characteristics of asymmetric period-1 and symmetric period-2 motions are discussed. The hardening Mathieu–Duffing oscillator is also numerically simulated to verify the approximate analytical solutions of periodic motions. Period-1 asymmetric and period-2 symmetric motions are illustrated for a better understanding of periodic motions in the hardening Mathieu–Duffing oscillator.

Complete Bifurcation Trees of Independent Period-2 Motions to Chaos in a Parametrically Excited Pendulum

2019 ◽  
Author(s):  
Yu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Bifurcation Analysis ◽  
Mapping Method ◽  
Eigenvalue Analysis ◽  
Periodic Motions ◽  
Parametrically Excited ◽  
Stability And Bifurcation ◽  
Period 2 ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis

Abstract In this paper, bifurcation trees of independent period-2 motions to chaos are investigated in a parametrically excited pendulum. The implicit discrete mapping method is employed to obtain periodic motions in such a system. Analytical predictions of periodic motions are based on the mapping structures and peroidicity. The bifurcation trees of independent period-2 motions to chaos are studied, and the corresponding stability and bifurcation analysis are completed through eigenvalue analysis. Finally, sampled period-2 motions are simulated numerically in comparison to the analytical predictions. The infinite bifurcation trees of independent period-2 motions to chaos can be obtained.

Analytical Solutions for Periodic Motions in a Hardening Mathieu-Duffing Oscillator

2013 ◽  
Author(s):  
Albert C. J. Luo ◽  
Dennis M. O’Connor
Keyword(s):  
Fourier Series ◽  
Bifurcation Analysis ◽  
Analytical Solutions ◽  
Duffing Oscillator ◽  
Series Solutions ◽  
Periodic Motions ◽  
Period 2 ◽  
Approximate Analytical Solutions ◽  
Stability And Bifurcation Analysis ◽  
The Stability

Analytical solutions for period-m motions in a hardening Mathieu-Duffing oscillator are obtained using the finite Fourier series solutions, and the stability and bifurcation analysis of such periodic motions are completed. To verify the approximate analytical solutions of periodic motions, numerical simulations of the hardening Mathieu-Duffing oscillator are presented. Period-1 asymmetric and period-2 symmetric motions are illustrated.

Periodic Motions in a Coupled Van Der Pol-Duffing Oscillator

2017 ◽  
Author(s):  
Yeyin Xu ◽  
Albert C. J. Luo
Keyword(s):  
Numerical Simulations ◽  
Bifurcation Analysis ◽  
Initial Conditions ◽  
Duffing Oscillator ◽  
Eigenvalue Analysis ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis

In this paper, periodic motions of a periodically forced, coupled van der Pol-Duffing oscillator are predicted analytically. The coupled van der Pol-Duffing oscillator is discretized for the discrete mapping. The periodic motions in such a coupled van der Pol-Duffing oscillator are obtained from specified mapping structures, and the corresponding stability and bifurcation analysis are carried out by eigenvalue analysis. Based on the analytical prediction, the initial conditions of periodic motions are used for numerical simulations.

Bifurcation Trees of Period-1 Motion to Chaos in a Duffing Oscillator With Double-Well Potential

2015 ◽  
Author(s):  
Yu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Differential Equation ◽  
Bifurcation Analysis ◽  
Duffing Oscillator ◽  
Eigenvalue Analysis ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
Discrete Maps ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis ◽  
Double Well Potential

In this paper, periodic motions of a periodically forced Duffing oscillator with double-well potential are analytically predicted through implicit discrete mappings. The implicit discrete maps are obtained from discretization of differential equation of the Duffing oscillator. From mapping structures, bifurcation trees of periodic motions are predicted analytically, and the corresponding stability and bifurcation analysis are carried out through eigenvalue analysis. From the analytical prediction, numerical results of periodic motions are performed to verify the analytical prediction.

Analytical Solutions for Periodic Motions in a Twin-Well Potential Mathieu-Duffing Oscillator

2016 ◽  
Author(s):  
Dennis M. O’Connor ◽  
Albert C. J. Luo
Keyword(s):  
Fourier Series ◽  
Bifurcation Analysis ◽  
Analytical Solutions ◽  
Duffing Oscillator ◽  
Series Solutions ◽  
Periodic Motions ◽  
Approximate Analytical Solutions ◽  
Stability And Bifurcation Analysis ◽  
Unstable Solutions ◽  
The Stability

Analytical solutions for periodic motion in a twin-well potential Mathieu-Duffing oscillator with damping are obtained using the finite Fourier series solutions, and the stability and bifurcation analysis of such periodic motions are completed. To verify the approximate analytical solutions of periodic motions, numerical simulations of the twin-well solutions are presented for Period-1 asymmetric motions. Further, simulation of the unstable solutions is considered to demonstrate the resonant bands separating the twin-wells.

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