Computer Algebra Recipes

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8.1. THE POINCAR ESECTION 321 for the ith solution, the evaluations being carried out in the following do loop. > for i from 1 to N1 do xx[i]:=X(i); yy[i]:=Y(i); end do: An operator Gr is created to graph a phase-plane point, represented by a size-16 blue cross, at time t = nT0 for the ith amplitude. > Gr:=(i,n)->pointplot(f[yy[i](n*T[0]),xx[i](n*T[0])]g color=blue,symbol=CROSS,symbolsize=16): Using this graphing operator in the following loop generates the Poincare section for each of the N1 = 4 amplitude values. To eliminate the transient part of the solution, a certain number of initial points must be removed in each plot. This number will vary for each numerical run and must usually be determined by trial and error. Here, the ¯rst 24 points have been removed. Since N2 =250, each plot still contains 225 points. A suitable viewing box is selected, which is the same for all four plots, and labels added and the minimum number of tick marks controlled. > for i from 1 to N1 do > display([seq(Gr(i,n),n=25..N2)],axes=boxed,view= [-0.4..0.8,-1.4..1.4],labels=["y","x"],tickmarks=[3,3]); > end do; 1 x –1 0 y 0.5 1 x –1 0 y 0.5 Figure 8.1: Period-1 (left ¯gure) and period-2 (right) Poincare sections. The plot on the left of Figure 8.1 corresponds to F1 =0:325, the one on the right to F2 =0:35. Despite the fact that 225 points were plotted, we see only a single cross in the F1 =0:325 graph, indicating that the system has settled down to a period-1 solution. For F2 =0:35, the ODE system oscillates back and forth between the two crosses, the Poincare section being characteristic of a period-2 solution. The observed periodicities agree with the results in Section 1.2.1. For F3 =0:356, Figure 8.2 indicates a period-4 solution, again agreeing with our earlier conclusion about the periodicity for this forcing amplitude. So it is clear that the Poincare section approach is a useful graphical tool for interpreting the periodicity of driven oscillator systems.
322 CHAPTER 8. NONLINEAR DIAGNOSTIC TOOLS 1 x –1 0 y 0.5 Figure 8.2: Period-4 Poincare section. Finally, for F4 =0:42, we had previously observed a localized chaotic trajectory, characteristic of a strange attractor. The corresponding \strange" Poincare section is shown in Figure 8.3. x 1 0 –1 0 y 0.5 Figure 8.3: Chaotic \rattler." Strange attractors are often given colorful or descriptive names. What name shallwegivetothechaoticPoincare section obtained above? This depends on the reader's experience and imagination. To coauthor Richard, it reminds him of an incident that occurred while he was hiking with his family in the Superstition Mountains of southern Arizona. While scrambling down a rocky scree slope, he had a feeling of impending chaos when his youngest daughter nearly stumbled into a coiled rattlesnake. To the author, this particular Poincare sec-
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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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Page 357 and 358: 356 BIBLIOGRAPHY [Dav62] H. T. Davi
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Page 361 and 362: Index acoustical waveguide, 261 ade
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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