Computer Algebra Recipes

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8.3. THE BIFURCATION DIAGRAM 341 PROBLEMS: Problem 8-16: Finer structure For the logistic map, produce the bifurcation diagram for the region a =3:54 to a =3:6, taking x0 =0:2, and dividing the a interval into 100 steps. Summarize the various periodic solutions that you observe. Problem 8-17: More ¯ne structure Explore the periodic window in the vicinity of a =3:8and report on what periodicities you observe at each a value sampled. Problem 8-18: Cubic map Produce a bifurcation diagram for the cubic map xn+1 = axn ¡ x 3 n over a suitable range of the parameter a. Determine the values of a in the diagram at which the periodicity changes. Problem 8-19: Quartic map With x0 =0:2, produce the bifurcation diagram for the quartic map xn+1 = axn (1 ¡ x 3 n ) over the range a =1:5 toa =2:0, taking as small an a step size as you can. Summarize the behavior of the map as a varies over the speci¯ed range. Problem 8-20: The tent map Produce the bifurcation diagram for the tent map xn+1 =2axn; 0
342 CHAPTER 8. NONLINEAR DIAGNOSTIC TOOLS 8.4 The Lyapunov Exponent Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov (1857{1918), the Lyapunov exponent ¸ is a measure of the very sensitive dependence on initial conditions that is characteristic of chaotic behavior in nonlinear di®erence equations (maps) and ODEs. The discussion that follows is for maps, a similar analysis applying to ODEs. Consider a one-dimensional map xn+1 = f(xn); (8.11) with f some speci¯ed functional form, and let x0 and y0 be two initial values very close to each other. In phase space, they would be represented by two very close points. After n iterations, the values of xn and yn will be given by xn = f (n) (x0); yn = f (n) (y0); (8.12) where f (n) denotes the nth iteration of the map. Because the chaotic regime is typically characterized by an extreme sensitivity to initial conditions, for a chaotic situation nearby initial points will rapidly separate. On the other hand, periodic solutions are insensitive to initial conditions, and nearby initial points rapidly converge. This suggests that for su±ciently large n, one might assume an approximately exponential dependence on n of the separation distance, viz., jxn ¡ ynj = jx0 ¡ y0je ¸n ; (8.13) with ¸>0 for the chaotic situation and ¸
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Richard H. Enns George C. McGuire C
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PREFACE A computer algebra system (
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viii CONTENTS 2.3.1 Finite Di®eren
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x CONTENTS 8.3 The Bifurcation Diag
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2 INTRODUCTION B. Computer Algebra
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4 INTRODUCTION (a) At what angle Á
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6 INTRODUCTION The negative answer
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8 INTRODUCTION D. Maple Help We tea
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Part I THE APPETIZERS A man ceases
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14 CHAPTER 1. PHASE-PLANE PORTRAITS
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48 CHAPTER 2. PHASE-PLANE ANALYSIS
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Chapter 3 Linear ODE Models Among a
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3.1. FIRST-ORDER MODELS 111 Therate
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3.1. FIRST-ORDER MODELS 113 Pressur
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3.1. FIRST-ORDER MODELS 115 3.1.2 G
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3.1. FIRST-ORDER MODELS 117 Evil Kn
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3.2. SECOND-ORDER MODELS 119 > tf:=
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3.2. SECOND-ORDER MODELS 121 3.2.2
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3.2. SECOND-ORDER MODELS 123 Loadin
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3.2. SECOND-ORDER MODELS 125 0.4 0.
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3.2. SECOND-ORDER MODELS 127 On ent
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3.2. SECOND-ORDER MODELS 129 Using
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3.3. SPECIAL FUNCTION MODELS 131 3.
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3.3. SPECIAL FUNCTION MODELS 133 1
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3.3. SPECIAL FUNCTION MODELS 135 Th
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3.3. SPECIAL FUNCTION MODELS 137 3.
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3.3. SPECIAL FUNCTION MODELS 139 th
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3.3. SPECIAL FUNCTION MODELS 141 PR
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3.3. SPECIAL FUNCTION MODELS 143 ¡
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3.3. SPECIAL FUNCTION MODELS 145 μ
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3.3. SPECIAL FUNCTION MODELS 147 >
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150 CHAPTER 4. NONLINEAR ODE MODELS
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208 CHAPTER 5. LINEAR PDE MODELS. P
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Chapter 6 Linear PDE Models. Part 2
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6.1. WAVE EQUATION MODELS 249 > sol
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6.1. WAVE EQUATION MODELS 251 6.1.2
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6.1. WAVE EQUATION MODELS 253 The p
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6.1. WAVE EQUATION MODELS 255 Recog
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6.1. WAVE EQUATION MODELS 257 PROBL
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6.1. WAVE EQUATION MODELS 259 Then
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6.2. SEMI-INFINITE AND INFINITE DOM
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6.3. NUMERICAL SIMULATION OF PDES 2
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Part III THE DESSERTS The way a chi
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288 CHAPTER 7. THE HUNT FOR SOLITON
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Page 357 and 358: 356 BIBLIOGRAPHY [Dav62] H. T. Davi
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Page 363 and 364: INDEX 363 eigenvalue, 130 Eiram Eir
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Page 367 and 368: INDEX 367 laplace, 121,267,268 lhs,
Page 369 and 370: INDEX 369 ¯sh harvesting, 100 ¯xe
Page 371 and 372: INDEX 371 quartic map, 345 Queen Di
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Computer Algebra Recipes

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