A Parameter Study on Periodic Motions in a Discontinuous Dynamical System with Two Circular Boundaries

2021 ◽  
Vol 10 (2) ◽  
pp. 289-309
Author(s):  
Siyu Guo ◽  
Albert C.J. Luo
Keyword(s):  
Dynamical System ◽  
Parameter Study ◽  
Periodic Motions ◽  
Discontinuous Dynamical System

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.

scholarly journals Analysis of Discontinuous Dynamical Behaviors of a Friction-Induced Oscillator with an Elliptic Control Law

2018 ◽  
Vol 2018 ◽  
pp. 1-33 ◽  
Author(s):  
Jinjun Fan ◽  
Ping Liu ◽  
Tianyi Liu ◽  
Shan Xue ◽  
Zhaoxia Yang
Keyword(s):  
Dynamical System ◽  
Vector Fields ◽  
Control Law ◽  
Periodic Motions ◽  
Separation Boundary ◽  
Discontinuous System ◽  
Dynamical Behaviors ◽  
Discontinuous Dynamical System ◽  
Discontinuous Boundary ◽  
Mass Spring

This paper develops the passability conditions of flow to the discontinuous boundary and the sticking or sliding and grazing conditions to the separation boundary in the discontinuous dynamical system of a friction-induced oscillator with an elliptic control law and the friction force acting on the mass M through the analysis of the corresponding vector fields and G-functions. The periodic motions of such a discontinuous system are predicted analytically through the mapping structure. Finally, the numerical simulations are given to illustrate the analytical results of motion for a better understanding of physics of motion in the mass-spring-damper oscillator.

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.

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.

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.

Bifurcation Trees of Period-1 Motions to Chaos in a Two-Degree-of-Freedom, Nonlinear Oscillator

2015 ◽  
Vol 25 (13) ◽  
pp. 1550179 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bo Yu
Keyword(s):  
Dynamical System ◽  
Fourier Series ◽  
Analytical Solutions ◽  
Nonlinear Oscillator ◽  
Degree Of Freedom ◽  
Harmonic Amplitude ◽  
Periodic Motions ◽  
Series Transformation ◽  
Two Degree Of Freedom ◽  
Analytical Bifurcation Trees

In this paper, analytical solutions for period-[Formula: see text] motions in a two-degree-of-freedom (2-DOF) nonlinear oscillator are developed through the finite Fourier series. From the finite Fourier series transformation, the dynamical system of coefficients of the finite Fourier series is developed. From such a dynamical system, the solutions of period-[Formula: see text] motions are obtained and the corresponding stability and bifurcation analyses of period-[Formula: see text] motions are carried out. Analytical bifurcation trees of period-1 motions to chaos are presented. Displacements, velocities and trajectories of periodic motions in the 2-DOF nonlinear oscillator are used to illustrate motion complexity, and harmonic amplitude spectrums give harmonic effects on periodic motions of the 2-DOF nonlinear oscillator.

Sign in / Sign up

Export Citation Format

Share Document