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E-grāmata: Ordinary Differential Equations: Basics and Beyond

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  • Formāts: PDF+DRM
  • Sērija : Texts in Applied Mathematics 65
  • Izdošanas datums: 10-Nov-2016
  • Izdevniecība: Springer-Verlag New York Inc.
  • Valoda: eng
  • ISBN-13: 9781493963898
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Ordinary Differential Equations: Basics and BeyondPalielināt
  • Formāts: PDF+DRM
  • Sērija : Texts in Applied Mathematics 65
  • Izdošanas datums: 10-Nov-2016
  • Izdevniecība: Springer-Verlag New York Inc.
  • Valoda: eng
  • ISBN-13: 9781493963898
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This book develops the theory of ordinary differential equations (ODEs), starting from an introductory level (with no prior experience in ODEs assumed) through to a graduate-level treatment of the qualitative theory, including bifurcation theory (but not chaos).  While proofs are rigorous, the exposition is reader-friendly, aiming for the informality of face-to-face interactions.  A unique feature of this book is the integration of rigorous theory with numerous applications of scientific interest.  Besides providing motivation, this synthesis clarifies the theory and enhances scientific literacy. Other features include:  (i) a wealth of exercises at various levels, along with commentary that explains why they matter; (ii) figures with consistent color conventions to identify nullclines, periodic orbits, stable and unstable manifolds; and (iii) a dedicated website with software templates, problem solutions, and other resources supporting thetext (www.math.duke.edu/ode-book).  Given its many applications, the book may be used comfortably in science and engineering courses as well as in mathematics courses.  Its level is accessible to upper-level undergraduates but still appropriate for graduate students. The thoughtful presentation, which anticipates many confusions of beginning students, makes the book suitable for a teaching environment that emphasizes self-directed, active learning (including the so-called inverted classroom).
1 Introduction
1(40)
1.1 Some Simple ODEs
1(3)
1.1.1 Examples
1(1)
1.1.2 Descriptive Concepts
2(2)
1.2 Solutions of ODEs
4(2)
1.2.1 Examples and Discussion
4(1)
1.2.2 Geometric Interpretation of Solutions
5(1)
1.3 Snippets of Solution Techniques
6(4)
1.3.1 Computer Solutions
6(1)
1.3.2 Linear Equations with Constant Coefficients
7(1)
1.3.3 First-Order Linear Equations
8(1)
1.3.4 Separable Equations
8(2)
1.4 Physically Based Second-Order ODEs
10(7)
1.4.1 Linear Spring-Mass Systems
10(2)
1.4.2 Nonlinearity, Part I: The Restoring Force
12(2)
1.4.3 Energy
14(3)
1.4.4 Nonlinearity, Part II: The Frictional Force
17(1)
1.5 Systems of ODEs
17(3)
1.6 Topics Covered in This Book
20(4)
1.6.1 General Overview
20(1)
1.6.2 A Case Study in the Qualitative Theory of ODEs
21(3)
1.7 Exercises
24(12)
1.7.1 Core Exercises
26(3)
1.7.2 Computational Exercises
29(2)
1.7.3 Anticipatory Exercises
31(2)
1.7.4 PHD Exercises
33(3)
1.8 Pearls of Wisdom
36(5)
1.8.1 Miscellaneous
36(1)
1.8.2 The Concept "Generic"
37(1)
1.8.3 A Comparison Theorem
38(3)
2 Linear Systems with Constant Coefficients
41(38)
2.1 Preview
41(1)
2.2 Definition and Properties of the Matrix Exponential
42(8)
2.2.1 Preliminaries About Norms
42(4)
2.2.2 Convergence
46(2)
2.2.3 The Main Theorem and Its Proof
48(2)
2.3 Calculation of the Matrix Exponential
50(7)
2.3.1 The Role of Similarity
50(2)
2.3.2 Two Problematic Cases
52(3)
2.3.3 Use of the Jordan Form
55(2)
2.4 Large-Time Behavior of Solutions of Linear Systems
57(3)
2.4.1 The Main Results
57(2)
2.4.2 Tests for Eigenvalues in the Left Half-Plane
59(1)
2.5 Classification and Phase Portraits for 2 × 2 Systems
60(3)
2.5.1 Saddles and Nodes
60(2)
2.5.2 Foci and Centers
62(1)
2.5.3 Additional Remarks
62(1)
2.6 Solution of Inhomogeneous Problems
63(1)
2.7 Exercises
64(10)
2.7.1 Core Exercises
64(4)
2.7.2 Practice with Linear Algebra
68(2)
2.7.3 Practice Sketching Phase Portraits
70(2)
2.7.4 PHD Exercises
72(2)
2.8 Pearls of Wisdom
74(5)
2.8.1 Alternative Norms
74(1)
2.8.2 Nondifferentiable Limits and the Cantor Set
75(2)
2.8.3 More on Generic Behavior
77(2)
3 Nonlinear Systems: Local Theory
79(32)
3.1 Two Counterexamples
79(2)
3.2 Local Existence Theory
81(10)
3.2.1 Statement of the Existence Theorem
81(1)
3.2.2 C1 Implies Lipschitz Continuity
82(3)
3.2.3 Reformulation of the IVP as an Integral Equation
85(1)
3.2.4 The Contraction-Mapping Principle
85(2)
3.2.5 Proof of the Existence Theorem
87(3)
3.2.6 An Illustrative Example and Picard Iteration
90(1)
3.2.7 Concluding Remarks
91(1)
3.3 Uniqueness Theory
91(5)
3.3.1 Gronwall's Lemma
91(2)
3.3.2 More on Lipschitz Functions
93(1)
3.3.3 The Uniqueness Theorem
94(2)
3.4 Generalization to Nonautonomous Systems
96(1)
3.4.1 Nonlinear Systems
96(1)
3.4.2 Linear Systems
96(1)
3.5 Exercises
97(9)
3.5.1 Core Exercises
97(5)
3.5.2 Linear ODEs with Periodic Coefficients
102(2)
3.5.3 PHD Exercises
104(2)
3.6 Pearls of Wisdom
106(5)
3.6.1 Miscellaneous
106(2)
3.6.2 Resonance
108(3)
4 Nonlinear Systems: Global Theory
111(50)
4.1 The Maximal Interval of Existence
112(1)
4.2 Two Sufficient Conditions for Global Existence
113(6)
4.2.1 Linear Growth of the RHS
113(1)
4.2.2 Trapping Regions
114(5)
4.3 Level Sets and Trapping Regions
119(4)
4.3.1 Introduction via Duffing's Equation
119(1)
4.3.2 The Chemostat
119(1)
4.3.3 The Torqued Pendulum and ODEs on Manifolds
120(3)
4.4 Nullclines and Trapping Regions
123(8)
4.4.1 Nullclines in the Chemostat
123(1)
4.4.2 An Activator--Inhibitor System
124(3)
4.4.3 Sel'kov's Model for Glycolysis
127(1)
4.4.4 Van der Pol's Equation
128(1)
4.4.5 Michaelis--Menten Kinetics
129(2)
4.5 Continuity Properties of the Solution
131(3)
4.5.1 The Main Issue: Continuous Dependence on Initial Conditions
131(2)
4.5.2 Some Associated Formalism
133(1)
4.5.3 Continuity with Respect to Parameters
134(1)
4.6 Differentiability Properties of the Solution
134(8)
4.6.1 Dependence on Initial Conditions
134(2)
4.6.2 The Perspective of Differentiability
136(1)
4.6.3 Examples
137(1)
4.6.4 The Order Notation
138(2)
4.6.5 Proof of Theorem 4.6.1
140(1)
4.6.6 Tying Up Loose Ends
141(1)
4.6.7 Generalizations
141(1)
4.7 Exercises
142(13)
4.7.1 Core Exercises
143(6)
4.7.2 Applying the Differentiation Theorems
149(2)
4.7.3 Some Mopping-Up Exercises
151(1)
4.7.4 Computing Exercise
152(1)
4.7.5 PHD Exercises
153(2)
4.8 Pearls of Wisdom
155(1)
4.9 Appendix: Euler's Method
156(5)
4.9.1 Introduction
156(1)
4.9.2 Theoretical Basis for the Approximation
157(1)
4.9.3 Convergence of the Numerical Solution
158(3)
5 Nondimensionalization and Scaling
161(34)
5.1 Classes of ODEs in Applications
162(3)
5.1.1 Mechanical Models
162(1)
5.1.2 Electrical Models
163(1)
5.1.3 "Bathtub" Models
163(2)
5.2 Scaling Example 0: Two Models from Ecology
165(4)
5.3 Scaling Example 1: A Nonlinear Oscillator
169(3)
5.4 Scaling Example 2: Sel'kov's Model for Glycolysis
172(2)
5.5 Scaling Example 3: The Chemostat
174(4)
5.5.1 ODEs Modeling Flow Through a Reactor
174(2)
5.5.2 The Chemostat
176(2)
5.6 Scaling Example 4: An Activator-Inhibitor System
178(3)
5.7 Scaling Example 5: Michaelis-Menten Kinetics
181(3)
5.8 Exercises
184(7)
5.8.1 Core Exercises
185(2)
5.8.2 Anticipatory Exercises
187(2)
5.8.3 PHD Exercises
189(2)
5.9 Pearls of Wisdom
191(4)
5.9.1 Making Scaling Work for You
191(3)
5.9.2 A Nod to Scientific Literacy
194(1)
6 Trajectories Near Equilibria
195(64)
6.1 Stability of Equilibria
196(3)
6.1.1 The Main Theorem
196(2)
6.1.2 An Easy Application
198(1)
6.2 Terminology to Classify Equilibria
199(7)
6.2.1 Terms Related to Theorem 6.1.1
199(2)
6.2.2 Other Terms Based on Eigenvalues
201(1)
6.2.3 Section 1.6 Revisited, Part I
202(3)
6.2.4 Two-Dimensional Equilibria and Slopes of Nullclines
205(1)
6.3 Activator--Inhibitor Systems and the Turing Instability
206(4)
6.3.1 Equilibria of the Activator-Inhibitor System
206(2)
6.3.2 The Turing Instability: Destabilization by Diffusion
208(2)
6.4 Feedback Stabilization of an Inverted Pendulum
210(5)
6.5 Lyapunov Functions
215(5)
6.5.1 The Main Results
215(2)
6.5.2 Lasalle's Invariance Principle
217(1)
6.5.3 Construction of Lyapunov Functions: An Example
218(2)
6.6 Stable and Unstable Manifolds
220(8)
6.6.1 A Linear Example
220(2)
6.6.2 Statement of the Local Theorem
222(1)
6.6.3 A Nonlinear Example
223(3)
6.6.4 Global Behavior of Stable/Unstable Manifolds
226(2)
6.7 Drawing Phase Portraits
228(5)
6.7.1 Example 1: The Chemostat
229(1)
6.7.2 Example 2: The Activator-Inhibitor
230(1)
6.7.3 Example 3: Section 1.6 Revisited, Part II
231(2)
6.8 Exercises
233(12)
6.8.1 Core Exercises
233(5)
6.8.2 Uses of a Lyapunov Function
238(2)
6.8.3 Phase-Portrait Exercises
240(1)
6.8.4 PHD Exercises
241(4)
6.9 Pearls of Wisdom
245(3)
6.9.1 Miscellaneous
245(1)
6.9.2 The Hartman--Grobman Theorem and Topological Conjugacy
246(1)
6.9.3 Structural Stability
247(1)
6.10 Appendix 1: Partial Proof of Theorem 6.6.1
248(6)
6.10.1 Reformulation of the IVP as an Integral Equation
248(2)
6.10.2 Fixed-Point Analysis
250(2)
6.10.3 The Stable Manifold
252(1)
6.10.4 Stable Manifolds at Nonhyperbolic Equilibria
253(1)
6.11 Appendix 2: Center Manifolds and Nonhyperbolicity
254(5)
6.11.1 First Example
255(1)
6.11.2 Second Example
256(3)
7 Oscillations in ODEs
259(68)
7.1 Periodic Solutions
259(8)
7.1.1 Basic Issues
259(2)
7.1.2 Examples of Periodic Solutions
261(4)
7.1.3 A Leisurely Overview of This
Chapter
265(2)
7.2 Special Behavior in Two Dimensions Mostly
267(9)
7.2.1 The Poincare--Bendixson Theorem: Minimal Version
267(1)
7.2.2 Applications of the Theorem
268(2)
7.2.3 Limit Sets (in Any Dimension)
270(4)
7.2.4 The Poincare--Bendixson Theorem: Strong Version
274(1)
7.2.5 Nonexistence: Dulac's Theorem
275(1)
7.2.6 Section 1.6 Revisited, Part III
275(1)
7.3 Stability of Periodic Orbits and the Poincare Map
276(7)
7.3.1 An Eigenvalue Test for Stability
276(2)
7.3.2 Basics of the Poincare Map
278(3)
7.3.3 Discrete-Time Dynamics and the Proof of Theorem 7.3.2
281(2)
7.4 Stability of the Limit Cycle in the Torqued Pendulum
283(2)
7.5 Van der Pol with Small β: Weakly Nonlinear Analysis
285(8)
7.5.1 Two Illustrative Examples of Perturbation Theory
286(5)
7.5.2 Application to the van der Pol Equation
291(2)
7.6 Van der Pol with Large β: Singular Perturbation Theory
293(8)
7.6.1 Two Sources of Guidance
293(2)
7.6.2 Approximation of the Initial Decay in (7.58)
295(3)
7.6.3 Phase-Plane Analysis of a Related Equation
298(3)
7.6.4 Concluding Remarks
301(1)
7.7 Stability of the van der Pol Limit Cycles
301(3)
7.7.1 Case 1: Small β
302(1)
7.7.2 Case 2: Large β
303(1)
7.8 Exercises
304(14)
7.8.1 Core Exercises
304(7)
7.8.2 The Poincare--Lindstedt Method
311(2)
7.8.3 Changes of Variables
313(3)
7.8.4 PHD Exercises
316(2)
7.9 Pearls of Wisdom
318(3)
7.9.1 Area and Dulac's Theorem
318(1)
7.9.2 Poincare-Like Maps in Constructing Periodic Solutions
319(1)
7.9.3 Stable/Unstable Manifolds in Other Contexts
320(1)
7.9.4 Miscellaneous
321(1)
7.10 Appendix: Stabilizing an Inverted Pendulum
321(6)
7.10.1 A Smidgen of Floquet Theory
321(2)
7.10.2 Some Stable Solutions of Mathieu's Equation
323(2)
7.10.3 Application to the Inverted Vibrated Pendulum
325(2)
8 Bifurcation from Equilibria
327(76)
8.1 Examples of Pitchfork Bifurcation
328(6)
8.1.1 Bead on a Rotating Hoop
328(2)
8.1.2 The Lorenz Equations
330(2)
8.1.3 A Laterally Supported Inverted Pendulum
332(2)
8.2 Perspectives on This
Chapter
334(2)
8.2.1 An Outline of the
Chapter
334(1)
8.2.2 A Bifurcation Theorem
334(2)
8.3 Examples of Transcritical Bifurcation
336(3)
8.3.1 Section 1.6 Revisited: Part IV
336(1)
8.3.2 The Chemostat
337(2)
8.4 Examples of Saddle-Node Bifurcation
339(2)
8.4.1 The Torqued Pendulum
339(1)
8.4.2 Activator--Inhibitor Systems
340(1)
8.5 The Lyapunov--Schmidt Reduction
341(14)
8.5.1 Bare Bones of the Reduction
341(3)
8.5.2 Proof of Theorem 8.2.2
344(4)
8.5.3 One-Dimensional Bifurcation Problems
348(2)
8.5.4 Exchange of Stability
350(1)
8.5.5 Symmetry and the Pitchfork Bifurcation
351(1)
8.5.6 Additional Parameters in Bifurcation Problems
352(3)
8.6 Steady-State Bifurcation in Two Applications
355(9)
8.6.1 The Two-Cell Turing Instability
355(5)
8.6.2 The CSTR
360(4)
8.7 Examples of Hopf Bifurcation
364(8)
8.7.1 An Academic Example
364(2)
8.7.2 The "Repressilator"
366(2)
8.7.3 Section 1.6 Revisited: Part V
368(1)
8.7.4 The "Denatured" Morris--Lecar System
369(3)
8.8 Theoretical Description of Hopf Bifurcation
372(4)
8.8.1 A Bifurcation Theorem
372(2)
8.8.2 The Activator-Inhibitor: Extreme Nongeneric Behavior
374(1)
8.8.3 Sub/Supercriticality in Two Dimensions
375(1)
8.9 Exercises
376(19)
8.9.1 Core Exercises
376(4)
8.9.2 Applications of Bifurcation Theory
380(5)
8.9.3 PHD Exercises
385(10)
8.10 Pearls of Wisdom
395(8)
8.10.1 Comments on Proving the Hopf Bifurcation Theorem
395(2)
8.10.2 High-Dimensional Bifurcation: Symmetry and Mode Competition
397(2)
8.10.3 Homeostasis, or "Antibifurcation"
399(4)
9 Examples of Global Bifurcation
403(48)
9.1 Homoclinic Bifurcation
403(6)
9.1.1 An Academic Example
403(2)
9.1.2 The van der Pol Equation with a Nonlinear Restoring Force
405(1)
9.1.3 Section 1.6 Revisited: Part VI
405(3)
9.1.4 Other Examples of Homoclinic and Heteroclinic Bifurcations
408(1)
9.2 Saddle-Node Bifurcation of Limit Cycles
409(5)
9.2.1 An Academic Example
409(1)
9.2.2 The Denatured Morris-Lecar Equation
409(2)
9.2.3 The Overdamped Torqued Pendulum
411(3)
9.3 Poincare Maps and Stability Loss of Limit Cycles
414(1)
9.4 Mutual Annihilation of Two Limit Cycles
415(2)
9.4.1 An Academic Example
415(1)
9.4.2 The Denatured Morris-Lecar Equation
415(2)
9.4.3 Phase-Locking in Coupled Oscillators
417(1)
9.5 Hopf-Like Bifurcation to an Invariant Torus
417(3)
9.5.1 An Academic Example
417(2)
9.5.2 The Periodically Forced van der Pol Equation
419(1)
9.5.3 Other Examples of Bifurcation to an Invariant Torus
420(1)
9.6 Period-Doubling
420(13)
9.6.1 Academic Example 1: Mappings
422(1)
9.6.2 Cardiac Alternans
423(3)
9.6.3 Academic Example 2: ODEs
426(2)
9.6.4 A Periodically Forced Pendulum
428(2)
9.6.5 Rossler's Equation
430(2)
9.6.6 Other Examples
432(1)
9.7 The Onset of Chaos in the Lorenz Equations
433(5)
9.8 Bursting in the Denatured Morris-Lecar Equations
438(2)
9.9 Exercises
440(8)
9.9.1 Core Exercises
440(4)
9.9.2 Computations to Support Claims in the Text
444(2)
9.9.3 Bifurcation in a Quadratic Map
446(1)
9.9.4 PHD Exercises
447(1)
9.10 Pearls of Wisdom
448(3)
9.10.1 Remarks on Heteroclinic Orbits
448(1)
9.10.2 Bifurcation in Fluid-Mechanics Problems
449(1)
9.10.3 Routes to Chaos
449(2)
10 Epilogue
451(36)
10.1 Boundary Value Problems
451(6)
10.1.1 An Overview Through Examples
451(4)
10.1.2 Eigenvalue Problems
455(2)
10.2 Stochastic Population Models
457(3)
10.3 Numerical Methods: Two Sobering Examples
460(3)
10.3.1 Stiff ODEs
460(1)
10.3.2 Unreasonable Behavior of Reasonable Methods
461(2)
10.4 ODEs on a Torus: Entrainment
463(4)
10.5 Delay Differential Equations
467(3)
10.6 A Peek at Chaos
470(9)
10.6.1 A One-Dimensional Mapping Model
470(3)
10.6.2 The Lorenz Equations
473(6)
10.7 Exercises
479(5)
10.7.1 Core Exercises
479(3)
10.7.2 PHD Exercises
482(2)
10.8 Pearls of Wisdom
484(3)
10.8.1 The Elastica
484(2)
10.8.2 A Bit More on Chaos
486(1)
A Guide to Commonly Used Notation
487(6)
A.1 Letter Choices
487(3)
A.2 Other Notations
490(1)
A.3 Other Conventions
491(2)
B Notions from Advanced Calculus
493(22)
B.1 Basic Issues
493(2)
B.2 Pointwise and Uniform Convergence
495(4)
B.2.1 Sequences
495(2)
B.2.2 Series
497(2)
B.2.3 Convergence of Integrals
499(1)
B.3 Selected Issues in Vector Calculus
499(7)
B.3.1 Differentiability
500(1)
B.3.2 The Implicit Function Theorem
501(2)
B.3.3 Surfaces and Manifolds
503(3)
B.4 Exercises
506(6)
B.4.1 Core Exercises
506(4)
B.4.2 PHD Exercises
510(2)
B.5 Pearls of Wisdom
512(3)
C Notions from Linear Algebra
515(14)
C.1 How to Work with Jordan Normal Forms
515(4)
C.2 The Real Canonical Form of a Matrix
519(1)
C.3 Eigenvalues as Continuous Functions of Matrix Entries
520(2)
C.4 The Routh--Hurwitz Criterion
522(2)
C.5 Exercises
524(5)
C.5.1 Core Exercises
524(2)
C.5.2 PHD Exercises
526(3)
Bibliography 529(8)
Index 537
David G. Schaeffer is Professor of Mathematics at Duke University.  His research interests include partial differential equations and granular flow.   John W. Cain is Professor of Mathematics at Harvard University. His background is in application-oriented mathematics with interest in applications to medicine, biology, and biochemistry.
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