Differential Equations, Dynamical Systems, and an Introduction to Chaos

by Hirsch; Smale; Devaney

• ISBN13: 9780123497031
• ISBN10: 0123497035
• Edition: 2nd
• Format: Hardcover
• Copyright: 2003-10-22
• Publisher: Elsevier Science

Summary

This best-selling, classic text by eminent mathematicians Hirsch and Smale has captured the beauty and relative accessibility of chaotic phenomena for three decades. This field has motivated scientists and engineers in many disciplines to understand how differential equations and dynamical systems phenomena appear in virtually every area of science: chemistry, electrical engineering, celestial mechanics, ecological systems, and beyond. Book jacket.

Author Biography

Morris W. Hirsch: University of California, Berkeley Stephen Smale: University of California, Berkeley Robert L. Devaney: Boston University

Table of Contents

Preface	x
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	(2)
1.4 Periodic Harvesting and Periodic Solutions	9	(3)
1.5 Computing the Poincaré Map	12	(3)
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	(14)
4.1 The Trace-Determinant Plane	61	(3)
4.2 Dynamical Classification	64	(7)
4.3 Exploration: A 3D Parameter Space	71	(4)
CHAPTER 5 Higher Dimensional Linear Algebra	75	(32)
5.1 Preliminaries from Linear Algebra	75	(8)
5.2 Eigenvalues and Eigenvectors	83	(3)
5.3 Complex Eigenvalues	86	(3)
5.4 Bases and Subspaces	89	(6)
5.5 Repeated Eigenvalues	95	(6)
5.6 Genericity	101	(6)
CHAPTER 6 Higher Dimensional Linear Systems	107	(32)
6.1 Distinct Eigenvalues	107	(7)
6.2 Harmonic Oscillators	114	(5)
6.3 Repeated Eigenvalues	119	(4)
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	(6)
CHAPTER 8 Equilibria in Nonlinear Systems	159	(30)
8.1 Some Illustrative Examples	159	(6)
8.2 Nonlinear Sinks and Sources	165	(3)
8.3 Saddles	168	(6)
8.4 Stability	174	(2)
8.5 Bifurcations	176	(6)
8.6 Exploration: Complex Vector Fields	182	(7)
CHAPTER 9 Global Nonlinear Techniques	189	(26)
9.1 Nullclines	189	(5)
9.2 Stability of Equilibria	194	(9)
9.3 Gradient Systems	203	(4)
9.4 Hamiltonian Systems	207	(3)
9.5 Exploration: The Pendulum with Constant Forcing	210	(5)
CHAPTER 10 Closed Orbits and Limit Sets	215	(20)
10.1 Limit Sets	215	(3)
10.2 Local Sections and Flow Boxes	218	(2)
10.3 The Poincaré Map	220	(2)
10.4 Monotone Sequences in Planar Dynamical Systems	222	(3)
10.5 The Poincaré-Bendixson Theorem	225	(2)
10.6 Applications of Poincaré-Bendixson	227	(3)
10.7 Exploration: Chemical Reactions That Oscillate	230	(5)
CHAPTER 11 Applications in Biology	235	(22)
11.1 Infectious Diseases	235	(4)
11.2 Predator/Prey Systems	239	(7)
11.3 Competitive Species	246	(6)
11.4 Exploration: Competition and Harvesting	252	(5)
CHAPTER 12 Applications in Circuit Theory	257	(20)
12.1 An RLC Circuit	257	(4)
12.2 The Lienard Equation	261	(1)
12.3 The van der Pol Equation	262	(8)
12.4 A Hopf Bifurcation	270	(2)
12.5 Exploration: Neurodynamics	272	(5)
CHAPTER 13 Applications in Mechanics	277	(26)
13.1 Newton's Second Law	277	(3)
13.2 Conservative Systems	280	(1)
13.3 Central Force Fields	281	(4)
13.4 The Newtonian Central Force System	285	(4)
13.5 Kepler's First Law	289	(3)
13.6 The Two-Body Problem	292	(1)
13.7 Blowing Up the Singularity	293	(4)
13.8 Exploration: Other Central Force Problems	297	(1)
13.9 Exploration: Classical Limits of Quantum Mechanical Systems	298	(5)
CHAPTER 14 The Lorenz System	303	(24)
14.1 Introduction to the Lorenz System	304	(2)
14.2 Elementary Properties of the Lorenz System	306	(4)
14.3 The Lorenz Attractor	310	(4)
14.4 A Model for the Lorenz Attractor	314	(5)
14.5 The Chaotic Attractor	319	(5)
14.6 Exploration: The Rössler Attractor	324	(3)
CHAPTER 15 Discrete Dynamital Systems	327	(32)
15.1 Introduction to Discrete Dynamital Systems	327	(5)
15.2 Bifurcations	332	(3)
15.3 The Discrete Logistic Model	335	(2)
15.4 Chaos	337	(5)
15.5 Symbolic Dynamics	342	(5)
15.6 The Shift Map	347	(2)
15.7 The Cantor Middle-Thirds Set	349	(3)
15.8 Exploration: Cubic Chaos	352	(1)
15.9 Exploration: The Orbit Diagram	353	(6)
CHAPTER 16 Homoclinic Phenomena	359	(24)
16.1 The Shil'nikov System	359	(7)
16.2 The Horseshoe Map	366	(6)
16.3 The Double Scroll Attractor	372	(3)
16.4 Homoclinic Bifurcations	375	(4)
16.5 Exploration: The Chua Circuit	379	(4)
CHAPTER 17 Existence and Uniqueness Revisited	383	(24)
17.1 The Existence and Uniqueness Theorem	383	(2)
17.2 Proof of Existence and Uniqueness	385	(7)
17.3 Continuous Dependence on Initial Conditions	392	(3)
17.4 Extending Solutions	395	(3)
17.5 Nonautonomous Systems	398	(2)
17.6 Differentiability of the Flow	400	(7)
Bibliography	407	(4)
Index	411