PARAMETRIC ANALYSIS OF BIFURCATION AND CHAOS IN A PERIODICALLY DRIVEN HORIZONTAL IMPACT PAIR

2012 ◽  
Vol 22 (11) ◽  
pp. 1250268 ◽  
Author(s):  
YU GUO ◽  
ALBERT C. J. LUO
Keyword(s):  
Dynamical Systems ◽  
Parametric Analysis ◽  
Periodic Motions ◽  
Bifurcation And Chaos ◽  
Chaotic Motions ◽  
Periodically Driven ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis ◽  
Different Types ◽  
The Impact

In this paper, complex motions and chaos in the periodically driven horizontal impact pair are investigated using the theory of switchability for discontinuous dynamical systems. Domains and boundaries are defined due to the discontinuity caused by impacts. Analytical conditions for switching of stick and grazing motions are derived in detail. Generic mappings are introduced to describe different periodic motions and to identify the mapping structures of chaos. The periodic motions in such impact pair are analytically predicted, and the corresponding stability and bifurcation analysis of periodic motions are carried out. Parameter maps with different types of motions are developed. Periodic and chaotic motions with different mapping structures are illustrated numerically for a better understanding of physics of ball motions in the impact pair.

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.

Dynamics of a Simplified van der Pol Oscillator Revisited

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 ◽  
Stability And Bifurcation Analysis

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 Dynamics of a Ball Bouncing on a Vibrating Table

2012 ◽  
Author(s):  
Yu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical Systems ◽  
Numerical Simulations ◽  
Analytical Dynamics ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Theory Of Flow ◽  
Mapping Structures ◽  
Ball Bouncing ◽  
Vibrating Table

In this paper, the theory of flow switchability for discontinuous dynamical systems is applied. Domains and boundaries for such a discontinuous problem are defined and analytical conditions for motion switching are developed. The conditions explain the important role of switching phase on the motion switchability in such a system. To describe different motions, the generic mappings and mapping structures are introduced. Bifurcation scenarios for periodic and chaotic motions are presented for different motions and switchability. Numerical simulations are provided for periodic motions with impacts only and with impact chatter to stick in the system.

Switching Mechanism and Complex Motions in an Extended Fermi-Acceleration Oscillator

2010 ◽  
Vol 5 (4) ◽  
Author(s):  
Albert C. J. Luo ◽  
Yu Guo
Keyword(s):  
Dynamical Systems ◽  
Extended Model ◽  
Time Varying ◽  
Fermi Acceleration ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Bouncing Ball ◽  
Mapping Structures ◽  
The Masses ◽  
First Time

In this paper, an extended model of the Fermi-acceleration oscillator is presented to describe impacting chatters, grazing, and sticking between the particle (or bouncing ball) and piston. The sticking phenomenon in such a system is investigated for the first time. Even in the traditional Fermi-oscillator, such a sticking phenomenon still exists but one often ignored it. In this paper, the analytical conditions for the grazing and sticking phenomena between the particle and piston in the Fermi-acceleration oscillator are developed from the theory of discontinuous dynamical systems. Compared with existing studies, the four exact mappings are used to analyze the motion behaviors of the Fermi-oscillator instead of one or two mappings. Mapping structures formed by generic mappings are adopted for the analytical predictions of periodic motions in the Fermi-acceleration oscillator. Periodic and chaotic motions in such an oscillator are illustrated to show motion complexity and grazing and sticking mechanism. Once the masses of the ball and primary mass are in the same quantity level, the model presented in this paper will be very useful and significant. This idea can apply to a system possessing two independent oscillators with impact, such as gear transmission systems, bearing systems, and time-varying billiard systems.

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.

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 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.

Bifurcation Analysis of a Fermi-Acceleration Oscillator Under Different Excitations

2011 ◽  
Author(s):  
Albert C. J. Luo ◽  
Yu Guo
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Analysis ◽  
Fermi Acceleration ◽  
Periodic Motions ◽  
Chaotic Motions

In this paper, switchability and bifurcation of motions in a double excited Fermi acceleration oscillator is discussed using the theory of discontinuous dynamical systems. The two oscillators are chosen to have different excitation and parameters. The analytical conditions for motion switching in such a Fermi-oscillator are presented. Bifurcation scenario for periodic and chaotic motions is presented, and the analytical predictions of periodic motions are presented. Finally, different motions in such an oscillator are illustrated.

Dynamic Mechanism of a Train Suspension System

2010 ◽  
Author(s):  
Albert C. J. Luo ◽  
Dennis O’Connor
Keyword(s):  
Numerical Simulations ◽  
System Parameter ◽  
Suspension System ◽  
Dynamic Mechanism ◽  
Impact Model ◽  
Periodic Motions ◽  
Chaotic Motions ◽  
Nonlinear Dynamical ◽  
Dynamical Behaviors ◽  
Mapping Structures

Nonlinear dynamical behaviors of a train suspension system with impacts are investigated. The suspension system is modelled through an impact model with possible stick between a bolster and two wedges. Based on the mapping structures, periodic motions of such a system are described. To understand the global dynamical behaviors of the train suspension system, system parameter maps are developed. Numerical simulations for periodic and chaotic motions are performed from the parameter maps.

The Synchronization of a Periodically Driven Pendulum With Periodic Motions in a Periodically Forced, Damped Duffing Oscillator

2010 ◽  
Author(s):  
Albert C. J. Luo ◽  
Fuhong Min
Keyword(s):  
Dynamical Systems ◽  
Duffing Oscillator ◽  
Periodic Motions ◽  
Invariant Domain ◽  
Periodically Driven

In this paper, the analytical conditions for the controlled pendulum synchronizing with periodic motions in Duffing oscillator is developed through the theory of discontinuous dynamical systems. The conditions for the synchronization invariant domain are obtained, and the partial and full synchronizations are illustrated for the analytical conditions.

Sign in / Sign up

Export Citation Format

Share Document