Computer Algebra Recipes

More documents

Recommendations

Info

1.2. THREE-DIMENSIONAL AUTONOMOUS SYSTEMS 45 trajectory that results if the initial condition is near the other ¯xed point is left as a problem. > ic:=[x(0)=0.1,y(0)=0.1,z(0)=0.1]: To enter the relevant ODEs, the dependent variables x, y, andz must be made time-dependent. This is done in the following command line. > vars:=fx=x(t),y=y(t),z=z(t)g: Substituting the variables into P , Q, andR, and equating to dx=dt, dy=dt, and dz=dt, yields the RÄossler system of ODEs. > sys:=diff(x(t),t)=subs(vars,P),diff(y(t),t)=subs(vars,Q), diff(z(t),t)=subs(vars,R); sys := d d x (t) =¡y(t) ¡ z(t); y(t) =x(t)+0:2y(t); dt dt d z (t) =0:2+z (t)(x (t) ¡ 5:7) dt Choosing the option scene=[x,y,z] in the DEplot3d command, we plot the trajectory in x-y-z space over the time interval t = 0 to 150, subject to the given initial condition. The step size is taken to be 0.01 in order to obtain a smooth curve. The trajectory is colored with the zhue shading option, and a particular orientation of the viewing box is chosen. > DEplot3d([sys],[x(t),y(t),z(t)],t=0..150,[ic],scene=[x,y,z], stepsize=0.01,shading=zhue,orientation=[-120,60], tickmarks=[3,3,3],thickness=1); 20 z 10 5 y 0 –5 –10 Figure 1.18: RÄossler's strange attractor. 0 x 10
46 CHAPTER 1. PHASE-PLANE PORTRAITS The resulting picture is shown in Figure 1.18. The trajectory unwinds in a spiral fashion from its starting point near the origin, indicating that the ¯xed point at the origin is an unstable focal point. As time progresses, the trajectory is attracted to a localized region of the phase space where it traces out a never-repeating (chaotic) path. This is an example of a strange attractor, the word strange being introduced historically because it was not like a \normal" attractor (e.g., a focal point). Strange attractors also have the property that they have noninteger, or fractal, dimensions. If you wish to learn more about strange attractors and fractal patterns, this topic is discussed at length in the Introductory Guide. PROBLEMS: Problem 1-21: Second ¯xed point Run the text recipe with an initial condition near the second ¯xed point. What is the probable nature of this ¯xed point? What is the nature of the resulting trajectory as time progresses? Problem 1-22: Varying c Holding all other parameters as in the text recipe, explore the behavior of the RÄossler system as the coe±cient c is varied. Interpret the results.
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 49: 44 CHAPTER 1. PHASE-PLANE PORTRAITS
Page 53 and 54: 48 CHAPTER 2. PHASE-PLANE ANALYSIS
Page 101 and 102:
96 CHAPTER 2. PHASE-PLANE ANALYSIS
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 279 and 280:
Page 281 and 282:
Page 283 and 284:
Page 285 and 286:
Page 287 and 288:
Part III THE DESSERTS The way a chi
Page 289 and 290:
288 CHAPTER 7. THE HUNT FOR SOLITON
Page 291 and 292:
Page 293 and 294:
Page 295 and 296:
Page 297 and 298:
Page 299 and 300:
Page 301 and 302:
Page 303 and 304:
Page 305 and 306:
Page 307 and 308:
Page 309 and 310:
Page 311 and 312:
Page 313 and 314:
Page 315 and 316:
Page 317 and 318:
Page 319 and 320:
Page 321 and 322:
320 CHAPTER 8. NONLINEAR DIAGNOSTIC
Page 323 and 324:
Page 325 and 326:
Page 327 and 328:
Page 329 and 330:
Page 331 and 332:
Page 333 and 334:
Page 335 and 336:
Page 337 and 338:
Page 339 and 340:
Page 341 and 342:
Page 343 and 344:
Page 345 and 346:
Page 347 and 348:
Page 349 and 350:
Page 351 and 352:
Page 353 and 354:
Page 355 and 356:
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?