Period-1 Motions to Chaos in a Parametrically Excited Pendulum

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

Bifurcation Trees of Periodic Motions in a Parametrically Excited Pendulum

10.1115/detc2017-67161 ◽

2017 ◽

Author(s):

Yu Guo ◽

Albert C. J. Luo

Keyword(s):

Algebraic Equations ◽

Periodic Motions ◽

Parametrically Excited ◽

Discrete Maps ◽

Parametric Pendulum ◽

Mapping Structures ◽

Nonlinear Algebraic Equations ◽

Specific Mapping ◽

Analytical Bifurcation Trees

In this paper, the bifurcation trees of periodic motions in a parametrically excited pendulum are studied using discrete implicit maps. From the discrete maps, mapping structures are developed for periodic motions in such a parametric pendulum. Analytical bifurcation trees of periodic motions to chaos are developed through the nonlinear algebraic equations of such implicit maps in the specific mapping structures. The corresponding stability and bifurcation analysis of periodic motions is carried out. Finally, numerical results of periodic motions are presented. Many new periodic motions in the parametrically excited pendulum are discovered.

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 ◽

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: 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 ◽

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 Motions to Chaos in a Quadratic Nonlinear Oscillator With Time-Delayed Displacement

Volume 4B: Dynamics, Vibration, and Control ◽

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: 11th International Conference on Multibody Systems, Nonlinear Dynamics, and Control ◽

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

10.1115/detc2015-48104 ◽

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 ◽

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.

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 ◽

Stability And Bifurcation ◽

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.

Dynamics of a Simplified van der Pol Oscillator Revisited

Volume 9: Mechanical Systems and Control, Parts A, B, and C ◽

10.1115/imece2007-42687 ◽

2007 ◽

Author(s):

Albert C. J. Luo ◽

Arun Rajendran

Keyword(s):

Bifurcation Analysis ◽

Local Stability ◽

Van Der Pol Oscillator ◽

Nonsmooth Dynamics ◽

Mapping Technique ◽

Periodic Motions ◽

Van Der Pol ◽

Chaotic Motions ◽

Mapping Structures ◽

In this paper, the dynamic characteristics of a simplified van der Pol oscillator are investigated. From the theory of nonsmooth dynamics, the structures of periodic and chaotic motions for such an oscillator are developed via the mapping technique. The periodic motions with a certain mapping structures are predicted analytically for m-cycles with n-periods. Local stability and bifurcation analysis for such motions are carried out. The (m:n)-periodic motions are illustrated. The further investigation of the stable and unstable periodic motions in such a system should be completed. The chaotic motion based on the Levinson donuts should be further discussed.

Analytical Bifurcation Trees of a Periodically Excited Pendulum

Volume 4B: Dynamics, Vibration, and Control ◽

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 ◽

Stability And Bifurcation ◽

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.

PERIODIC MOTIONS AND CHAOS WITH IMPACTING CHATTER AND STICK IN A GEAR TRANSMISSION SYSTEM

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127409023858 ◽

2009 ◽

Vol 19 (06) ◽

pp. 1975-1994 ◽

Cited By ~ 22

Author(s):

ALBERT C. J. LUO ◽

DENNIS O'CONNOR

Keyword(s):

Stability Analysis ◽

Bifurcation Analysis ◽

Local Stability ◽

Transmission System ◽

Gear Transmission ◽

Periodic Motions ◽

Local Stability Analysis ◽

Gear Transmission System ◽

Mapping Structures ◽

This paper focuses on periodic motions and chaos relative to the impacting chatter and stick in order to find the origin of noise and vibration in such a gear transmission system. Such periodic motions are predicted analytically through mapping structures, and the corresponding local stability and bifurcation analysis are carried out. The grazing and stick conditions presented in [Luo & O'Connor, 2007] are adopted to determine the existence of periodic motions, which cannot be achieved from the local stability analysis. Numerical simulations are performed to illustrate periodic motions and stick motion criteria. Such an investigation may provide some clues to reduce the noise in gear transmission systems.