Phase Portraits of the Autonomous Duffing Single-Degree-of-Freedom Oscillator with Coulomb Dry Friction

<table>Discriminant <svg height="9.64612pt" id="M82" style="vertical-align:-0.2537899pt" version="1.1" viewbox="-0.0498162 -9.39233 12.1347 9.64612" width="12.1347pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0136,0,0,-0.0136,0,0)"><path d="M810 587C770 574 761 568 725 543C654 630 540 687 408 687C217 687 69 577 69 430C69 319 160 247 276 247C369 247 511 319 511 495C511 510 512 523 509 538H490C490 354 393 277 274 277C174 277 110 344 110 422C110 559 219 655 415 655C524 655 636 602 695 519C628 463 572 392 516 303C433 170 377 97 313 57C252 89 199 109 143 109C79 109 36 87 36 54C36 15 82 -9 165 -9C222 -9 276 0 326 18C373 -1 423 -15 482 -15C645 -15 804 133 804 324C804 393 783 458 747 514C776 546 781 551 818 567L810 587ZM756 336C756 135 623 16 478 16C439 16 402 24 367 36C386 45 401 56 420 68C505 127 569 206 616 302C647 362 679 427 719 480C744 435 756 387 756 336ZM276 39C243 26 207 21 164 21C106 21 77 32 77 54C77 75 125 79 144 79C191 79 227 66 276 39Z" id="g198-5"></path><glyph.data ascent="687" descent="-226" horiz-adv-x="885" vert-adv-y="885"></glyph.data></g></svg> value in the <svg height="9.7821pt" id="M83" style="vertical-align:-3.57194pt" version="1.1" viewbox="-0.0498162 -6.21016 17.3295 9.7821" width="17.3295pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M478 372C478 418 458 448 431 448C409 448 389 431 389 410C389 404 391 400 394 395C398 388 406 371 406 348C406 253 308 122 251 51H249C254 122 249 257 231 336C212 421 189 448 159 448C126 448 75 412 23 327L48 306C83 354 103 371 115 371C125 371 134 360 144 334C185 224 192 64 183 -19C146 -100 116 -202 110 -244L125 -261C154 -259 208 -234 222 -220C222 -194 225 -84 235 -23C247 -3 273 36 308 79C379 165 478 288 478 372Z" id="g113-225"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="501" vert-adv-y="501"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,6.791,0)"><path d="M300 253H71L57 198H286L300 253Z" id="g190-46"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="356" vert-adv-y="356"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,11.632,0)"><path d="M387 375C387 402 357 448 257 448C172 448 82 404 82 326C82 289 108 255 156 241V239C85 223 23 181 23 116C23 39 89 -12 182 -12C265 -12 336 31 378 91L361 114C320 73 269 47 216 47C157 47 115 82 115 137C115 191 160 219 218 219C243 219 262 218 272 217L304 259L302 266C295 265 281 264 255 264C195 264 163 294 163 335C163 377 200 416 249 416C293 416 321 389 329 342C331 332 335 329 341 329C355 329 387 352 387 375Z" id="g113-247"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="410" vert-adv-y="410"></glyph.data></g></svg> parameter plane. Actual parameters’ values: <svg height="13.6986pt" id="M84" style="vertical-align:-1.5015pt" version="1.1" viewbox="-0.0498162 -12.1971 41.2455 13.6986" width="41.2455pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.0135,0,0,-0.0135,0,0)"><path d="M615 290C615 392 563 448 529 448C511 448 491 441 484 434L483 425C518 400 546 349 546 273C546 158 504 46 417 46C381 46 356 77 356 142C356 184 380 293 392 334L387 340L307 323C305 288 296 228 287 188C265 94 221 52 180 52C144 52 112 89 112 171C112 278 164 372 241 430L226 454C106 394 23 288 23 155C23 40 79 -12 139 -12C184 -12 245 32 285 85C292 31 322 -12 376 -12C423 -12 470 18 501 45C570 104 615 184 615 290Z" id="g113-246"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="638" vert-adv-y="638"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,12.43,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="587" vert-adv-y="587"></glyph.data></g><g transform="matrix(.0135,0,0,-0.0135,24.168,-1.526)"><path d="M782 731H740L476 -79H474L280 352L133 274L151 239L242 286L473 -219L782 731Z" id="g119-37"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="764" vert-adv-y="764"></glyph.data></g><rect height="0.679945" width="6.52751" x="34.5574" y="-11.4673"></rect><g transform="matrix(.0135,0,0,-0.0135,34.557,0)"><path d="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 579 325 635 236 635H235C182 635 136 610 108 579L65 516L85 496C110 533 150 575 205 575C258 575 300 543 300 481C300 407 232 369 141 339L147 310C163 315 188 321 211 321C268 321 338 284 338 192C338 94 288 40 217 40C160 40 119 68 93 91C85 98 77 97 69 91C60 84 47 71 46 58C44 46 48 35 62 22C75 10 116 -12 162 -12C234 -12 424 62 424 224C424 297 373 359 285 376V378Z" id="g113-52"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g></svg>.</table>

Advances in Acoustics and Vibration

fig1

Figure 1

Figure 1: Phase Portraits of the Autonomous Duffing Single-Degree-of-Freedom Oscillator with Coulomb Dry Friction 