Computer Algebra Recipes

More documents

Recommendations

Info

8.1. THE POINCAR ESECTION 327 The output informs us of the input H (energy) value, the initial conditions, the maximum percentage deviation from the input H value, and the computer time taken to produce a plot of the trajectory, which is shown in Figure 8.5. The trajectory executes quasiperiodic motion on the surface of a twisted torus, referred to as the Kolmogorov{Moser{Arnold (KAM) torus. The Poincare section for the q1 = 0 plane shown in Figure 8.5 can be obtained with the following poincare command. In addition to the same information as before, the number of points (96 here) in the q2 -p2 plane where the trajectory crosses the plane is also given. > poincare(H,t=0..300,ic,stepsize=.05,iterations=3, scene=[q2,p2]); H = :62500000e ¡ 1 ; Initial conditions :; t=0:; p1 =0:2667708130; p2 = ¡0:05; q1 = ¡0:1; q2 = ¡0:2 Number of points found crossing the (q2; p2) plane : 96 Maximum H deviation : :9800000000e ¡ 6 % p2 0.2 0.1 0 –0.1 –0.2 Time consumed : 1 seconds H = .6250000000e–1; –0.3 –0.2 –0.1 0 0.1 0.2 0.3 q2 Figure 8.6: Poincare section for E =1=16. The Poincare section can be qualitatively understood. One can think of the KAM torus as a twisted \donut," with the trajectories con¯ned to the donut's surface. The con¯guration of points in the Poincare section resembles two distorted ellipses that one might expect from \slicing" the donut through its cross section with the q1 =0plane. If the total energy is increased to 1 8 and the viewing box increased in size, the \chaotic" trajectory shown in Figure 8.7 results.
328 CHAPTER 8. NONLINEAR DIAGNOSTIC TOOLS q1 0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –0.5 0 p2 H = .1250000000; 0.6 0.4 0.2 0 –0.2 q2 –0.4 –0.6 Figure 8.7: Chaotic trajectory for E = 1 8 . A great deal of mathematics research has gone into understanding the onset of chaos for the nonlinear Henon{Heiles ODE system. The interested reader is referred to the nonlinear dynamics texts by Jackson [Jac90] and by Hilborn [Hil94]. PROBLEMS: Problem 8-6: A di®erent potential Replace the potential in the text recipe with V = 1 2 q2 1 1 + 2 q2 2 + q4 1 q2 ¡ 1 4 q3 2 : (a) Execute the modi¯ed recipe with E =1=16 and initial conditions as in the text recipe. Discuss the resulting plots. (b) Explore other initial conditions for the same total energy as in part (a). Discuss the results. (c) Explore what happens as the energy is increased with the same initial conditions as in part (a). Discuss the results. (d) Explore other potential energy functions and discuss the results. Problem 8-7: Toda potential The Toda potential [Jac90] is given by V = 1 μ e 3 (q2 + p 3 q1) (q2 + e ¡ p 3 q1) (¡2 q2) + e ¡ 1: (a) Create two- and three-dimensional contour plots of V , choosing suitable potential energy contours and viewing ranges.
Page 1 and 2:
Richard H. Enns George C. McGuire C
Page 3 and 4:
PREFACE A computer algebra system (
Page 5 and 6:
viii CONTENTS 2.3.1 Finite Di®eren
Page 7 and 8:
x CONTENTS 8.3 The Bifurcation Diag
Page 9 and 10:
2 INTRODUCTION B. Computer Algebra
Page 11 and 12:
4 INTRODUCTION (a) At what angle Á
Page 13 and 14:
6 INTRODUCTION The negative answer
Page 15 and 16:
8 INTRODUCTION D. Maple Help We tea
Page 17 and 18:
Part I THE APPETIZERS A man ceases
Page 19 and 20:
14 CHAPTER 1. PHASE-PLANE PORTRAITS
Page 21 and 22:
Page 23 and 24:
Page 25 and 26:
Page 27 and 28:
Page 29 and 30:
Page 31 and 32:
Page 33 and 34:
Page 35 and 36:
Page 37 and 38:
Page 39 and 40:
Page 41 and 42:
Page 43 and 44:
Page 45 and 46:
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
Page 53 and 54:
48 CHAPTER 2. PHASE-PLANE ANALYSIS
Page 55 and 56:
Page 57 and 58:
Page 59 and 60:
Page 61 and 62:
Page 63 and 64:
Page 65 and 66:
Page 67 and 68:
Page 69 and 70:
Page 71 and 72:
Page 73 and 74:
Page 75 and 76:
Page 77 and 78:
Page 79 and 80:
Page 81 and 82:
Page 83 and 84:
Page 85 and 86:
Page 87 and 88:
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
Page 95 and 96:
Page 97 and 98:
Page 99 and 100:
Page 101 and 102:
Page 103 and 104:
Page 105 and 106:
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Chapter 3 Linear ODE Models Among a
Page 115 and 116:
3.1. FIRST-ORDER MODELS 111 Therate
Page 117 and 118:
3.1. FIRST-ORDER MODELS 113 Pressur
Page 119 and 120:
3.1. FIRST-ORDER MODELS 115 3.1.2 G
Page 121 and 122:
3.1. FIRST-ORDER MODELS 117 Evil Kn
Page 123 and 124:
3.2. SECOND-ORDER MODELS 119 > tf:=
Page 125 and 126:
3.2. SECOND-ORDER MODELS 121 3.2.2
Page 127 and 128:
3.2. SECOND-ORDER MODELS 123 Loadin
Page 129 and 130:
3.2. SECOND-ORDER MODELS 125 0.4 0.
Page 131 and 132:
3.2. SECOND-ORDER MODELS 127 On ent
Page 133 and 134:
3.2. SECOND-ORDER MODELS 129 Using
Page 135 and 136:
3.3. SPECIAL FUNCTION MODELS 131 3.
Page 137 and 138:
3.3. SPECIAL FUNCTION MODELS 133 1
Page 139 and 140:
3.3. SPECIAL FUNCTION MODELS 135 Th
Page 141 and 142:
3.3. SPECIAL FUNCTION MODELS 137 3.
Page 143 and 144:
3.3. SPECIAL FUNCTION MODELS 139 th
Page 145 and 146:
3.3. SPECIAL FUNCTION MODELS 141 PR
Page 147 and 148:
3.3. SPECIAL FUNCTION MODELS 143 ¡
Page 149 and 150:
3.3. SPECIAL FUNCTION MODELS 145 μ
Page 151 and 152:
3.3. SPECIAL FUNCTION MODELS 147 >
Page 153 and 154:
150 CHAPTER 4. NONLINEAR ODE MODELS
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
Page 183 and 184:
Page 185 and 186:
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
Page 193 and 194:
Page 195 and 196:
Page 197 and 198:
Page 199 and 200:
Page 201 and 202:
Page 203 and 204:
Page 205 and 206:
Page 207 and 208:
Page 209 and 210:
Page 211 and 212:
208 CHAPTER 5. LINEAR PDE MODELS. P
Page 213 and 214:
Page 215 and 216:
Page 217 and 218:
Page 219 and 220:
Page 221 and 222:
Page 223 and 224:
Page 225 and 226:
Page 227 and 228:
Page 229 and 230:
Page 231 and 232:
Page 233 and 234:
Page 235 and 236:
Page 237 and 238:
Page 239 and 240:
Page 241 and 242:
Page 243 and 244:
Page 245 and 246:
Page 247 and 248:
Page 249 and 250:
Chapter 6 Linear PDE Models. Part 2
Page 251 and 252:
6.1. WAVE EQUATION MODELS 249 > sol
Page 253 and 254:
6.1. WAVE EQUATION MODELS 251 6.1.2
Page 255 and 256:
6.1. WAVE EQUATION MODELS 253 The p
Page 257 and 258:
6.1. WAVE EQUATION MODELS 255 Recog
Page 259 and 260:
6.1. WAVE EQUATION MODELS 257 PROBL
Page 261 and 262:
6.1. WAVE EQUATION MODELS 259 Then
Page 263 and 264:
6.2. SEMI-INFINITE AND INFINITE DOM
Page 265 and 266:
Page 267 and 268:
Page 269 and 270:
Page 271 and 272:
Page 273 and 274:
Page 275 and 276:
Page 277 and 278: 6.3. NUMERICAL SIMULATION OF PDES 2
Page 287 and 288: Part III THE DESSERTS The way a chi
Page 289 and 290: 288 CHAPTER 7. THE HUNT FOR SOLITON
Page 321 and 322: 320 CHAPTER 8. NONLINEAR DIAGNOSTIC
Page 327: 326 CHAPTER 8. NONLINEAR DIAGNOSTIC
Page 357 and 358: 356 BIBLIOGRAPHY [Dav62] H. T. Davi
Page 359 and 360: 358 BIBLIOGRAPHY [Nyq28] H. Nyquist
Page 361 and 362: Index acoustical waveguide, 261 ade
Page 363 and 364: INDEX 363 eigenvalue, 130 Eiram Eir
Page 365 and 366: INDEX 365 Help, Full Text Search, 8
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
show all

Computer Algebra Recipes

Create successful ePaper yourself

Delete template?

Save as template?