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.

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.

scholarly journals Discrete Normal Vector Field Approximation via Time Scale Calculus

2020 ◽  
Vol 5 (1) ◽  
pp. 349-360
Author(s):  
Ömer Akgandüller ◽  
Sibel Paşalı Atmaca
Keyword(s):  
Time Scales ◽  
Vector Fields ◽  
Geometric Analysis ◽  
Field Approximation ◽  
Normal Vector ◽  
Normal Vector Field ◽  
Time Scale Calculus ◽  
Vertex Set ◽  
Rectangular Grids ◽  
Mimetic Discretization

AbstractThe theory of time scales calculus have long been a subject to many researchers from different disciplines. Beside the unification and the extension aspects of the theory, it emerge as a powerful tool for mimetic discretization process. In this study, we present a framework to find normal vector fields of discrete point sets in ℝ3 by using symmetric differential on time scales. A surface parameterized by the tensor product of two time scales can be analogously expressed as the vertex set of non-regular rectangular grids. If the time scales are dense, then the discrete grid structure vanishes. If the time scales are isolated, then the further geometric analysis can be executed by using symmetric dynamic differential. Moreover, we present an algorithmic procedure to determine the symmetric dynamic differential structure on the neighborhood of points in surfaces. Our results indicate that the method we present has good approximation to unit normal vector fields of parameterized surfaces rather than the Delaunay triangulation for some points.

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 Defining the space in a general spacetime

2016 ◽  
Vol 13 (03) ◽  
pp. 1650031 ◽  
Author(s):  
Mayeul Arminjon
Keyword(s):  
Vector Field ◽  
Vector Fields ◽  
Differentiable Manifold ◽  
Normal Vector ◽  
Closed Set ◽  
Normal Vector Field ◽  
Dimension Formula ◽  
Global Space ◽  
Local Space ◽  
Global Vector

A global vector field [Formula: see text] on a “spacetime” differentiable manifold [Formula: see text], of dimension [Formula: see text], defines a congruence of world lines: the maximal integral curves of [Formula: see text], or orbits. The associated global space [Formula: see text] is the set of these orbits. A “[Formula: see text]-adapted” chart on [Formula: see text] is one for which the [Formula: see text] vector [Formula: see text] of the “spatial” coordinates remains constant on any orbit [Formula: see text]. We consider non-vanishing vector fields [Formula: see text] that have non-periodic orbits, each of which is a closed set. We prove transversality theorems relevant to such vector fields. Due to these results, it can be considered plausible that, for such a vector field, there exists in the neighborhood of any point [Formula: see text] a chart [Formula: see text] that is [Formula: see text]-adapted and “nice”, i.e. such that the mapping [Formula: see text] is injective — unless [Formula: see text] has some “pathological” character. This leads us to define a notion of “normal” vector field. For any such vector field, the mappings [Formula: see text] build an atlas of charts, thus providing [Formula: see text] with a canonical structure of differentiable manifold (when the topology defined on [Formula: see text] is Hausdorff, for which we give a sufficient condition met in important physical situations). Previously, a local space manifold [Formula: see text] had been associated with any “reference frame” [Formula: see text], defined as an equivalence class of charts. We show that, if [Formula: see text] is made of nice [Formula: see text]-adapted charts, [Formula: see text] is naturally identified with an open subset of the global space manifold [Formula: see text].

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 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.

AN ANALYTICAL PREDICTION OF PERIODIC FLOWS IN THE CHUA CIRCUIT SYSTEM

2009 ◽  
Vol 19 (07) ◽  
pp. 2165-2180 ◽  
Author(s):  
ALBERT C. J. LUO ◽  
BING XUE
Keyword(s):  
Local Stability ◽  
Normal Vector ◽  
Analytical Prediction ◽  
Separation Boundary ◽  
Normal Vector Field ◽  
Periodic Flows ◽  
System Parameters ◽  
Chua Circuit ◽  
Discontinuous Boundary ◽  
Mapping Structures

In this paper, periodic and chaotic behaviors in the Chua circuit system are investigated, and the analytical prediction of periodic flows in such a system is carried out. The solutions of the system in different regions with different parameters are first obtained. The switching boundaries are introduced for systems switching because of different system parameters in different domains. In the vicinity of the switching boundaries, the normal vector-field product is introduced to measure flow switching on the separation boundary, and the conditions for grazing and passable flows to the discontinuous boundary are presented. The basic mappings are defined and periodic responses of such a system are predicted analytically from mapping structures. The local stability and bifurcation analysis are carried out.

Nonlinear Dynamics of the Chua’s Circuit System Revisited

2008 ◽  
Author(s):  
Albert C. J. Luo ◽  
Bing Xue
Keyword(s):  
Local Stability ◽  
Normal Vector ◽  
Chua’S Circuit ◽  
Separation Boundary ◽  
Normal Vector Field ◽  
Chua's Circuit ◽  
System Parameters ◽  
Discontinuous Boundary ◽  
Mapping Structures ◽  
Stability And Bifurcation Analysis

In this paper, periodic and chaotic behaviors in the Chua’s circuit system are discussed. The solutions of the system in different regions with different parameters are obtained. The switching boundaries are introduced for systems switching because of different system parameters. In the vicinity of the switching boundary, the normal vector-field product is introduced to measure the flow switching on the separation boundary, and the grazing and passable conditions to the discontinuous boundary are presented. The basic mappings are defined and periodic responses of such a system are predicted analytically from the mapping structures. The local stability and bifurcation analysis are carried out.

Immersions with non-zero normal vector fields

1992 ◽  
Vol 112 (2) ◽  
pp. 281-285 ◽  
Author(s):  
Bang-He Li ◽  
Gui-Song Li
Keyword(s):  
Vector Fields ◽  
Normal Vector ◽  
Homotopy Classes ◽  
Natural Action ◽  
Singularity Method ◽  
One To One ◽  
Regular Homotopy

Let M be a smooth n-manifold, X be a smooth (2n − 1)-manifold, and g:M → X be a map. It was proved in [6] that g is always homotopic to an immersion. The set of homotopy classes of monomorphisms from TM into g*TX, which is denoted by Sg, may be enumerated either by the method of I. M. James and E. Thomas or by the singularity method of U. Koschorke (see [1] and references therein). When the natural action of π1(XM, g) on Sg is trivial, for example, if X is euclidean, the set Sg is in one-to-one correspondence with the set of regular homotopy classes of immersions homotopic to g (see e.g. [4]).

Planar quadratic vector fields with finite saddle connection on a straight line (convex case)

2005 ◽  
Vol 6 (2) ◽  
pp. 187-204 ◽  
Author(s):  
Paulo César Carrião ◽  
Maria Elasir Seabra Gomes ◽  
Antonio Augusto Gaspar Ruas
Keyword(s):  
Vector Fields ◽  
Saddle Connection ◽  
Quadratic Vector Fields ◽  
Straight Line ◽  
Convex Case
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