Differential Equations, Dynamical Systems, and an Introduction to ...

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for all t. Suppose there are const<strong>an</strong>ts p, q such that f (t, p) > 0, f (t, q) < 0 Exercises 19 for all t. Prove that there is a periodic solution x(t) for this equation with p
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Page 2 and 3: DIFFERENTIAL EQUATIONS, DYNAMICAL S
Page 4 and 5: DIFFERENTIAL EQUATIONS, DYNAMICAL S
Page 6 and 7: Preface x Contents CHAPTER 1 First-
Page 8 and 9: CHAPTER 9 Global Nonlinear Techniqu
Page 10 and 11: Bibliography 407 Index 411 17.3 Con
Page 12 and 13: Preface xi the 1970s. We have accor
Page 14 and 15: Preface xiii We then return to nonl
Page 16 and 17: 1 First-Order Equations The purpose
Page 18 and 19: 1.1 The Simplest Example 3 qualitat
Page 20 and 21: Therefore the logistic equation red
Page 22 and 23: 0.8 1.3 Constant Harvesting and Bif
Page 24 and 25: x 1.4 Periodic Harvesting and Perio
Page 26 and 27: 1.4 Periodic Harvesting and Periodi
Page 28 and 29: 1.5 Computing the Poincaré Map 13
Page 30 and 31: 1 1.6 Exploration: A Two-Parameter
Page 32 and 33: Exercises 17 (a) Sketch the phase l
Page 36 and 37: 2 Planar Linear Systems In this cha
Page 38 and 39: 2.1 Second-Order Differential Equat
Page 40 and 41: Example. The curve x(t) = y(t) fo
Page 42 and 43: 2.3 Preliminaries from Algebra 27 w
Page 44 and 45: (1/2, 1/2), because 1 0 0 1 = 1
Page 46 and 47: 2.5 Eigenvalues and Eigenvectors 31
Page 48 and 49: 2.6 Solving Linear Systems 33 or
Page 50 and 51: We therefore have shown the followi
Page 52 and 53: 1. 2. 3. 4. Figure 2.2 Match these
Page 54 and 55: 3 Phase Portraits for Planar System
Page 56 and 57: 3.1 Real Distinct Eigenvalues 41 sy
Page 58 and 59: 3.1 Real Distinct Eigenvalues 43 (a
Page 60 and 61: to λ = iβ. We therefore solve
Page 62 and 63: 3.3 Repeated Eigenvalues 47 Example
Page 64 and 65: Thus, if y = 0, we must have y(t) =
Page 66 and 67: 3.4 Changing Coordinates 51 serves
Page 68 and 69: Finally, we compute T −1 AT = 3.4
Page 70 and 71: 3.4 Changing Coordinates 55 α > 0.
Page 72 and 73: Exercises 57 for some constants μ,
Page 74 and 75: Exercises 59 (a) For which values o
Page 76 and 77: 4 Classification of Planar Systems
Page 78 and 79: Thus we have 4.1 The Trace-Determin
Page 80 and 81: 4.2 Dynamical Classification 65 We
Page 82 and 83: 4.2 Dynamical Classification 67 2.
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t A r (x, y) S 1 4.2 Dynamical Clas
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4.3 Exploration: A 3D Parameter Spa
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Exercises 73 6. Prove that any two
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5 Higher Dimensional Linear Algebra
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5.1 Preliminaries from Linear Algeb
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5.2 Eigenvalues and Eigenvectors 83
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5.2 Eigenvalues and Eigenvectors 85
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Let V denote the complex conjugate
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Therefore the matrix associated to
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By the previous observation, the sy
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5.4 Bases and Subspaces 93 Hence th
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5.5 Repeated Eigenvalues 95 linearl
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5.5 Repeated Eigenvalues 97 A −
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5.5 Repeated Eigenvalues 99 an exer
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The characteristic equation for A i
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5.6 Genericity 103 that we can find
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5. Put the following matrices in ca
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6 Higher Dimensional Linear Systems
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x z y 6.1 Distinct Eigenvalues 109
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where αj βj Bj = 6.1 Distinct Ei
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Figure 6.4 A spiral saddle in canon
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6.2 Harmonic Oscillators 115 We cou
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Restricted to this torus, the equat
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6.3 Repeated Eigenvalues 119 in the
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Solving as above, we find Altogethe
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6.4 The Exponential of a Matrix 6.4
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6.4 The Exponential of a Matrix 125
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and ⎛ m αm = ⎝ j=0 6.4 The Exp
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6.4 The Exponential of a Matrix 129
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6.5 Nonautonomous Linear Systems 13
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6.5 Nonautonomous Linear Systems 13
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Exercises 135 and using the fact th
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k 1 m 1 k 2 m 2 k 1 Figure 6.10 A c
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7 Nonlinear Systems In this chapter
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7.1 Dynamical Systems 141 “trajec
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7.2 The Existence and Uniqueness Th
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7.2 The Existence and Uniqueness Th
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7.3 Continuous Dependence of Soluti
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7.4 The Variational Equation 149 Th
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nonautonomous linear equation 7.4 T
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7.5 Exploration: Numerical Methods
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7.5 Exploration: Numerical Methods
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Exercises 157 8. Construct an examp
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8 Equilibria in Nonlinear Systems T
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8.1 Some Illustrative Examples 161
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8.1 Some Illustrative Examples 163
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8.2 Nonlinear Sinks and Sources 165
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The linearized system is now given
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8.3 Saddles 169 satisfies −μ < 0
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8.3 Saddles 171 Let C + M denote th
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8.3 Saddles 173 if η(0) =−1, the
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x W s (0) zW u (0) Figure 8.5 The p
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8.5 Bifurcations 177 has a single e
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x x 0 Figure 8.6 The bifurcation di
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8.5 Bifurcations 181 We denote thes
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8.6 Exploration: Complex Vector Fie
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Exercises 185 (b) Describe the phas
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Exercises 187 12. In the definition
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9 Global Nonlinear Techniques In th
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A D C (a) B 9.1 Nullclines 191 (b)
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(a) (b) 9.1 Nullclines 193 Figure 9
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containing X ∗ . Suppose further
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9.2 Stability of Equilibria 197 For
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L 1 (c 3) 9.2 Stability of Equilibr
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9.2 Stability of Equilibria 201 Bef
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9.3 Gradient Systems 203 First, φt
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9.3 Gradient Systems 205 X ∈ V
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9.4 Hamiltonian Systems 207 equilib
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Thus we have: 9.4 Hamiltonian Syste
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Exercises 211 where we assume that
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(c) y sin x (d) 2x 2 − 2xy + 5y 2
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10 Closed Orbits and Limit Sets In
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(0, 0) 10.1 Limit Sets 217 (o, o) F
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10.2 Local Sections and Flow Boxes
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10.3 The Poincaré Map 221 entirely
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X 0 X 2 X 1 10.4 Monotone Sequences
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Hence there is a sequence 10.5 The
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10.6 Applications of Poincaré-Bend
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10.6 Applications of Poincaré-Bend
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Exercises 231 equilibrium. The conc
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A Figure 10.12 The region A is posi
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11 Applications in Biology In this
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Hence we have I 11.1 Infectious Dis
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11.2 Predator/Prey Systems 239 Biol
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y 11.2 Predator/Prey Systems 241 xc
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11.2 Predator/Prey Systems 243 xc/d
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11.2 Predator/Prey Systems 245 and
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Hence ∂M ∂y < 0 and 11.3 Compet
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m S B R Q Figure 11.9 The basic reg
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l m P 11.3 Competitive Species 251
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consider the system x ′ = x(1 −
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Exercises 255 7. Two species x, y a
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12 Applications in Circuit Theory I
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12.1 An RLC Circuit 259 the voltage
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coordinates (iL, vC): 12.2 The Lien
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12.3 The van der Pol Equation 263 d
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g 12.3 The van der Pol Equation 26
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12.3 The van der Pol Equation 267 W
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12.3 The van der Pol Equation 269 y
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and the eigenvalues are λ± = 1
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Exercises 273 explicitly solving th
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R L C Figure 12.11 Exercises 275 ve
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13 Applications in Mechanics We tur
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13.1 Newton’s Second Law 279 for
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13.3 Central Force Fields 281 in me
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13.3 Central Force Fields 283 Consi
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13.4 The Newtonian Central Force Sy
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13.4 The Newtonian Central Force Sy
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since, when r =α, we have Figure 1
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Finally, 13.5 Kepler’s First Law
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13.7 Blowing Up the Singularity 293
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13.7 Blowing Up the Singularity 295
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u r u h 13.8 Exploration: Other Cen
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y ′′ = −y (μx 2 +y 2 ) 3/2 E
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and grad j (U ) = ∂U ∂x j 1 ,
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14 The Lorenz System So far, in all
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z x P 1 14.1 Introduction to the Lo
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14.2 Elementary Properties of the L
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the ellipsoid 14.2 Elementary Prope
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U Figure 14.3 The interval on the x
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Q 14.3 The Lorenz Attractor 313 (0
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x 1 z 14.4 A Model for the Lorenz A
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14.4 A Model for the Lorenz Attract
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14.5 The Chaotic Attractor 319 sinc
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14.5 The Chaotic Attractor 321 prei
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W yc 0 W 1 Q1 14.5 The Chaotic Attr
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Exercises 325 viewpoints in R 3 . W
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15 Discrete Dynamical Systems Our g
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15.1 Introduction to Discrete Dynam
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15.1 Introduction to Discrete Dynam
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15.2 Bifurcations 333 c0.35 c0.25 c
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15.3 The Discrete Logistic Model 33
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15.4 Chaos 337 1/2 1/2 1/2 y 0 y 1
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15.4 Chaos 339 Figure 15.9 The grap
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15.4 Chaos 341 f m (x1) ∈ h −1
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A 1 A 0 (a) A 1 15.5 Symbolic Dynam
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15.5 Symbolic Dynamics 345 The impo
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15.6 The Shift Map 347 This proves
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15.7 The Cantor Middle-Thirds Set 3
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15.7 The Cantor Middle-Thirds Set 3
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15.9 Exploration: The Orbit Diagram
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Exercises 355 (b) List all points w
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Exercises 357 20. Recall that compr
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16 Homoclinic Phenomena In this cha
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16.1 The Shil’nikov System 361 le
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16.1 The Shil’nikov System 363 Th
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16.1 The Shil’nikov System 365 fo
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D 2 S D 1 F 16.2 The Horseshoe Map
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Let Figure 16.8 The second iterate
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We define the stable set of X to be
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f 16.3 The Double Scroll Attractor
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16.4 Homoclinic Bifurcations 375 Th
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where ɛ > 0. As in Section 16.1, w
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16.5 Exploration: The Chua Circuit
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Exercises 381 homoclinic to (0). Pr
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17 Existence and Uniqueness Revisit
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17.2 Proof of Existence and Uniquen
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17.3 Continuous Dependence on Initi
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17.4 Extending Solutions 395 This i
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17.4 Extending Solutions 397 The fo
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17.5 Nonautonomous Systems 399 As u
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17.6 Differentiability of the Flow
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17.6 Differentiability of the Flow
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(c) f (x)=1/x,1≤x ≤∞ (d) f (x
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Bibliography 1. Abraham, R., and Ma
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References 409 38. Rudin, W. Princi
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Index Page numbers followed by “f
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equations algebra, 26-29 Cauchy-Rie
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mechanics classical, 277 conservati
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spiral sink, 47, 48f spiral source,
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