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El. knyga: Piecewise-smooth Dynamical Systems: Theory and Applications

3.50/5 (4 ratings by Goodreads)
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  • Formatas: PDF+DRM
  • Serija: Applied Mathematical Sciences 163
  • Išleidimo metai: 01-Jan-2008
  • Leidėjas: Springer London Ltd
  • Kalba: eng
  • ISBN-13: 9781846287084
Kitos knygos pagal šią temą:
Piecewise-smooth Dynamical Systems: Theory and ApplicationsDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Applied Mathematical Sciences 163
  • Išleidimo metai: 01-Jan-2008
  • Leidėjas: Springer London Ltd
  • Kalba: eng
  • ISBN-13: 9781846287084
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This book presents a coherent framework for understanding the dynamics of piecewise-smooth and hybrid systems. An informal introduction expounds the ubiquity of such models via numerous. The results are presented in an informal style, and illustrated with many examples. The book is aimed at a wide audience of applied mathematicians, engineers and scientists at the beginning postgraduate level. Almost no mathematical background is assumed other than basic calculus and algebra.

Traditional analysis of dynamical systems has restricted its attention to smooth problems, but it has become increasingly clear that there are distinctive phenomena unique to discontinuous systems that can be analyzed mathematically but which fall outside the usual methodology for smooth dynamical systems.The primary purpose of this book is to present a coherent framework for understanding the dynamics of piecewise-smooth and hybrid systems. An informal introduction asserts the ubiquity of such models with examples drawn from mechanics, electronics, control theory and physiology. The main thrust is to classify complex behavior via bifurcation theory in a systematic yet applicable way. The key concept is that of a discontinuity-induced bifurcation, which generalizes diverse phenomena such as grazing, border-collision, sliding, chattering and the period-adding route to chaos. The results are presented in an informal style and illustrated with copious examples, both theoretical and experimental.Aimed at a wide audience of applied mathematicians, engineers and scientists at the early postgraduate level, the book assumes only the standard background of basic calculus and linear algebra for most of the presentation and will be an indispensable resource for students and researchers. The inclusion of a comprehensive bibliography and many open questions will also serve as a stimulus for future research.

Recenzijos

From the reviews:



"This book is undoubtedly a strong contribution to the field of bifurcation and chaos analysis and more generally to the field of nonsmooth dynamical systems analysis. The authors have made a remarkable effort in mixing intricate technical developments with numerous examples, numerical results, and experimental results."



IEEE Control Systems Magazine



"PSDS presents a valuable compendium of information about the bifurcations of different types of piecewise-smooth systems, but it stops short of completely specifying the mathematical context within which the bifurcation phenomena it discusses are generic. That leaves lots of interesting work to do in studying piecewise-smooth dynamical systems. PSDS is an excellent starting point where one can find extensive analysis of diverse examples."



SIAM Book Reviews



"This book treats dynamical systems that have a piecewise smooth right-hand-side. Graphical sketches are abundant, supporting the presentation of the essential ideas behind arguments and techniques. Overall, the level of presentation makes the book useful as a source of theoretical background knowledge for researchers and postgraduate students in engineering and applied mathematics. It is also suitable as a reference for undergraduate projects or advanced undergraduate reading groups." (Jan Sieber, Zentralblatt MATH, Vol. 1146, 2008)



"The book providing motivation by showing a variety of applications and new phenomena. This book offers a very good survey of the rapidly developing area of the dynamics of non-smooth systems . the authors succeed in building up a systematic framework which is based on relevant applications. This is a book rich in content and an excellent introduction to this new area. The book is configured as a compiled introduction for graduate students and researchers interested in this area." (Tassilo Küpper, Mathematical Reviews, Issue 2009i)

Introduction
1(47)
Why piecewise smooth?
1(2)
Impact oscillators
3(25)
Case study I: A one-degree-of-freedom impact oscillator
6(7)
Periodic motion
13(5)
What do we actually see?
18(8)
Case study II: A bilinear oscillator
26(2)
Other examples of piecewise-smooth systems
28(11)
Case study III: Relay control systems
28(4)
Case study IV: A dry-friction oscillator
32(2)
Case study V: A DC-DC converter
34(5)
Non-smooth one-dimensional maps
39(8)
Case study VI: A simple model of irregular heartbeats
39(3)
Case study VII: A square-root map
42(2)
Case study VIII: A continuous piecewise-linear map
44(3)
Qualitative theory of non-smooth dynamical systems
47(74)
Smooth dynamical systems
47(24)
Ordinary differential equations (flows)
49(4)
Iterated maps
53(5)
Asymptotic stability
58(1)
Structural stability
59(4)
Periodic orbits and Poincare maps
63(4)
Bifurcations of smooth systems
67(4)
Piecewise-smooth dynamical systems
71(12)
Piecewise-smooth maps
71(2)
Piecewise-smooth ODEs
73(2)
Filippov systems
75(3)
Hybrid dynamical systems
78(5)
Other formalisms for non-smooth systems
83(10)
Complementarity systems
83(5)
Differential inclusions
88(3)
Control systems
91(2)
Stability and bifurcation of non-smooth systems
93(10)
Asymptotic stability
94(2)
Structural stability and bifurcation
96(4)
Types of discontinuity-induced bifurcations
100(3)
Discontinuity mappings
103(11)
Transversal intersections; a motivating calculation
105(2)
Transversal intersections; the general case
107(4)
Non-transversal (grazing) intersections
111(3)
Numerical methods
114(7)
Direct numerical simulation
115(3)
Path-following
118(3)
Border-collision in piecewise-linear continuous maps
121(50)
Locally piecewise-linear continuous maps
121(7)
Definitions
124(1)
Possible dynamical scenarios
125(2)
Border-collision normal form map
127(1)
Bifurcation of the simplest orbits
128(9)
A general classification theorem
128(3)
Notation for bifurcation classification
131(6)
Equivalence of border-collision classification methods
137(6)
Observer canonical form
137(3)
Proof of Theorem 3.1
140(3)
One-dimensional piecewise-linear maps
143(11)
Periodic orbits of the map
145(2)
Bifurcations between higher modes
147(2)
Robust chaos
149(5)
Two-dimensional piecewise-linear normal form maps
154(5)
Border-collision scenarios
155(2)
Complex bifurcation sequences
157(2)
Maps that are noninvertible on one side
159(10)
Robust chaos
159(5)
Numerical examples
164(5)
Effects of nonlinear perturbations
169(2)
Bifurcations in general piecewise-smooth maps
171(48)
Types of piecewise-smooth maps
171(3)
Piecewise-smooth discontinuous maps
174(14)
The general case
174(2)
One-dimensional discontinuous maps
176(4)
Periodic behavior:l = -1, ν1 > 0, ν2 < 1
180(5)
Chaotic behavior: l = -1, ν1 > 0, 1 < ν2 < 2
185(3)
Square-root maps
188(22)
The one-dimensional square-root map
188(5)
Quasi one-dimensional behavior
193(6)
Periodic orbits bifurcating from the border-collision
199(6)
Two-dimensional square-root maps
205(5)
Higher-order piecewise-smooth maps
210(9)
Case I: γ = 2
211(2)
Case II: γ = 3/2
213(1)
Period-adding scenarios
214(3)
Location of the saddle-node bifurcations
217(2)
Boundary equilibrium bifurcations in flows
219(34)
Piecewise-smooth continuous flows
219(14)
Classification of simplest BEB scenarios
221(4)
Existence of other attractors
225(1)
Planar piecewise-smooth continuous systems
226(3)
Higher-dimensional systems
229(3)
Global phenomena for persistent boundary equilibria
232(1)
Filippov flows
233(12)
Classification of the possible cases
235(2)
Planar Filippov systems
237(5)
Some global and non-generic phenomena
242(3)
Equilibria of impacting hybrid systems
245(8)
Classification of the simplest BEB scenarios
246(3)
The existence of other invariant sets
249(4)
Limit cycle bifurcations in impacting systems
253(54)
The impacting class of hybrid systems
253(12)
Examples
255(6)
Poincare maps related to hybrid systems
261(4)
Discontinuity mappings near grazing
265(14)
The geometry near a grazing point
266(5)
Approximate calculation of the discontinuity mappings
271(1)
Calculating the PDM
271(2)
Approximate calculation of the ZDM
273(1)
Derivation of the ZDM and PDM using Lie derivatives
274(5)
Grazing bifurcations of periodic orbits
279(16)
Constructing compound Poincare maps
280(4)
Unfolding the dynamics of the map
284(1)
Examples
285(10)
Chattering and the geometry of the grazing manifold
295(7)
Geometry of the stroboscopic map
295(1)
Global behavior of the grazing manifold G
296(3)
Chattering and the set G(∞)
299(3)
Multiple collision bifurcation
302(5)
Limit cycle bifurcations in piecewise-smooth flows
307(48)
Definitions and examples
307(11)
Grazing with a smooth boundary
318(22)
Geometry near a grazing point
319(2)
Discontinuity mappings at grazing
321(4)
Grazing bifurcations of periodic orbits
325(2)
Examples
327(7)
Detailed derivation of the discontinuity mappings
334(6)
Boundary-intersection crossing bifurcations
340(15)
The discontinuity mapping in the general case
341(5)
Derivation of the discontinuity mapping in the corner-collision case
346(1)
Examples
347(8)
Sliding bifurcations in Filippov systems
355(54)
Four possible cases
355(9)
The geometry of sliding bifurcations
356(3)
Normal form maps for sliding bifurcations
359(5)
Motivating example: a relay feedback system
364(9)
An adding-sliding route to chaos
366(2)
An adding-sliding bifurcation cascade
368(2)
A grazing-sliding cascade
370(3)
Derivation of the discontinuity mappings
373(10)
Crossing-sliding bifurcation
375(2)
Grazing-sliding bifurcation
377(4)
Switching-sliding bifurcation
381(1)
Adding-sliding bifurcation
382(1)
Mapping for a whole period: normal form maps
383(13)
Crossing-sliding bifurcation
384(6)
Grazing-sliding bifurcation
390(3)
Switching-sliding bifurcation
393(2)
Adding-sliding bifurcation
395(1)
Unfolding the grazing-sliding bifurcation
396(7)
Non-sliding period-one orbits
396(1)
Sliding orbit of period-one
397(2)
Conditions for persistence or a non-smooth fold
399(1)
A dry-friction example
399(4)
Other cases
403(6)
Grazing-sliding with a repelling sliding region --- catastrophe
403(1)
Higher-order sliding
404(5)
Further applications and extensions
409(50)
Experimental impact oscillators: noise and parameter sensitivity
409(13)
Noise
410(2)
An impacting pendulum: experimental grazing bifurcations
412(7)
Parameter uncertainty
419(3)
Rattling gear teeth: the similarity of impacting and piecewise-smooth systems
422(12)
Equations of motion
423(2)
An illustrative case
425(1)
Using an impacting contact model
426(5)
Using a piecewise-linear contact model
431(3)
A hydraulic damper: non-smooth invariant tori
434(14)
The model
436(2)
Grazing bifurcations
438(3)
A grazing bifurcation analysis for invariant tori
441(7)
Two-parameter sliding bifurcations in friction oscillators
448(11)
A degenerate crossing-sliding bifurcation
449(4)
Fold bifurcations of grazing-sliding limit cycles
453(2)
Two simultaneous grazings
455(4)
References 459(16)
Index 475


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