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

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Therefore the logistic equation reduces to 1.2 The Logistic Population Model 5 x ′ = fa(x) = ax(1 − x). This is an example of a first-order, autonomous, nonlinear differential equation. It is first order since only the first derivative of x appears in the equation. It is autonomous since the right-hand side of the equation depends on x alone, not on time t. And it is nonlinear since fa(x) is a nonlinear function of x. The previous example, x ′ = ax, is a first-order, autonomous, linear differential equation. The solution of the logistic differential equation is easily found by the triedand-true calculus method of separation and integration: dx x(1 − x) = adt. The method of partial fractions allows us to rewrite the left integral as 1 1 + dx. x 1 − x Integrating both sides and then solving for x yields x(t) = Keat 1 + Ke at where K is the arbitrary constant that arises from integration. Evaluating this expression at t = 0 and solving for K gives K = x(0) 1 − x(0) . Using this, we may rewrite this solution as x(0)eat . 1 − x(0) + x(0)eat So this solution is valid for any initial population x(0). When x(0) = 1, we have an equilibrium solution, since x(t) reduces to x(t) ≡ 1. Similarly, x(t) ≡ 0 is also an equilibrium solution. Thus we have “existence” of solutions for the logistic differential equation. We have no guarantee that these are all of the solutions to this equation at this stage; we will return to this issue when we discuss the existence and uniqueness problem for differential equations in Chapter 7.
6 Chapter 1 First-Order Equations Figure 1.3 The slope field, solution graphs, and phase line for x ′ = ax(1 – x). x t x1 x0 To get a qualitative feeling for the behavior of solutions, we sketch the slope field for this equation. The right-hand side of the differential equation determines the slope of the graph of any solution at each time t. Hence we may plot little slope lines in the tx–plane as in Figure 1.3, with the slope of the line at (t, x) given by the quantity ax(1 − x). Our solutions must therefore have graphs that are everywhere tangent to this slope field. From these graphs, we see immediately that, in agreement with our assumptions, all solutions for which x(0) > 0 tend to the ideal population x(t) ≡ 1. For x(0) < 0, solutions tend to −∞, although these solutions are irrelevant in the context of a population model. Note that we can also read this behavior from the graph of the function fa(x) = ax(1 − x). This graph, displayed in Figure 1.4, crosses the x-axis at the two points x = 0 and x = 1, so these represent our equilibrium points. When 0
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 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 34 and 35: for all t. Suppose there are consta
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
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3.4 Changing Coordinates 55 α > 0.
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Exercises 57 for some constants μ,
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Exercises 59 (a) For which values o
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4 Classification of Planar Systems
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Thus we have 4.1 The Trace-Determin
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4.2 Dynamical Classification 65 We
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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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