Periodic Motions in a Discontinuous Dynamical System With Two Circular Boundaries

2019 ◽  
Author(s):  
Siyu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Numerical Simulations ◽  
Periodic Motions ◽  
Discontinuous Dynamical System ◽  
Mapping Structures

Abstract In this paper, periodic motions in a discontinuous dynamical systems are studied. The discontinuous dynamical system has three domains partitioned through two circular boundaries. On the three domains, there are three distinct dynamical systems. From the G-functions, the switchability conditions of a flow from one domain to anther domain at the boundary are developed. The flow mappings from a boundary to a bounbary are developed for each domain and boundary. From the mapping structures, periodic motions in the discontinuous dynamical system are predicted. Numerical simulations of periodic motions and motion switchability at boundaries are presented in the discontinuous dynamical system.

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.

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.

Complex Dynamics of Bouncing Motions at Boundaries and Corners in a Discontinuous Dynamical System

2017 ◽  
Vol 12 (6) ◽  
Author(s):  
Jianzhe Huang ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Vector Fields ◽  
Complex Dynamics ◽  
Singularity Theory ◽  
Local Theory ◽  
Periodic Motions ◽  
Border Collision Bifurcations ◽  
Discontinuous Dynamical System ◽  
Stability And Bifurcations

In this paper, from the local theory of flow at the corner in discontinuous dynamical systems, obtained are analytical conditions for switching impact-alike chatter at corners. The objective of this investigation is to find the dynamics mechanism of border-collision bifurcations in discontinuous dynamical systems. Multivalued linear vector fields are employed, and generic mappings are defined among boundaries and corners. From mapping structures, periodic motions switching at the boundaries and corners are determined, and the corresponding stability and bifurcations of periodic motions are investigated by eigenvalue analysis. However, the grazing and sliding bifurcations are determined by the local singularity theory of discontinuous dynamical systems. From such analytical conditions, the corresponding parameter map is developed for periodic motions in such a multivalued dynamical system in the single domain with corners. Numerical simulations of periodic motions are presented for illustrations of motions complexity and catastrophe in such a discontinuous dynamical system.

On Existence and Bifurcations of Periodic Motions in Discontinuous Dynamical Systems

2021 ◽  
Vol 31 (04) ◽  
pp. 2150063
Author(s):  
Siyu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical Systems ◽  
Mechanical Model ◽  
System Parameter ◽  
Period Doubling ◽  
Periodic Motions ◽  
Discontinuous System ◽  
Saddle Node Bifurcation ◽  
Discontinuous Dynamical System ◽  
Mapping Structures ◽  
Parametric Maps

In this paper, the existence and bifurcations of periodic motions in a discontinuous dynamical system is studied through a discontinuous mechanical model. One can follow the study presented herein to investigate other discontinuous dynamical systems. Such a sampled discontinuous system consists of two subsystems on boundaries and three subsystems in subdomains. From the theory of discontinuous dynamical systems, switchability conditions of a flow at and on the boundaries are developed. From such switchability conditions, grazing motions of a flow at boundaries are discussed, and sliding motions of a flow on boundaries are presented. Based on the motions in each domain and on each boundary, generic mappings are introduced. Using the generic mappings, mapping structures for specific periodic motions are developed. Based on the grazing conditions and appearance and vanishing conditions of sliding motions, parametric dynamics of the existences of the specific periodic motions are presented. In addition, the traditional saddle-node bifurcation, Neimark bifurcations and period-doubling bifurcation are used for parametric dynamics of periodic motions. Bifurcation trees of periodic motions varying with a system parameter are presented first, and phase trajectories of periodic motions are illustrated. The [Formula: see text]-functions are presented for the illustration of the motion switchability at the boundaries and sliding motions on the boundaries. Codimension-2 parametric dynamics of periodic motions are studied and how to develop the 2D parametric maps for specific periodic motions are presented. In the end, periodic motions with grazing are illustrated.

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.

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.

Periodic Motions With Grazing in a Discontinuous Dynamical System With Two Circular Boundaries

2019 ◽  
Author(s):  
Siyu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical System ◽  
Periodic Motion ◽  
Necessary Conditions ◽  
Periodic Motions ◽  
Discontinuous System ◽  
Discontinuous Dynamical System ◽  
Sufficient And Necessary Conditions

Abstract In this paper, studied are periodic motions with grazing in a discontinuous dynamical system with two circular boundaries. The grazing motion is for a periodic motion switching to another periodic motions. Thus, the sufficient and necessary conditions of motion switching, grazing and sliding on the boundaries are discussed first. Periodic motions with grazing in the discontinuous system are presented for illustration of motions switching.

Analytical Predication of Complex Motion of a Ball in a Periodically Shaken Horizontal Impact Pair

2011 ◽  
Vol 7 (2) ◽  
Author(s):  
Yu Guo ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical System ◽  
Bifurcation Analysis ◽  
Periodic Motion ◽  
Analytical Prediction ◽  
Periodic Excitation ◽  
Complex Motion ◽  
Chaotic Motions ◽  
Discontinuous Dynamical System ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis

In this paper, complex motions of a ball in the horizontal impact pair with a periodic excitation are studied analytically using the theory of discontinuous dynamical system. Analytical conditions for motion switching caused by impacts are developed, and generic mapping structures are introduced to describe different periodic and chaotic motions. Analytical prediction of complex periodic motion of the ball in the periodically shaken impact pair is completed, and the corresponding stability and bifurcation analysis are also carried out. Numerical illustrations of periodic and chaotic motions are given.

On Symmetric Periodic Motions With Different Excitation Periods in a Discontinuous System With a Hyperbolic Boundary

2020 ◽  
Author(s):  
Albert C. J. Luo ◽  
Chuanping Liu
Keyword(s):  
Dynamic System ◽  
Numerical Simulations ◽  
Periodic Motions ◽  
Discontinuous System ◽  
Mapping Structures ◽  
Specific Mapping ◽  
Hyperbolic Boundary

Abstract In this paper, symmetric periodic motions with different excitation periods in a discontinuous dynamic system with a hyperbolic boundary are presented analytically. The switchability conditions of flows at the hyperbolic boundaries are developed. Periodic motions with specific mapping structures are predicted analytically, and numerical simulations of periodic motions are carried out. The corresponding G-functions are presented for illustration of motion switchability at the hyperbolic boundaries.

Periodic Motions in the Periodically Forced Oscillator With Multiple Discontinuities

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 ◽  
Stability And Bifurcation Analysis ◽  
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.

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