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

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.

Bifurcation Trees of Period-1 Motions to Chaos in a Quadratic Nonlinear Oscillator With Time-Delayed Displacement

Stability And Bifurcation Analysis

10.1115/imece2015-50029 ◽

2015 ◽

Author(s):

Albert C. J. Luo ◽

Siyuan Xing

Keyword(s):

Differential Equation ◽

Bifurcation Analysis ◽

Nonlinear Oscillator ◽

Numerical Results ◽

Eigenvalue Analysis ◽

Analytical Prediction ◽

Periodic Motions ◽

Discrete Maps ◽

Mapping Structures ◽

In this paper, periodic motions in a periodically forced, damped, quadratic nonlinear oscillator with time-delayed displacement are analytically predicted through implicit discrete mappings. The implicit discrete maps are obtained from discretization of differential equation of such a quadratic nonlinear oscillator. From mapping structures, bifurcation trees of periodic motions are achieved analytically, and the corresponding stability and bifurcation analysis are completed through eigenvalue analysis. From the analytical prediction, numerical results of periodic motions are illustrated to verify such an analytical prediction.

Volume 6: 13th International Conference on Multibody Systems, Nonlinear Dynamics, and Control ◽

Periodic Motions in a Coupled Van Der Pol-Duffing Oscillator

10.1115/detc2017-67563 ◽

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

10.1115/imece2015-50027 ◽

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.

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

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127413300097 ◽

2013 ◽

Vol 23 (03) ◽

pp. 1330009 ◽

Cited By ~ 5

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 ◽

Mapping Structures ◽

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.

Analytical Bifurcation Trees of a Periodically Excited Pendulum

10.1115/imece2016-65916 ◽

2016 ◽

Author(s):

Yu Guo ◽

Albert C. J. Luo

Keyword(s):

Analytical Prediction ◽

Periodic Motions ◽

Discrete Maps ◽

Implicit Mapping ◽

Mapping Structures ◽

Forced Pendulum ◽

First Time ◽

Analytical Bifurcation Trees ◽

Mapping Systems

In this paper, the bifurcation trees of a periodically excited pendulum are investigated. Implicit discrete maps for such a pendulum are developed to construct discrete mapping structures. Bifurcation trees of the corresponding periodic motions are predicted semi-analytically through the discrete mapping structure. The corresponding stability and bifurcation analysis are carried out through eigenvalue analysis. Finally, numerical illustrations of various periodic motions are given to verify the analytical prediction. The accurate periodic motions in the periodically forced pendulum are predicted for the first time through the implicit mapping systems, and the corresponding bifurcation trees of periodic motion to chaos are obtained.

Periodic Motions in the Periodically Forced Oscillator With Multiple Discontinuities

Design Engineering and Computers and Information in Engineering, Parts A and B ◽

10.1115/imece2006-13827 ◽

2006 ◽

Author(s):

Albert C. J. Luo ◽

Mehul T. Patel

Keyword(s):

Dynamical System ◽

Bifurcation Analysis ◽

Analytical Prediction ◽

Periodic Motions ◽

Mapping Structures ◽

Forced Oscillator ◽

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.

Period-1 Motions to Chaos in a Parametrically Excited Pendulum

10.1115/imece2017-70775 ◽

2017 ◽

Cited By ~ 1

Author(s):

Yu Guo ◽

Albert C. J. Luo

Keyword(s):

Numerical Methods ◽

Numerical Simulations ◽

Bifurcation Analysis ◽

Periodic Motions ◽

Parametrically Excited ◽

Pendulum System ◽

Discrete Maps ◽

Mapping Structures ◽

Simulation Results

In this paper, bifurcation trees of period-1 motions to chaos are investigated in a parametrically excited pendulum. To construct discrete mapping structures of periodic motions, implicit discrete maps are developed for such a pendulum system. The bifurcation trees from period-1 motions to chaos are predicted semi-analytically through period-1 to period-4 motions. The corresponding stability and bifurcation analysis are carried out through eigenvalue analysis. Finally, numerical simulations of periodic motions can be completed through numerical methods. Such simulation results are illustrated for verification of the analytical predictions.

Volume 6: 12th International Conference on Multibody Systems, Nonlinear Dynamics, and Control ◽

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

10.1115/detc2016-59343 ◽

2016 ◽

Author(s):

Albert C. J. Luo ◽

Siyuan Xing

Keyword(s):

Differential Equation ◽

Bifurcation Analysis ◽

Analytical Method ◽

Particle Physics ◽

Duffing Oscillator ◽

Periodic Motions ◽

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.

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

Volume 8: 31st Conference on Mechanical Vibration and Noise ◽

10.1115/detc2019-97260 ◽

2019 ◽

Author(s):

Yu Guo ◽

Albert C. J. Luo

Keyword(s):

Bifurcation Analysis ◽

Mapping Method ◽

Eigenvalue Analysis ◽

Periodic Motions ◽

Parametrically Excited ◽

Stability And Bifurcation Analysis

Period 2 ◽

Mapping Structures ◽

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.

Volume 6: 12th International Conference on Multibody Systems, Nonlinear Dynamics, and Control ◽

Bifurcation Tree of Period-1 Motion to Chaos in a Pendulum With Periodic Excitation

10.1115/detc2016-59331 ◽

2016 ◽

Author(s):

Yu Guo ◽

Albert C. J. Luo

Keyword(s):

Harmonic Amplitude ◽

Analytical Prediction ◽

Algebraic Equations ◽

Periodic Motions ◽

Bifurcation Tree ◽

Discrete Maps ◽

Periodically Driven ◽

Mapping Structures ◽

Nonlinear Algebraic Equations ◽

Analytical Bifurcation Trees

In this paper, the bifurcation trees of period-1 motions to chaos are presented in a periodically driven pendulum. Discrete implicit maps are obtained through a mid-time scheme. Using these discrete maps, mapping structures are developed to describe different types of motions. Analytical bifurcation trees of periodic motions to chaos are obtained through the nonlinear algebraic equations of such implicit maps. Eigenvalue analysis is carried out for stability and bifurcation analysis of the periodic motions. Finally, numerical simulation results of various periodic motions are illustrated in verification to the analytical prediction. Harmonic amplitude characteristics are also be presented.