Period-1 and Period-2 Motions in a Brusselator With a Harmonic Diffusion

2017 ◽  
Author(s):  
Albert C. J. Luo ◽  
Siyu Guo
Keyword(s):  
Numerical Simulations ◽  
Analytical Solutions ◽  
Eigenvalue Analysis ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Stable Period ◽  
Period 2 ◽  
Approximate Analytical Solutions ◽  
Stability And Bifurcations

In this paper, the analytical solutions of periodic evolution of Brusselator are investigated through the general harmonic balanced method. Both stable and unstable, period-1 and period-2 solutions of the Brussellator are presented. Stability and bifurcations of the periodic evolution are determined by the eigenvalue analysis. Numerical simulations of stable period-1 and period-2 motions of Brusselator are completed. The harmonic amplitude spectrums show harmonic effects on periodic motions, and the corresponding accuracy of approximate analytical solutions can be prescribed specifically.

Analytical Solutions of Period-1 to Period-2 Motions in a Periodically Diffused Brusselator

2018 ◽  
Vol 13 (9) ◽  
Author(s):  
Albert C. J. Luo ◽  
Siyu Guo
Keyword(s):  
Analytical Solutions ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Stable Period ◽  
Period 2 ◽  
Approximate Analytical Solutions ◽  
Amplitude Spectra ◽  
Unstable Solutions ◽  
Stability And Bifurcations

In this paper, the analytical solutions of periodic evolutions of the periodically diffused Brusselator are obtained through the generalized harmonic balanced method. Stable and unstable solutions of period-1 and period-2 evolutions in the Brusselator are presented. Stability and bifurcations of the periodic evolution are determined by the eigenvalue analysis, and the corresponding Hopf bifurcations are presented on the analytical bifurcation tree of the periodic motions. Numerical simulations of stable period-1 and period-2 motions of Brusselator are completed. The harmonic amplitude spectra show harmonic effects on periodic motions, and the corresponding accuracy of approximate analytical solutions can be prescribed specifically.

Bifurcation Trees of a Periodically Forced, Two-Degree-of-Freedom Oscillator With a Nonlinear Hardening Spring

2015 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Analytical Solutions ◽  
Degree Of Freedom ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Stable Period ◽  
Period 2 ◽  
Approximate Analytical Solutions ◽  
The Stability ◽  
Nonlinear Hardening ◽  
Two Degree Of Freedom

In this paper, analytical solutions of periodic motions in a periodically forced, damped, two-degree-of-freedom oscillator with a nonlinear hardening spring are obtained. The bifurcation trees of periodic motions are presented, and the stability and bifurcation of the periodic motion are determined through the eigenvalue analysis. Numerical simulations of stable period-1 and period-2 motions in the two-degree-of-freedom systems are presented, and the harmonic amplitude spectrums are presented to show the harmonic effects on periodic motions, and the accuracy of approximate analytical solutions can be estimated through the harmonic amplitudes.

Period-3 Motions in a Two Degree-of-Freedom Nonlinear Oscillator

2016 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Analytical Solutions ◽  
Nonlinear Oscillator ◽  
Degrees Of Freedom ◽  
Degree Of Freedom ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Approximate Analytical Solutions ◽  
Amplitude Spectra ◽  
Stability And Bifurcations ◽  
Two Degree Of Freedom

In this paper, periodic motions of a two-degree-of-freedom nonlinear oscillator are studied by using general harmonic balanced method. Stable and unstable period-3 motions are obtained. The corresponding stability and bifurcations of the period-3 motions are determined through the eigenvalue analysis. Both symmetric and asymmetric period-3 motions are found in the system with a certain set of parameter. Numerical simulations of both stable and unstable period-3 motions in the two degrees of freedom systems are illustrated. The harmonic amplitude spectra show the harmonic effects on periodic motions, and the corresponding accuracy of approximate analytical solutions can be observed.

Periodic Motion in a Nonlinear Vibration Isolator Under Harmonic Excitation

2017 ◽  
Author(s):  
Bo Yu ◽  
Albert C. J. Luo
Keyword(s):  
Nonlinear Vibration ◽  
Analytical Solutions ◽  
Periodic Motion ◽  
Harmonic Excitation ◽  
Eigenvalue Analysis ◽  
Harmonic Amplitude ◽  
Vibration Isolator ◽  
Periodic Motions ◽  
Approximate Analytical Solutions ◽  
Forced Oscillator

In this paper, periodic motions of a periodically forced oscillator with a nonlinear isolator are studied through generalized harmonic balanced method. Both symmetric and asymmetric period-1 motions are obtained. Stability and bifurcation of the periodic motions are determined through eigenvalue analysis. Numerical illustrations of both symmetric and asymmetric are given. From the harmonic amplitude spectrums, the harmonic effects on periodic motions are determined, and the corresponding accuracy of approximate analytical solutions can be observed.

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.

An Approximate Solution for Period-1 Motions in a Periodically Forced Van Der Pol Oscillator

2014 ◽  
Vol 9 (3) ◽  
Author(s):  
Albert C. J. Luo ◽  
Arash Baghaei Lakeh
Keyword(s):  
Approximate Solution ◽  
Fourier Series ◽  
Analytical Solutions ◽  
Amplitude Spectrum ◽  
Van Der Pol Oscillator ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Van Der Pol ◽  
Approximate Analytical Solutions ◽  
Series Expression

In this paper the approximate analytical solutions of period-1 motion in the periodically forced van der Pol oscillator are obtained by the generalized harmonic balance (HB) method. Such an approximate solution of periodic motion is given by the Fourier series expression, and the convergence of such an expression is guaranteed by the Fourier series theory of periodic functions. The approximate solution is different from traditional, approximate solution because the number of total harmonic terms (N) is determined by the precision of harmonic amplitude quantity level, set by the investigator (e.g., AN≤ɛ and ɛ=10-8). The stability and bifurcation analysis of the period-1 solutions is completed through the eigenvalue analysis of the coefficient dynamical systems of the Fourier series expressions of periodic solutions, and numerical illustrations of period-1 motions are compared to verify the analytical solutions of periodic motions. The trajectories and analytical harmonic amplitude spectrum for stable and unstable periodic motions are presented. The harmonic amplitude spectrum shows the harmonic term effects on periodic motions, and one can directly know which harmonic terms contribute on periodic motions and the convergence of the Fourier series expression is clearly illustrated.

Regenerative Cutting Dynamics for a Periodically Forced Machine-Tool System

2019 ◽  
Author(s):  
Siyuan Xing ◽  
Albert C. J. Luo
Keyword(s):  
Machine Tool ◽  
Numerical Simulations ◽  
Analytical Method ◽  
Orthogonal Cutting ◽  
Periodic Oscillation ◽  
Eigenvalue Analysis ◽  
Stable Period ◽  
Machine Tool Chatter ◽  
Stability And Bifurcations ◽  
Cutting System

Abstract In this paper, a nonlinear, regenerative, orthogonal cutting system with a weak periodic oscillation of workpiece is considered. Period-1 motions in such a system are studied through a semi-analytical method, and the corresponding stability and bifurcations of the period-1 motions are analyzed via the eigenvalue analysis. The vibration of machine-tool varying with excitation is studied, and excitation effects on machine-tool chatters are discussed. Numerical simulations of unstable and stable period-1 motions are completed from analytical predictions. The machine-tool chatter can emerge from the saddle-node or Neimark bifurcation of period-1 motions.

Period Motions and Limit Cycle in a Periodically Forced, Plunged Galloping Oscillator

2017 ◽  
Author(s):  
Bo Yu ◽  
Albert C. J. Luo
Keyword(s):  
Hopf Bifurcation ◽  
Numerical Simulations ◽  
Limit Cycle ◽  
Analytical Solutions ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Hopf Bifurcations ◽  
Periodic Motions ◽  
Stability And Bifurcations ◽  
Generalized Harmonic Balance Method

In this paper, periodic motions of a periodically forced, plunged galloping oscillator are investigated. The analytical solutions of stable and unstable periodic motions are obtained by the generalized harmonic balance method. Stability and bifurcations of the periodic motions are discussed through the eigenvalue analysis. The saddle-node and Hopf bifurcations of periodic motions are presented through frequency-amplitude curves. The Hopf bifurcation generates the quasiperiodic motions. Numerical simulations of stable and unstable periodic motions are illustrated.

Analytical Solutions for Period-1 Motions in a Nonlinear Jeffcott Rotor System

2014 ◽  
Author(s):  
Jianzhe Huang ◽  
Albert C. J. Luo
Keyword(s):  
Analytical Solutions ◽  
Periodic Motion ◽  
Rotor System ◽  
Eigenvalue Analysis ◽  
Hopf Bifurcations ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Rotor Systems ◽  
Jeffcott Rotor

In this paper, the analytical solutions of period-1 solutions are developed, and the corresponding stability and bifurcation are also analyzed by eigenvalue analysis. 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 nonlinear Jeffcott rotor systems are illustrated, and harmonic amplitude spectrums are presented for harmonic effects on periodic motions of the nonlinear rotor.

On Periodic Motions in a Parametric Hardening Duffing Oscillator

2014 ◽  
Vol 24 (01) ◽  
pp. 1430004 ◽  
Author(s):  
Albert C. J. Luo ◽  
Dennis M. O'Connor
Keyword(s):  
Fourier Series ◽  
Bifurcation Analysis ◽  
Analytical Solutions ◽  
Duffing Oscillator ◽  
Periodic Motions ◽  
Period 2 ◽  
Approximate Analytical Solutions ◽  
Stability And Bifurcation Analysis ◽  
Series Expression ◽  
Amplitude Characteristics

In this paper, analytical solutions for periodic motions in a parametric hardening Duffing oscillator are presented using the finite Fourier series expression, and the corresponding stability and bifurcation analysis for such periodic motions are carried out. The frequency-amplitude characteristics of asymmetric period-1 and symmetric period-2 motions are discussed. The hardening Mathieu–Duffing oscillator is also numerically simulated to verify the approximate analytical solutions of periodic motions. Period-1 asymmetric and period-2 symmetric motions are illustrated for a better understanding of periodic motions in the hardening Mathieu–Duffing oscillator.

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