Periodic Motions and Bifurcation Trees in a Parametric Duffing Oscillator

2017 ◽  
Author(s):  
Albert C. J. Luo ◽  
Haolin Ma
Keyword(s):  
Differential Equation ◽  
Numerical Simulations ◽  
Duffing Oscillator ◽  
Eigenvalue Analysis ◽  
Analytic Method ◽  
Periodic Motions ◽  
Implicit Mapping ◽  
Stability And Bifurcation

This paper studies bifurcation trees of periodic motions in a parametric, damped Duffing oscillator. From the semi-analytic method, the corresponding differential equation is discretized to obtain the implicit mapping. From implicit mapping structure, the periodic nodes of periodic motions are computed, and the bifurcation trees of period-1 to period-4 motions are presented and the corresponding stability and bifurcation are carried out by eigenvalue analysis. From the analytical predictions, numerical simulations are completed, and the trajectory, harmonic amplitudes and phases of period-1 to period-4 motions are illustrated.

Period-3 Motions in a Parametrically Exited Inverted Pendulum

2020 ◽  
Author(s):  
Albert C. J. Luo ◽  
Chuan Guo
Keyword(s):  
Numerical Simulations ◽  
Inverted Pendulum ◽  
Period Doubling ◽  
Mapping Method ◽  
Eigenvalue Analysis ◽  
Symmetric Motion ◽  
Bifurcation Tree ◽  
Implicit Mapping ◽  
Stability And Bifurcation ◽  
Period Doubling Bifurcations

Abstract In this paper, period-3 motions in a parametrically exited inverted pendulum are analytically investigated through a discrete implicit mapping method. The corresponding stability and bifurcation conditions of the period-3 motions are predicted through eigenvalue analysis. The symmetric and asymmetric period-3 motions are obtained on the bifurcation tree, and the period-doubling bifurcations of the asymmetric period-3 motions are observed. The saddle-node and Neimark bifurcations for symmetric period-3 motions are obtained. The saddle-bifurcations of the symmetric period-3 motions are for symmetric motion appearance (or vanishing) and onsets of asymmetric period-3 motion. Numerical simulations of the period-3 motions in the inverted pendulum are completed from analytical predictions for illustration of motion complexity and characteristics.

Period Motions in a Periodically Forced, Damped Double Pendulum

2018 ◽  
Author(s):  
Albert C. J. Luo ◽  
Chuan Guo
Keyword(s):  
Differential Equation ◽  
Numerical Simulation ◽  
Mapping Method ◽  
Eigenvalue Analysis ◽  
Double Pendulum ◽  
Implicit Mapping ◽  
Stability And Bifurcation

In this paper, period motions in a periodically forced, damped, double pendulum are analytically predicted through a discrete implicit mapping method. The implicit mapping is established via the discretized differential equation. The corresponding stability and bifurcation conditions of the period motions are predicted through eigenvalue analysis. Numerical simulation of the period motions in the double pendulum is completed from analytical predictions.

Analytical Prediction of Period-1 Motions in a Time-Delayed, Softening Duffing Oscillator

2017 ◽  
Author(s):  
Albert C. J. Luo ◽  
Siyuan Xing
Keyword(s):  
Differential Equation ◽  
Numerical Simulation ◽  
Duffing Oscillator ◽  
Mapping Method ◽  
Eigenvalue Analysis ◽  
Analytical Prediction ◽  
Implicit Mapping ◽  
Stability And Bifurcations ◽  
The Stability

In this paper, symmetric and asymmetric period-1 motions in a periodically forced, time-delayed, softening Duffing oscillator is analytically predicted through a discrete implicit mapping method. Such a method is based on the discretization of the corresponding differential equation. The stability and bifurcations of the symmetric and asymmetric period-1 motions are determined through eigenvalue analysis. Numerical simulation of the period-1 motions in the time-delayed softening Duffing oscillator is presented for verification of the analytical prediction.

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.

Period-1 Motions in a Time-Delayed Duffing Oscillitor With Periodic Excitation

2014 ◽  
Author(s):  
Albert C. J. Luo ◽  
Hanxiang Jin
Keyword(s):  
Fourier Series ◽  
Analytical Solutions ◽  
Duffing Oscillator ◽  
Eigenvalue Analysis ◽  
Periodic Excitation ◽  
Periodic Motions ◽  
Stability And Bifurcation ◽  
The Stability ◽  
Amplitude Characteristics

In this paper, analytical solutions of period-1 motions in a time-delayed Duffing oscillator with a periodic excitation are investigated through the Fourier series, and the stability and bifurcation of such periodic motions are discussed by eigenvalue analysis. The symmetric and asymmetric period-1 motions in such time-delayed Duffing oscillator are obtained analytically, and the frequency-amplitude characteristics of period-1 motions in such a time-delayed Duffing oscillator are investigated. Numerical illustrations of period-1 motions are given by numerical and analytical solutions.

Periodic Motions in a 2-DOF Self-Excited Duffing Oscillator

2017 ◽  
Author(s):  
Yeyin Xu ◽  
Albert C. J. Luo
Keyword(s):  
Numerical Simulations ◽  
Analytical Solutions ◽  
Analytical Method ◽  
Periodic Motion ◽  
Initial Conditions ◽  
Duffing Oscillator ◽  
Excitation Frequency ◽  
The Self ◽  
Periodic Motions ◽  
Implicit Mapping

In this paper, analytical solutions of periodic motions in a 2-DOF self-excited Duffing oscillator are investigated through a semi-analytical method. The semi-analytical method discretizes the self-excited Duffing oscillator for the discrete implicit mappings. Through the implicit mapping, period-1 motion varying with excitation frequency are presented, and the corresponding stability and bifurcation are discussed via the eigenvalues analysis. The Neimark and saddle-node bifurcations of the periodic motion are obtained. Initial conditions for numerical simulations are from analytical solutions. Numerical and analytical solutions of periodic motions are illustrated for comparison.

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.

An Anaytical Prediction of Period-1 to Chaos in a Periodically Forced, Damped, Hardening Duffing Oscillator

2015 ◽  
Author(s):  
Yu Guo ◽  
Albert Luo
Keyword(s):  
Differential Equation ◽  
Periodic Motion ◽  
Duffing Oscillator ◽  
Numerical Results ◽  
Eigenvalue Analysis ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
Discrete Maps ◽  
Mapping Structures ◽  
Stability And Bifurcations

In this paper, periodic motions of a periodically forced, damped Duffing oscillator are analytically predicted by use of implicit discrete mappings. The implicit discrete maps are achieved by the discretization of the differential equation of the periodically forced, damped Duffing oscillator. Periodic motion is constructed by mapping structures, and bifurcation trees of periodic motions are developed analytically, and the corresponding stability and bifurcations of periodic motion are determined through eigenvalue analysis. Finally, from the analytical prediction, numerical results of periodic motions are presented to show the complexity of periodic motions.

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.

Period-3 Motions in a Periodically Forced, Damped, Double-Well Duffing Oscillator With Time-Delay

2017 ◽  
Author(s):  
Albert C. J. Luo ◽  
Siyuan Xing
Keyword(s):  
Differential Equation ◽  
Time Delay ◽  
Analytical Method ◽  
Duffing Oscillator ◽  
Eigenvalue Analysis ◽  
Algebraic Equations ◽  
Implicit Mapping ◽  
Mapping Structures ◽  
Nonlinear Algebraic Equations ◽  
Amplitude Characteristics

In this paper, period-3 motions in a double-well Duffing oscillator with time-delay are predicted by a semi-analytical method. The implicit mapping structures of period-3 motions are constructed through the implicit mappings obtained by discretization of the corresponding differential equation. Complex period-3 motions are predicted through nonlinear algebraic equations of the implicit mappings in the mapping structures and the corresponding stability and bifurcation are carried out through eigenvalue analysis. Numerical and analytical results of complex period-3 motions are obtained and the corresponding frequency-amplitude characteristics are presented.

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