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E-raamat: Differential Equations, Dynamical Systems, and an Introduction to Chaos

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(University of Wisconsin, Madison, USA), (Department of Mathematics, University of California, Berkeley, USA), (Department of Mathematics, Boston University, MA, USA)
  • Formaat: PDF+DRM
  • Ilmumisaeg: 12-Mar-2012
  • Kirjastus: Academic Press Inc
  • Keel: eng
  • ISBN-13: 9780123820112
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Differential Equations, Dynamical Systems, and an Introduction to ChaosSuurem pilt
  • Formaat: PDF+DRM
  • Ilmumisaeg: 12-Mar-2012
  • Kirjastus: Academic Press Inc
  • Keel: eng
  • ISBN-13: 9780123820112
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Hirsch, Devaney, and Smale’s classic Differential Equations, Dynamical Systems, and an Introduction to Chaos has been used by professors as the primary text for undergraduate and graduate level courses covering differential equations. It provides a theoretical approach to dynamical systems and chaos written for a diverse student population among the fields of mathematics, science, and engineering. Prominent experts provide everything students need to know about dynamical systems as students seek to develop sufficient mathematical skills to analyze the types of differential equations that arise in their area of study. The authors provide rigorous exercises and examples clearly and easily by slowly introducing linear systems of differential equations. Calculus is required as specialized advanced topics not usually found in elementary differential equations courses are included, such as exploring the world of discrete dynamical systems and describing chaotic systems.

  • Classic text by three of the world’s most prominent mathematicians
  • Continues the tradition of expository excellence
  • Contains updated material and expanded applications for use in applied studies


Muu info

This classic book presents the mathematical rigor, theory, and applications needed to understand dynamical systems in an attempt to solve problems in engineering, science, and biology.
Preface to the Third Edition ix
Preface xi
Chapter 1 First-Order Equations
1(20)
1.1 The Simplest Example
1(3)
1.2 The Logistic Population Model
4(3)
1.3 Constant Harvesting and Bifurcations
7(3)
1.4 Periodic Harvesting and Periodic Solutions
10(1)
1.5 Computing the Poincare Map
11(4)
1.6 Exploration: A Two-Parameter Family
15(6)
Chapter 2 Planar Linear Systems
21(18)
2.1 Second-Order Differential Equations
23(1)
2.2 Planar Systems
24(2)
2.3 Preliminaries from Algebra
26(3)
2.4 Planar Linear Systems
29(1)
2.5 Eigenvalues and Eigenvectors
30(3)
2.6 Solving Linear Systems
33(3)
2.7 The Linearity Principle
36(3)
Chapter 3 Phase Portraits for Planar Systems
39(22)
3.1 Real Distinct Eigenvalues
39(5)
3.2 Complex Eigenvalues
44(3)
3.3 Repeated Eigenvalues
47(2)
3.4 Changing Coordinates
49(12)
Chapter 4 Classification of Planar Systems
61(12)
4.1 The Trace-Determinant Plane
61(3)
4.2 Dynamical Classification
64(7)
4.3 Exploration: A 3D Parameter Space
71(2)
Chapter 5 Higher-Dimensional Linear Algebra
73(34)
5.1 Preliminaries from Linear Algebra
73(9)
5.2 Eigenvalues and Eigenvectors
82(3)
5.3 Complex Eigenvalues
85(3)
5.4 Bases and Subspaces
88(5)
5.5 Repeated Eigenvalues
93(7)
5.6 Genericity
100(7)
Chapter 6 Higher-Dimensional Linear Systems
107(32)
6.1 Distinct Eigenvalues
107(7)
6.2 Harmonic Oscillators
114(6)
6.3 Repeated Eigenvalues
120(3)
6.4 The Exponential of a Matrix
123(7)
6.5 Nonautonomous Linear Systems
130(9)
Chapter 7 Nonlinear Systems
139(20)
7.1 Dynamical Systems
140(2)
7.2 The Existence and Uniqueness Theorem
142(5)
7.3 Continuous Dependence of Solutions
147(2)
7.4 The Variational Equation
149(4)
7.5 Exploration: Numerical Methods
153(3)
7.6 Exploration: Numerical Methods and Chaos
156(3)
Chapter 8 Equilibria in Nonlinear Systems
159(28)
8.1 Some Illustrative Examples
159(6)
8.2 Nonlinear Sinks and Sources
165(3)
8.3 Saddles
168(6)
8.4 Stability
174(1)
8.5 Bifurcations
175(7)
8.6 Exploration: Complex Vector Fields
182(5)
Chapter 9 Global Nonlinear Techniques
187(26)
9.1 Nullclines
187(5)
9.2 Stability of Equilibria
192(10)
9.3 Gradient Systems
202(4)
9.4 Hamiltonian Systems
206(3)
9.5 Exploration: The Pendulum with Constant Forcing
209(4)
Chapter 10 Closed Orbits and Limit Sets
213(20)
10.1 Limit Sets
213(3)
10.2 Local Sections and Flow Boxes
216(2)
10.3 The Poincare Map
218(2)
10.4 Monotone Sequences in Planar Dynamical Systems
220(2)
10.5 The Poincare-Bendixson Theorem
222(3)
10.6 Applications of Poincare-Bendixson
225(3)
10.7 Exploration: Chemical Reactions that Oscillate
228(5)
Chapter 11 Applications in Biology
233(24)
11.1 Infectious Diseases
233(4)
11.2 Predator-Prey Systems
237(7)
11.3 Competitive Species
244(6)
11.4 Exploration: Competition and Harvesting
250(1)
11.5 Exploration: Adding Zombies to the SIR Model
251(6)
Chapter 12 Applications in Circuit Theory
257(20)
12.1 An RLC Circuit
257(4)
12.2 The Lienard Equation
261(2)
12.3 The van der Pol Equation
263(7)
12.4 A Hopf Bifurcation
270(2)
12.5 Exploration: Neurodynamics
272(5)
Chapter 13 Applications in Mechanics
277(28)
13.1 Newton's Second Law
277(3)
13.2 Conservative Systems
280(2)
13.3 Central Force Fields
282(3)
13.4 The Newtonian Central Force System
285(5)
13.5 Kepler's First Law
290(3)
13.6 The Two-Body Problem
293(1)
13.7 Blowing Up the Singularity
294(4)
13.8 Exploration: Other Central Force Problems
298(1)
13.9 Exploration: Classical Limits of Quantum Mechanical Systems
299(2)
13.10 Exploration: Motion of a Glider
301(4)
Chapter 14 The Lorenz System
305(24)
14.1 Introduction
306(2)
14.2 Elementary Properties of the Lorenz System
308(4)
14.3 The Lorenz Attractor
312(4)
14.4 A Model for the Lorenz Attractor
316(5)
14.5 The Chaotic Attractor
321(5)
14.6 Exploration: The Rossler Attractor
326(3)
Chapter 15 Discrete Dynamical Systems
329(32)
15.1 Introduction
329(5)
15.2 Bifurcations
334(3)
15.3 The Discrete Logistic Model
337(3)
15.4 Chaos
340(4)
15.5 Symbolic Dynamics
344(5)
15.6 The Shift Map
349(2)
15.7 The Cantor Middle-Thirds Set
351(3)
15.8 Exploration: Cubic Chaos
354(1)
15.9 Exploration: The Orbit Diagram
355(6)
Chapter 16 Homoclinic Phenomena
361(24)
16.1 The Shilnikov System
361(7)
16.2 The Horseshoe Map
368(7)
16.3 The Double Scroll Attractor
375(2)
16.4 Homoclinic Bifurcations
377(4)
16.5 Exploration: The Chua Circuit
381(4)
Chapter 17 Existence and Uniqueness Revisited
385(26)
17.1 The Existence and Uniqueness Theorem
385(2)
17.2 Proof of Existence and Uniqueness
387(7)
17.3 Continuous Dependence on Initial Conditions
394(3)
17.4 Extending Solutions
397(4)
17.5 Nonautonomous Systems
401(3)
17.6 Differentiability of the Flow
404(7)
Bibliography 411(4)
Index 415
Morris W. Hirsch works at the University of Wisconsin, Madison, USA. Stephen Smale works in the Department of Mathematics at University of California, Berkeley, USA. Robert L. Devaney works in the Department of Mathematics at Boston University, MA, USA.
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