Singularity and Dynamics on Discontinuous Vector Fields (eBook)

Albert C.J. Luo (Autor)

eBook Download: EPUB

2006 | 1. Auflage
310 Seiten
Elsevier Science (Verlag)
978-0-08-048093-0 (ISBN)

Infos

This book discussed fundamental problems in dynamics, which extensively exist in engineering, natural and social sciences. The book presented a basic theory for the interactions among many dynamical systems and for a system whose motions are constrained naturally or artificially. The methodology and techniques presented in this book are applicable to discontinuous dynamical systems in physics, engineering and control. In addition, they may provide useful tools to solve non-traditional dynamics in biology, stock market and internet network et al, which cannot be easily solved by the traditional Newton mechanics. The new ideas and concepts will stimulate ones' thought and creativities in corresponding subjects. The author also used the simple, mathematical language to write this book. Therefore, this book is very readable, which can be either a textbook for senior undergraduate and graduate students or a reference book for researches in dynamics.

? Challenging continuous Newton's dynamics
? Original theory and seeds of new researches in the field
? Wide spectrum of applications in science and engineering
? Systematic presentation and clear illustrations

Inhalt

Cover 1
Copyright page 5
Preface 8
Contents 10
Introduction 12
Smooth dynamics 12
Nonsmooth dynamics 15
Book layout 18
Flow Passability and Tangential Flows 22
Domain accessibility 22
Discontinuous dynamic systems 23
Oriented boundary and singular sets 25
Local singularity and tangential flows 34
A piecewise linear system 44
A friction-induced oscillator 51
Flow Switching Bifurcations 82
Set-valued vector fields 82
Switching bifurcations 84
Sliding fragmentation 94
Sliding conditions in a friction oscillator 104
Sliding criteria for a friction oscillator 106
Transversal Singularity and Bouncing Flows 124
Transversal tangential flows 124
Cusped and inflexed tangential flows 138
Nonpassable tangential flows 144
Bouncing flows 146
A controlled piecewise linear system 152
Real and Imaginary Flows 158
Singularity on boundary 158
Hyperbolicity and parabolicity 159
Boundary formation 166
Real flows 173
Imaginary flows 179
An example 191
Discontinuous Vector Fields with Flow Barriers 198
Flow barriers 198
Switching bifurcations 204
Sliding fragmentation 208
A friction oscillator with flow barrier 212
Transport Laws and Mapping Dynamics 224
Classification of discontinuity 224
Transport laws 229
Mapping dynamics 238
An impacting piecewise system 243
Symmetry and Fragmentized Strange Attractor 262
Symmetric discontinuity 262
Switching sets and mappings 264
Grazing and mappings symmetry 268
Steady-state flow symmetry 278
Strange attractor fragmentation 287
Fragmentized strange attractors 294
Appendix A 300
References 304
Subject Index 310

Textauszug

Chapter 2

Flow Passability and Tangential Flows

A.C.J. Luo    Department of Mechanical and Industrial Engineering Southern Illinois University Edwardsville Edwardsville, Illinois, USA

In this chapter, the singularity in the vicinity of the discontinuous boundary will be presented. The accessible and inaccessible subdomains will be introduced for development of a theory of nonsmooth dynamic systems on connectable and accessible subdomains. On the accessible domain, the corresponding dynamic systems are introduced. The oriented boundary sets and singular sets caused by the separation boundary will be discussed. The local singularity and tangency of a flow on the separation boundary will be investigated. The necessary and sufficient conditions for such a local singularity and tangency will be presented. The grazing flows in piecewise linear systems and friction-induced oscillators will be investigated, and the grazing conditions of the flows will be determined.

2.1 Domain accessibility

Before development of a general theory for nonsmooth dynamical systems on a universal domain ⊂Rn in phase space, the subdomains ii=1,2,…) of the domain are introduced, and the dynamics on the subdomains are defined differently.

DEFINITION 2.1 A subdomain in the universal domain is termed the accessible subdomain on which a specific, continuous dynamical system can be defined.

DEFINITION 2.2 A subdomain in a universal domain is termed the inaccessible subdomain on which no dynamical system can be defined.

Since the dynamical system can be defined differently on each accessible subdomain, the dynamical behaviors of the system in those accessible subdomains i can be different from each other in the sense of Newton's mechanics. These different behaviors cause the complexity of motion in the universal domain . Owing to the accessible and inaccessible subdomains, the universal domain is classified into the connectable and separable ones. The connectable domain is defined as:

DEFINITION 2.3 A domain in phase space is termed the connectable domain if all the accessible subdomains of the universal domain can be connected without any inaccessible subdomain.

Similarly, a definition of the separable domain is:

DEFINITION 2.4 A domain is termed the separable domain if the accessible subdomains in the universal domain are separated by inaccessible domains.

The boundary between two adjacent, accessible subdomains is a bridge of dynamical behaviors in two domains for motion continuity. For the connectable domain, it is bounded by the universal boundary surface ⊂Rrr≤n−1), and each subdomain is bounded by the subdomain boundary surface ij⊂Rri,j∈{1,2,…}) with or without the partial universal boundary. For instance, consider an n-D connectable domain in phase space, as shown in Fig. 2.1(a) through an 1-dimensional, subvector n1 and an n−n1)-dimensional, subvector n−n1. The shaded area i is a specific subdomain, and other subdomains are white. The dark, solid curve represents the original boundary of the domain . In the separable domain, there is at least an inaccessible subdomain to separate the accessible subdomains. The union of inaccessible subdomains is also called the “sea”. The sea is the complement of the accessible subdomains to the universal (original) domain . That is determined by 0=℧∖⋃iΩi. The accessible subdomains in the domain are also called the “islands”. For illustration of such a definition, an n-D separable domain is shown in Fig. 2.1(b). The dashed surface is the boundary of the universal domain, and the gray area is the sea. The white regions are the accessible domains (or islands). The diagonal line shaded region represents a specific accessible subdomain (island). From one island to another, the transport is needed for motion continuity. The transport laws will be discussed in Chapter 7. For an accessible domain, grazing flows in discontinuous systems will be presented in this chapter.

Figure 2.1 Phase space: (a) connectable and (b) separable domains.

2.2 Discontinuous dynamic systems

To demonstrate the basic concepts of nonsmooth dynamical system theory, the development of the theory in this chapter is restricted to an n-dimensional, nonsmooth dynamical system. Consider a dynamic system consisting of N subdynamic systems in a universal domain ⊂Rn. The universal domain is divided into N accessible subdomains i, and the union of all the accessible subdomains i=1NΩi and the universal domain =⋃i=1NΩi∪Ξ, as shown in Fig. 2.1 through an 1-dimensional, subvector n1 and an n−n1)-dimensional, subvector n−n1. 0 is the union of the inaccessible domains. For the connectable domain in Fig. 2.1(a), 0=∅. In Fig. 2.1(b), the union of the inaccessible subdomains is the sea, 0=℧∖⋃i=1mΩi is the complement of the union of the accessible subdomain. On the ith open subdomain i, there is a r-continuous system r≥1) in the form of

˙≡F(i)(x,t,μi)∈Rn,x=(x1,x2,…,xn)T∈Ωi.

  (2.1)

The time is t and ˙=dx/dt. In an accessible subdomain i, the vector field (i)(x,t,μi) with parameter vectors i=(μi(1),μi(2),…,μi(l))T∈Rl is r-continuous r⩾1) in x and for all time t; and the continuous flow in Eq. (2.1)(i)(t)=Φ(i)(x(i)(t0),t,μi) with (i)(t0)=Φ(i)(x(i)(t0),t0,μi) is r+1-continuous for time t.

The nonsmooth dynamic theory developed in this paper holds for the following conditions:

(A1) The switching between two adjacent subsystems possesses time-continuity.

(A2) For an unbounded, accessible subdomain i, there is a bounded domain i⊂Ωi and the corresponding vector field and its flow are bounded, i.e.,

F(i)‖≤K1(const)and

  (2.2)

Φ(i)‖≤K2(const)onDi for t∈[0,∞).

(A3) For a bounded, accessible domain i, there is a bounded domain i⊂Ωi and the corresponding vector field is bounded, but the flow may be unbounded, i.e.,

F(i)‖≤K1(const)and‖Φ(i)‖<∞onDi for t∈[0,∞).

  (2.3)

2.3 Oriented boundary and singular sets

Since dynamical systems on the different accessible subdomains are distinguishing, the relation between flows in the two subdomains should be developed herein for flow continuity. For a subdomain i, there are i-segment boundaries (i≤N−1). Consider a boundary set of any two subdomains, formed by the intersection of the closed subdomains, i.e., Ωij=Ω¯i∩Ω¯j(i,j∈{1,2,…,N},j≠i), as shown in Fig. 2.2.

Figure 2.2 Subdomains i and j, the corresponding boundary Ωij.

DEFINITION 2.5 The boundary in the n-D phase space is defined as

ij≡∂Ωij=Ω¯i∩Ω¯j={x|φij(x,t)=0whereφijisCr-continuous(r≥1)}⊂Rn−1.

  (2.4)

DEFINITION 2.6 The two subdomains i and j are disjoint if the boundary Ωij is an empty set (i.e., Ωij=∅).

The boundary values (α)=(x1(α),x2(α),…,xn(α))T,α∈{i,j}, pertain to the open domains i and j,...

Produkt-Details

• Erscheint lt. Verlag: 7.7.2006
• Sprache: englisch
• Themenwelt: Sachbuch/Ratgeber
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Naturwissenschaften ► Physik / Astronomie ► Quantenphysik
• Themenwelt: Technik
• Themenwelt: Wirtschaft
• ISBN-10: 0-08-048093-4 / 0080480934
• ISBN-13: 978-0-08-048093-0 / 9780080480930