Phase Plane Analysis Method of Nonlinear Traffic Phenomena

<div>(a) The density-time variation curve on the fiftieth road section; (b) the phase plane diagram of <svg height="12.7178pt" id="M73" style="vertical-align:-3.42947pt" version="1.1" viewbox="-0.0498162 -9.28833 50.6764 12.7178" width="50.6764pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path d="M471 401C471 442 455 448 440 448C421 448 389 439 366 424C314 390 233 331 165 242H163L187 345C193 372 197 393 197 409C197 435 189 448 174 448C146 448 80 408 23 349L40 327C64 351 97 373 107 373C115 373 118 366 111 337L29 -4L36 -12C56 -4 83 3 110 9C123 69 136 126 150 172C227 283 336 381 377 381C392 381 396 370 379 298L326 74C290 -77 273 -172 273 -197C273 -219 316 -261 346 -261L353 -246C348 -227 352 -174 374 -66L457 310C467 352 471 381 471 401Z" id="g113-229"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="494" vert-adv-y="494"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,10.921,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z" id="g113-45"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,16.064,0)"><path d="M471 379C471 433 467 511 427 578C379 658 305 712 208 712C154 712 98 699 70 684C70 668 95 634 111 634C141 651 180 665 234 665C306 665 350 627 372 575C391 530 393 480 393 442V415C371 436 344 448 313 448C280 448 227 437 176 411C100 372 23 279 23 148C23 67 64 -12 180 -12C293 -12 360 46 407 129C450 205 471 310 471 379ZM382 335C382 142 289 50 238 35C226 31 213 28 200 28C139 28 110 91 110 164C110 310 191 382 223 395C245 404 259 409 280 409C305 409 332 404 359 388C379 376 382 362 382 335Z" id="g113-222"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="494" vert-adv-y="494"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,22.88,0)"><path d="M471 401C471 442 455 448 440 448C421 448 389 439 366 424C314 390 233 331 165 242H163L187 345C193 372 197 393 197 409C197 435 189 448 174 448C146 448 80 408 23 349L40 327C64 351 97 373 107 373C115 373 118 366 111 337L29 -4L36 -12C56 -4 83 3 110 9C123 69 136 126 150 172C227 283 336 381 377 381C392 381 396 370 379 298L326 74C290 -77 273 -172 273 -197C273 -219 316 -261 346 -261L353 -246C348 -227 352 -174 374 -66L457 310C467 352 471 381 471 401Z" id="g113-229"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="494" vert-adv-y="494"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,29.303,0)"><path d="M368 703H309L44 -163H104L368 703Z" id="g113-48"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="412" vert-adv-y="412"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,34.664,0)"><path d="M471 379C471 433 467 511 427 578C379 658 305 712 208 712C154 712 98 699 70 684C70 668 95 634 111 634C141 651 180 665 234 665C306 665 350 627 372 575C391 530 393 480 393 442V415C371 436 344 448 313 448C280 448 227 437 176 411C100 372 23 279 23 148C23 67 64 -12 180 -12C293 -12 360 46 407 129C450 205 471 310 471 379ZM382 335C382 142 289 50 238 35C226 31 213 28 200 28C139 28 110 91 110 164C110 310 191 382 223 395C245 404 259 409 280 409C305 409 332 404 359 388C379 376 382 362 382 335Z" id="g113-222"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="494" vert-adv-y="494"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,41.48,0)"><path d="M324 430H196L233 583L223 592L145 529L120 430H54L29 396L31 388H111L56 126C33 15 54 -12 77 -12C137 -12 214 57 250 95L233 119C208 92 155 59 138 59C126 59 120 70 131 125L186 390L298 394L324 430Z" id="g113-117"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="341" vert-adv-y="341"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,45.913,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g></svg> on the fiftieth road section.</div>

Journal of Control Science and Engineering

Phase Plane Analysis Method of Nonlinear Traffic Phenomena

Figure 2