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Principles of Discontinuous Dynamical Systems

:	Marat Akhmet
:	Principles of Discontinuous Dynamical Systems
:	Springer-Verlag
:	9781441965813
:	1
:	CHF 47.50
:

:	Analysis
:	English

:	176
:	Wasserzeichen
:	PC/MAC/eReader/Tablet
:	PDF

Discontinuous dynamical systems have played an important role in both theory and applications during the last several decades. This is still an area of active research and techniques to make the applications more effective are an ongoing topic of interest.Principles of Discontinuous Dynamical Systems is devoted to the theory of differential equations with variable moments of impulses. It introduces a new strategy of implementing an equivalence to systems whose solutions have prescribed moments of impulses and utilizing special topologies in spaces of piecewise continuous functions. The achievements obtained on the basis of this approach are described in this book. The text progresses systematically, by covering preliminaries in the first four chapters. This is followed by more complex material and special topics such as Hopf bifurcation, Devaney's chaos, and the shadowing property are discussed in the last two chapters. This book is suitable for researchers and graduate students in mathematics and also in diverse areas such as biology, computer science, and engineering who deal with real world problems.

	Principles of Discontinuous Dynamical Systems	1
	Preface	7
	Contents	7
	9	7
	1 Introduction	13
	2 Description of the System with Fixed Moments of Impulses and Its Solutions	19
	2.1 Spaces of Piecewise Continuous Functions	19
	2.2 Description of the System	21
	2.3 Description of Solutions	22
	2.4 Equivalent Integral Equations	26
	2.5 The Gronwall--Bellman Lemma for Piecewise Continuous Functions	28
	2.6 Existence and Uniqueness Theorems	30
	2.7 Continuity	32
	Notes	35
	3 Stability and Periodic Solutions of Systems with Fixed Moments of Impulses	37
	3.1 Definitions of Stability	37
	3.2 Basics of Periodic Systems	39
	Notes	42
	4 Basics of Linear Systems	43
	4.1 Linear Homogeneous Systems	43
	4.2 Linear Nonhomogeneous Systems	53
	4.3 Linear Periodic Systems	59
	Notes	64
	5 Nonautonomous Systems with Variable Moments of Impulses	67
	5.1 Description of Systems	67
	5.2 Existence, Uniqueness, and Extension	68
	5.3 Beating Phenomena and Related Properties	71
	5.4 The Topology on the Set of Discontinuous Functions	74
	5.5 B-Equivalence: General Case	75
	5.6 Continuity Properties	80
	5.7 Generalities of Stability	82
	5.8 B-Equivalence: Quasilinear Systems	85
	5.9 Poincaré Criterion and Periodic Solutions of Quasilinear Systems	90
	Notes	91
	6 Differentiability Properties of Nonautonomous Systems	93
	6.1 Differentiability with Respect to Initial Conditions	94
	6.2 Differentiability with Respect to Parameters	100
	6.3 Higher Order B-Derivatives	102
	6.4 B-Analyticity Property	105
	6.5 B-Asymptotic Approximation of Solutions	107
	Notes	109
	7 Periodic Solutions of Nonlinear Systems	111
	7.1 Quasilinear Systems: the Noncritical Case	111
	7.2 The Critical Case	118
	Notes	122
	8 Discontinuous Dynamical Systems	124
	8.1 Generalities	124
	8.2 Local Existence and Uniqueness	129
	8.3 Extension of Solutions	130
	8.4 The Group Property	137
	8.5 Continuity Properties	139
	8.6 B-Equivalence	141
	8.7 Differentiability Properties	144
	8.8 Conclusion	146
	8.9 Examples	146
	Notes	148
	9 Perturbations and Hopf Bifurcation of a Discontinuous Limit Cycle	149
	9.1 The Nonperturbed System	149
	9.2 The Perturbed System	152
	9.3 Foci of the D-System	154
	9.4 The Center and Focus Problem	157
	9.5 Bifurcation of a Discontinuous Limit Cycle	159
	9.6 Examples	163
	Notes	163
	10 Chaos and Shadowing	165
	10.1 Introduction and Preliminaries	165
	10.2 The Devaney's Chaos	167
	10.3 Shadowing Property	172
	10.4 Simulations	173
	Notes	175
	References	176
	Index	183