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

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.

Bifurcation Trees of Periodic Motions in a Parametrically Excited Pendulum

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 ◽  
Stability And Bifurcation Analysis ◽  
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.

Period-1 Motions to Chaos With Varying Excitation Frequency in a Parametrically Driven Pendulum

2019 ◽  
Author(s):  
Yu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Differential Equations ◽  
Excitation Frequency ◽  
Analytical Prediction ◽  
Algebraic Equations ◽  
Periodic Motions ◽  
Discrete Maps ◽  
Period 2 ◽  
Mapping Structures ◽  
Nonlinear Algebraic Equations ◽  
Stability And Bifurcations

Abstract In this paper, with varying excitation frequency, period-1 motions to chaos in a parametrically driven pendulum are presented through period-1 to period-4 motions. Using the implicit discrete maps of the corresponding differential equations, discrete mapping structures are developed for different periodic motions, and the corresponding nonlinear algebraic equations of such mapping structures are solved for analytical predictions of bifurcation trees of periodic motions. Both period-1 static points to period-2 motions and period-1 motions to period-4 motions are illustrated. The corresponding stability and bifurcations are studied. Finally, numerical illustrations of various periodic motions on the bifurcation trees are presented in verification of the analytical prediction.

Analytical Bifurcation Trees of a Periodically Excited Pendulum

2016 ◽  
Author(s):  
Yu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Analytical Prediction ◽  
Periodic Motions ◽  
Discrete Maps ◽  
Implicit Mapping ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis ◽  
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.

On Periodic Motions in a Periodically Driven van der Pol-Duffing Oscillator

2020 ◽  
Author(s):  
Yeyin Xu ◽  
Albert C. J. Luo
Keyword(s):  
Analytical Solutions ◽  
Analytical Method ◽  
Duffing Oscillator ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Bifurcation Tree ◽  
Periodically Driven ◽  
Implicit Mapping ◽  
Mapping Structures ◽  
Bifurcation And Stability

Abstract In this paper, the symmetric and asymmetric period-1 motions on the bifurcation tree are obtained for a periodically driven van der Pol-Duffing hardening oscillator through a semi-analytical method. Such a semi-analytical method develops an implicit mapping with prescribed accuracy. Based on the implicit mapping, the mapping structures are used to determine periodic motions in the van der Pol-Duffing oscillator. The symmetry breaks of period-1 motion are determined through saddle-node bifurcations, and the corresponding asymmetric period-1 motions are generated. The bifurcation and stability of period-1 motions are determined through eigenvalue analysis. To verify the semi-analytical solutions, numerical simulations are also carried out.

Periodic Motions to Chaos in Pendulum

2016 ◽  
Vol 26 (09) ◽  
pp. 1650159 ◽  
Author(s):  
Albert C. J. Luo ◽  
Yu Guo
Keyword(s):  
Complex Dynamics ◽  
Traditional Approach ◽  
Analytical Prediction ◽  
Periodic Motions ◽  
Pendulum System ◽  
Linear Dynamical Systems ◽  
Mapping Structures ◽  
Forced Pendulum ◽  
And Mathematics ◽  
Analytical Bifurcation Trees

It is not easy to find periodic motions to chaos in a pendulum system even though the periodically forced pendulum is one of the simplest nonlinear systems. However, the inherent complex dynamics of the periodically forced pendulum is much beyond our imaginations through the traditional approach of linear dynamical systems. Until now, we did not know complex motions of pendulum yet. What are the mechanism and mathematics of such complex motions in the pendulum? The results presented herein give a new view of complex motions in the periodically forced pendulum. Thus, in this paper, periodic motions to chaos in a periodically forced pendulum are predicted analytically by a semi-analytical method. The method is based on discretization of differential equations of the dynamical system to obtain implicit maps. Using the implicit maps, mapping structures for specific periodic motions are developed, and the corresponding periodic motions can be predicted analytically through such mapping structures. Analytical bifurcation trees of periodic motions to chaos are obtained, and the corresponding stability and bifurcation analysis of periodic motions to chaos are carried out by eigenvalue analysis. From the analytical prediction of periodic motions to chaos, the corresponding frequency-amplitude characteristics are obtained for a better understanding of motions’ complexity in the periodically forced pendulum. Finally, numerical simulations of selected periodic motions are illustrated. The nontravelable and travelable periodic motions on the bifurcation trees are discovered. Through this investigation, the periodic motions to chaos in the periodically forced pendulums can be understood further. Based on the perturbation method, one cannot achieve the adequate solutions presented herein for periodic motions to chaos in the periodically forced pendulum.

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

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 ◽  
Stability And Bifurcation Analysis

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.

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.

Periodic Motions and Bifurcation Trees in a Buckled, Nonlinear Jeffcott Rotor System

2015 ◽  
Vol 25 (01) ◽  
pp. 1550002 ◽  
Author(s):  
Jianzhe Huang ◽  
Albert C. J. Luo
Keyword(s):  
Analytical Solutions ◽  
Degrees Of Freedom ◽  
Rotor System ◽  
Hopf Bifurcations ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Rotor Systems ◽  
Jeffcott Rotor ◽  
Analytical Bifurcation Trees

In this paper, analytical solutions for period-m motions in a buckled, nonlinear Jeffcott rotor system are obtained. This nonlinear Jeffcott rotor system with two-degrees of freedom is excited periodically from the rotor eccentricity. The analytical solutions of period-m solutions are developed, and the corresponding stability and bifurcation are also analyzed by eigenvalue analysis. Analytical bifurcation trees of period-1 motions to chaos are presented. The Hopf bifurcations of periodic motions cause not only the bifurcation tree but quasi-periodic motions. The quasi-periodic motion can be stable or unstable. Displacement orbits of periodic motions in the buckled, nonlinear Jeffcott rotor systems are illustrated, and harmonic amplitude spectrums are presented for harmonic effects on periodic motions of the nonlinear rotor. Coexisting periodic motions exist in such a buckled nonlinear Jeffcott rotor.

Periodic Motions in an Autonomous System With a Discontinuous Vector Field

2020 ◽  
Author(s):  
Siyu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Vector Field ◽  
Dynamical Systems ◽  
Numerical Simulations ◽  
Autonomous System ◽  
Parameter Study ◽  
Spring Stiffness ◽  
Algebraic Equations ◽  
Periodic Motions ◽  
Mapping Structures ◽  
The Stability

Abstract In this paper, periodic motions in an autonomous system with a discontinuous vector field are discussed. The periodic motions are obtained by constructing a set of algebraic equations based on motion mapping structures. The stability of periodic motions is investigated through eigenvalue analysis. The grazing bifurcations are presented by varying the spring stiffness. Once the grazing bifurcation occurs, periodic motions switches from the old motion to a new one. Numerical simulations are conducted for motion illustrations. The parameter study helps one understand autonomous discontinuous dynamical systems.

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