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.

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.

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.

Period-1 Motions to Chaos in a Parametrically Excited Pendulum

2017 ◽  
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 ◽  
Stability And Bifurcation Analysis ◽  
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.

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.

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.

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.

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.

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 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.

On Period-1 and Period-2 Motions of a Jeffcott Rotor System

2019 ◽  
Author(s):  
Yeyin Xu ◽  
Albert C. J. Luo
Keyword(s):  
Analytical Solutions ◽  
Mapping Method ◽  
Rotor System ◽  
Periodic Motions ◽  
Period 2 ◽  
Jeffcott Rotor ◽  
Mapping Structures ◽  
Point Scheme ◽  
Stability And Bifurcations ◽  
Specific Mapping

Abstract In this paper, the semi-analytical solutions of period-1 and period-2 motions in a nonlinear Jeffcott rotor system are presented through the discrete mapping method. The periodic motions in the nonlinear Jeffcott rotor system are obtained through specific mapping structures with a certain accuracy. A bifurcation tree of period-1 to period-2 motion is achieved, and the corresponding stability and bifurcations of periodic motions are analyzed. For verification of semi-analytical solutions, numerical simulations are carried out by the mid-point scheme.

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