Discontinuous Dynamical Systems on Time-varying Domains is the first monograph focusing on this topic. While in the classic theory of dynamical systems the focus is on dynamical systems on time-invariant domains, this book presents discontinuous dynamical systems on time-varying domains where the corresponding switchability of a flow to the time-varying boundary in discontinuous dynamical systems is discussed. From such a theory, principles of dynamical system interactions without any physical connections are presented. Several discontinuous systems on time-varying domains are analyzed in detail to show how to apply the theory to practical problems. The book can serve as a reference book for researchers, advanced undergraduate and graduate students in mathematics, physics and mechanics.

Dr. Albert C. J. Luo is a professor at Southern Illinois University Edwardsville, USA. His research is involved in the nonlinear theory of dynamical systems. His main contributions are in the following aspects: a stochastic and resonant layer theory in nonlinear Hamiltonian systems, singularity on discontinuous dynamical systems, and approximate nonlinear theories for a deformable-body.

1 Introduction

1.1 Discontinuous systems

1.2 Book layout

References

2 Flow Switchability

2.1 Discontinuous dynamic systems

2.2 G-functions

2.3 Passable flows

2.4 Non-passable flows

2.5 Tangential flows

2.6 Switching bifurcations

References

3 Transversality and Sliding Phenomena

3.1 A controlled system

3.2 Transversality conditions

3.3 Mappings and predictions

3.4 Periodic and chaotic motions

References

4 A Frictional Oscillator on Time-varying Belt

4.1 Mechanical model

4.2 Analytical conditions

4.2.1 Equations of motion

4.2.2 Passable flows to boundary

4.2.3 Sliding flows on boundary

4.2.4 Grazing flows to boundary

4.3 Generic mappings and force product criteria...

4.3.1 Generic mappings

4.3.2 Sliding flows and fragmentation

4.3.3 Grazing flows

4.4 Periodic motions

4.4.1 Mapping structures

4.4.2 Illustrations

4.5 Numerical simulations

References

5 Two Oscillators with Impacts and Stick

5.1 Physical problem

5.1.1 Introduction to problem

5.1.2 Equations of motion

5.2 Domains and vector fields

5.2.1 Absolute motion description

5.2.2 Relative motion description

5.3 Mechanism of stick and grazing

5.3.1 Analytical conditions

5.3.2 Physical interpretation

5.4 Mapping structures and motions

5.4.1 Switching sets and basic mappings

5.4.2 Mapping equations

5.4.3 Mapping structures

5.4.4 Bifurcation scenario

5.5 Periodic motion prediction

5.5.1 Approach

5.5.2 Impacting chatter

5.5.3 Impacting chatter with stick

5.5.4 Parameter maps

5.6 Numerical illustrations

5.6.1 Impacting chatter

5.6.2 Impacting chatter with stick

5.6.3 Further illustrations

References

6 Dynamical Systems with Frictions

6.1 Problem statement

6.2 Switching and stick motions

6.2.1 Equations of motion

6.2.2 Analytical conditions

6.3 Periodic motions

6.3.1 Switching planes and mappings

6.3.2 Mapping structures and motions

6.3.3 Bifurcation scenario

6.4 Numerical illustrations

6.4.1 Periodic motion without stick

6.4.2 Periodic motion with stick

6.4.3 Periodic motion with stick only

References

7　Principles for System Interactions

7.1 Two dynamical systems

7.1.1 Dynamical systems with interactions

7.1.2 Discontinuous description

7.1.3 Resultant dynamical systems

7.2 Fundamental interactions

7.3 Interactions with singularity

7.4 Interactions with comer singularity

References

Appendix

A.1 Basic solution

A.2 Stability and bifurcation

Index