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.

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.

Periodic Orbits in a Second-Order Discontinuous System with an Elliptic Boundary

2016 ◽  
Vol 26 (13) ◽  
pp. 1650224 ◽  
Author(s):  
Liping Li ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Periodic Orbits ◽  
Vector Fields ◽  
Sliding Mode ◽  
Elliptic Boundary ◽  
Second Order ◽  
Discontinuous System ◽  
System A ◽  
Discontinuous Dynamical System

This paper develops the analytical conditions for the onset and disappearance of motion passability and sliding along an elliptic boundary in a second-order discontinuous system. A periodically forced system, described by two different linear subsystems, is considered mainly to demonstrate the methodology. The passable, sliding and grazing conditions of a flow to the elliptic boundary in the discontinuous dynamical system are provided through the analysis of the corresponding vector fields and [Formula: see text]-functions. Moreover, by constructing appropriate generic mappings, periodic orbits in such a discontinuous system are predicted analytically. Finally, three different cases are discussed to illustrate the existence of periodic orbits with passable and/or sliding flows. The results obtained in this paper can be applied to the sliding mode control in discontinuous dynamical systems.

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.

Switching Dynamics of a Periodically Forced Discontinuous System With an Inclined Boundary

2007 ◽  
Author(s):  
Albert C. J. Luo ◽  
Brandon M. Rapp
Keyword(s):  
Dynamical System ◽  
Vector Fields ◽  
Normal Vector ◽  
Periodic Motions ◽  
Separation Boundary ◽  
Discontinuous System ◽  
Normal Vector Field ◽  
Periodically Driven ◽  
Switching Dynamics ◽  
Straight Line

This paper presents the switching dynamics of flow from one domain into another one in the periodically driven, discontinuous dynamical system. The simple inclined straight line boundary in phase space is considered as a control law for the dynamical system to switch. The normal vector-field product for flow switching on the separation boundary is introduced, and the passability condition of flow to the discontinuous boundary is presented. The sliding and grazing conditions to the separation boundary are presented as well. Using mapping structures, periodic motions of such a discontinuous system are predicted, and the local stability and bifurcation analysis are carried out. Numerical illustrations of periodic motions with grazing to the boundary and/or sliding on the boundary are given, and the normal vector fields are illustrated to show the analytical criteria.

Sequential Periodic Motions in a Vibration-Assisted, Regenerative, Nonlinear Turning-Tool System

2021 ◽  
Vol 31 (12) ◽  
pp. 2150186
Author(s):  
Siyuan Xing ◽  
Albert C. J. Luo
Keyword(s):  
Time Delay ◽  
Machine Tool ◽  
Complex Dynamics ◽  
Metal Removal ◽  
Removal Rate ◽  
Mapping Method ◽  
Strong Nonlinearity ◽  
Periodic Motions ◽  
Machine Vibration ◽  
Stability And Bifurcations

This paper studies the dynamics and bifurcations of a vibration-assisted, regenerative, nonlinear turning-tool system using an implicit mapping method. Machine vibration has been studied for a century for the improvement of machine accuracy and metal removal rate. In fact, this problem is unsolved yet. This is because such dynamical systems are involved in nonlinearity, discontinuity and time-delay. Thus, a comprehensive understanding of nonlinear machining dynamics with time-delay is indispensable. In this paper, period-[Formula: see text] motions in the turning machine-tool system are studied through specific mapping structures, and the corresponding stability and bifurcations of the period-[Formula: see text] motion are determined through the eigenvalue analysis. The analytical bifurcation scenarios for two sets of sequential period-[Formula: see text] motions in a turning-tool system are presented. Numerical simulations of period-[Formula: see text] motions are carried out to verify the prediction of periodic motions. The complex dynamics of vibration-assisted machining with strong nonlinearity are presented, which can provide a good overview for nonlinear dynamics of machine-tool systems.

On a Global Sequential Scenario of Bifurcation Trees to Chaos in a First-Order, Periodically Excited, Time-Delayed System

2019 ◽  
Vol 29 (10) ◽  
pp. 1950141
Author(s):  
Siyuan Xing ◽  
Albert C. J. Luo
Keyword(s):  
Dynamical System ◽  
Nonlinear Dynamical System ◽  
Global Bifurcation ◽  
Periodic Motions ◽  
Bifurcation Tree ◽  
Nonlinear Dynamical ◽  
First Order ◽  
Delayed System ◽  
The Rich ◽  
Stability And Bifurcations

In this paper, the global sequential scenario of bifurcation trees of periodic motions to chaos is studied for a first-order, time-delayed, nonlinear dynamical system with periodic excitation. The periodic motions of such a first-order time-delayed system is obtained semi-analytically, and the corresponding stability and bifurcations are determined by eigenvalue analysis. A global sequential scenario of bifurcation trees is given by [Formula: see text] where [Formula: see text] is a global bifurcation tree of an asymmetric period-[Formula: see text] motion to chaos, and [Formula: see text] is a global bifurcation tree of a symmetric period-[Formula: see text] motion to chaos. Each bifurcation tree of a specific periodic motion to chaos is presented in detail. Numerical simulations of periodic motions are performed from analytical predictions. From finite Fourier series, harmonic amplitudes and phases for periodic motions are obtained for frequency analysis. Through this study, the rich dynamics of the first-order, time-delayed, nonlinear dynamical system is presented.

Maximizing Sensitivity Vector Fields: A Parametric Study

2014 ◽  
Vol 9 (2) ◽  
Author(s):  
Andrew R. Sloboda ◽  
Bogdan I. Epureanu
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Vector Fields ◽  
Nonlinear Feedback ◽  
Baseline System ◽  
Nominal Parameter ◽  
Construction Play ◽  
Sensitivity Vector ◽  
System Attractor ◽  
Made In

Sensitivity vector fields (SVFs) have proven to be an effective method for identifying parametric variations in dynamical systems. These fields are constructed using information about how a dynamical system's attractor deforms under prescribed parametric variations. Once constructed, they can be used to quantify any additional variations from the nominal parameter set as they occur. Since SVFs are based on attractor deformations, the geometry and other qualities of the baseline system attractor impact how well a set of SVFs will perform. This paper examines the role attractor characteristics and the choices made in SVF construction play in determining the sensitivity of SVFs. The use of nonlinear feedback to change a dynamical system with the intent of improving SVF sensitivity is explored. These ideas are presented in the context of constructing SVFs for several dynamical systems.

scholarly journals Local Bifurcations of Reversible Piecewise Smooth Planar Dynamical Systems

2020 ◽  
pp. 1-15
Author(s):  
V. Sh. Roitenberg
Keyword(s):  
Dynamical System ◽  
Vector Field ◽  
Dynamical Systems ◽  
Singular Point ◽  
Vector Fields ◽  
Singular Points ◽  
Piecewise Smooth ◽  
Smooth Vector ◽  
Local Bifurcations ◽  
Piecewise Smooth Vector Fields

There are quite a few works, which consider local bifurcations of piecewise-smooth vector fields on the plane. A number of papers also studied  the local bifurcations of smooth vector fields on the plane that are reversible with respect to involution. In the paper, we introduce reversible dynamical systems defined by piecewise-smooth vector fields on the coordinate plane (x, y) for which the discontinuity line y = 0 coincides with the set of fixed points of the system involution. We consider the generic one-parameter perturbations of such a vector field. The bifurcations of the singular point O lying on this line are described in two cases. In the first case, the point O is a rough saddle of the smooth vector fields that coincide with a piecewise smooth vector field in the half-planes y > 0 and y < 0. The parameter can be chosen so that for parameter values less than or equal to zero, the dynamical system has a unique singular point with four hyperbolic sectors in a vicinity of the point O. For positive values of the parameter in the vicinity of the point O, there are three singular points, a quasi-centre and two saddles, the separatrixes of which form a simple closed contour that bounds the cell from closed trajectories. In the second case, O is a rough node of the corresponding vector fields. The parameter can be chosen so that for values of the parameter less than or equal to zero, the dynamical system has a unique singular point in a vicinity of the point O, and all other trajectories are closed. For positive values of the parameter in the vicinity of the point O, there are three singular points, two nodes and a quasi-saddle, whose two separatrixes go to the nodes.

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