Dynamic Behaviors Analysis of a Chaotic Circuit Based on a Novel Fractional-Order Generalized Memristor

<p>Phase trajectories of variable of fractional-order memristive circuits when <span class="nowrap"><svg height="11.7782pt" id="M286" style="vertical-align:-3.42938pt" version="1.1" viewbox="-0.0498162 -8.34882 43.1952 11.7782" width="43.1952pt" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M474 429L457 433C435 440 389 448 367 448C348 448 323 446 309 443C266 434 195 406 148 366C78 307 23 210 23 101C23 35 55 -12 92 -12C118 -12 146 1 196 35C247 70 311 130 346 173H348L281 -148C268 -211 256 -221 208 -229L191 -232L187 -257L433 -245L437 -219L411 -216C357 -210 350 -205 362 -140L427 207C447 315 461 381 474 429ZM387 387C379 337 363 262 355 236C318 180 201 57 142 57C126 57 112 81 112 128C112 205 150 321 220 376C244 395 280 403 312 403C345 403 370 396 387 387Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.015,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,21.278,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 635ZM238 602C329 602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z"></path></g><g transform="matrix(.013,0,0,-0.013,27.518,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z"></path></g><g transform="matrix(.013,0,0,-0.013,30.482,0)"><path d="M244 635C114 635 38 519 38 422C38 317 111 240 217 240C236 240 255 244 277 256L345 292C311 140 203 39 59 15L64 -15C89 -15 150 -5 204 17C339 72 440 202 440 386C440 521 368 635 244 635ZM228 602C326 602 352 479 352 390C352 370 351 347 348 324C327 308 293 296 258 296C174 296 124 369 124 458C124 517 152 602 228 602Z"></path></g><g transform="matrix(.013,0,0,-0.013,36.723,0)"><path d="M244 635C114 635 38 519 38 422C38 317 111 240 217 240C236 240 255 244 277 256L345 292C311 140 203 39 59 15L64 -15C89 -15 150 -5 204 17C339 72 440 202 440 386C440 521 368 635 244 635ZM228 602C326 602 352 479 352 390C352 370 351 347 348 324C327 308 293 296 258 296C174 296 124 369 124 458C124 517 152 602 228 602Z"></path></g></svg>;</span> (a) projection on the <svg height="11.927pt" id="M287" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 35.2043 11.927" width="35.2043pt" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M219 86C216 168 211 250 206 337C201 410 189 448 163 448C131 448 79 396 43 344L60 322C91 359 110 375 118 375S132 358 136 298C141 238 152 81 155 -12H182C242 62 331 177 390 258C435 321 451 360 451 391C450 424 432 448 408 448C390 448 372 435 366 419C362 410 362 401 366 394C373 383 376 367 376 350C376 283 262 138 221 86H219Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.811,3.132)"><path d="M690 629C653 643 577 667 483 667C225 667 24 528 24 292C24 92 174 -17 367 -17C448 -17 534 1 570 10C592 34 642 131 658 174C648 180 639 186 629 192C562 89 499 21 382 21C223 21 130 138 130 298C130 467 255 627 462 627C582 627 643 589 650 462L686 466C689 517 688 580 690 629Z"></path></g><g transform="matrix(.013,0,0,-0.013,15.398,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,25.935,0)"><path d="M244 607C244 633 228 655 200 655C166 655 146 618 146 594C146 564 166 546 191 546C221 546 244 574 244 607ZM222 91L209 114C184 94 148 66 133 66C127 66 124 73 130 96L201 370C213 416 211 448 191 448C162 448 88 407 29 352L42 328C73 354 104 371 114 371C120 371 119 365 115 345L53 92C32 5 45 -12 68 -12C103 -12 186 50 222 91Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,29.419,3.132)"><path d="M567 167L534 181C505 122 482 94 454 70C428 46 385 42 330 42C288 42 248 44 235 55C216 70 213 91 220 130L295 528C310 609 314 611 401 618L409 650H140L131 618C222 611 224 609 209 528L133 126C118 47 108 41 25 33L18 0H497C514 33 554 126 567 167Z"></path></g></svg> plane; (b) projection on the <svg height="13.9769pt" id="M288" style="vertical-align:-5.341001pt" version="1.1" viewbox="-0.0498162 -8.6359 41.8484 13.9769" width="41.8484pt" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M219 86C216 168 211 250 206 337C201 410 189 448 163 448C131 448 79 396 43 344L60 322C91 359 110 375 118 375S132 358 136 298C141 238 152 81 155 -12H182C242 62 331 177 390 258C435 321 451 360 451 391C450 424 432 448 408 448C390 448 372 435 366 419C362 410 362 401 366 394C373 383 376 367 376 350C376 283 262 138 221 86H219Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.811,3.132)"><path d="M690 629C653 643 577 667 483 667C225 667 24 528 24 292C24 92 174 -17 367 -17C448 -17 534 1 570 10C592 34 642 131 658 174C648 180 639 186 629 192C562 89 499 21 382 21C223 21 130 138 130 298C130 467 255 627 462 627C582 627 643 589 650 462L686 466C689 517 688 580 690 629Z"></path></g><g transform="matrix(.013,0,0,-0.013,15.398,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,25.935,0)"><path d="M244 607C244 633 228 655 200 655C166 655 146 618 146 594C146 564 166 546 191 546C221 546 244 574 244 607ZM222 91L209 114C184 94 148 66 133 66C127 66 124 73 130 96L201 370C213 416 211 448 191 448C162 448 88 407 29 352L42 328C73 354 104 371 114 371C120 371 119 365 115 345L53 92C32 5 45 -12 68 -12C103 -12 186 50 222 91Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,29.419,3.132)"><path d="M567 167L534 181C505 122 482 94 454 70C428 46 385 42 330 42C288 42 248 44 235 55C216 70 213 91 220 130L295 528C310 609 314 611 401 618L409 650H140L131 618C222 611 224 609 209 528L133 126C118 47 108 41 25 33L18 0H497C514 33 554 126 567 167Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,35.298,5.233)"><path d="M798 93L780 124C747 88 714 75 706 75C699 75 699 79 706 108L752 301C786 442 748 454 723 454C697 454 685 448 656 433C617 413 538 363 464 262H462L471 298C504 429 469 454 440 454C409 454 394 449 368 435C319 408 245 343 179 257H177L197 341C214 413 214 454 181 454C153 454 90 421 25 344L43 314C57 332 106 375 118 375C127 375 129 374 121 336C97 225 67 111 32 -6L42 -12C66 -5 95 2 122 6L160 166C192 225 325 384 381 384C403 384 405 375 390 295C371 194 348 91 323 -6L327 -12C347 -6 379 3 407 6C417 61 432 125 445 170C479 232 607 384 659 384C679 384 690 370 677 322L616 88C600 26 608 -12 640 -12C667 -12 736 23 798 93Z"></path></g></svg> plane; (c) projection on the <svg height="13.9769pt" id="M289" style="vertical-align:-5.341001pt" version="1.1" viewbox="-0.0498162 -8.6359 38.4531 13.9769" width="38.4531pt" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M244 607C244 633 228 655 200 655C166 655 146 618 146 594C146 564 166 546 191 546C221 546 244 574 244 607ZM222 91L209 114C184 94 148 66 133 66C127 66 124 73 130 96L201 370C213 416 211 448 191 448C162 448 88 407 29 352L42 328C73 354 104 371 114 371C120 371 119 365 115 345L53 92C32 5 45 -12 68 -12C103 -12 186 50 222 91Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,3.484,3.132)"><path d="M567 167L534 181C505 122 482 94 454 70C428 46 385 42 330 42C288 42 248 44 235 55C216 70 213 91 220 130L295 528C310 609 314 611 401 618L409 650H140L131 618C222 611 224 609 209 528L133 126C118 47 108 41 25 33L18 0H497C514 33 554 126 567 167Z"></path></g><g transform="matrix(.013,0,0,-0.013,12.012,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,22.548,0)"><path d="M244 607C244 633 228 655 200 655C166 655 146 618 146 594C146 564 166 546 191 546C221 546 244 574 244 607ZM222 91L209 114C184 94 148 66 133 66C127 66 124 73 130 96L201 370C213 416 211 448 191 448C162 448 88 407 29 352L42 328C73 354 104 371 114 371C120 371 119 365 115 345L53 92C32 5 45 -12 68 -12C103 -12 186 50 222 91Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,26.032,3.132)"><path d="M567 167L534 181C505 122 482 94 454 70C428 46 385 42 330 42C288 42 248 44 235 55C216 70 213 91 220 130L295 528C310 609 314 611 401 618L409 650H140L131 618C222 611 224 609 209 528L133 126C118 47 108 41 25 33L18 0H497C514 33 554 126 567 167Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,31.911,5.233)"><path d="M798 93L780 124C747 88 714 75 706 75C699 75 699 79 706 108L752 301C786 442 748 454 723 454C697 454 685 448 656 433C617 413 538 363 464 262H462L471 298C504 429 469 454 440 454C409 454 394 449 368 435C319 408 245 343 179 257H177L197 341C214 413 214 454 181 454C153 454 90 421 25 344L43 314C57 332 106 375 118 375C127 375 129 374 121 336C97 225 67 111 32 -6L42 -12C66 -5 95 2 122 6L160 166C192 225 325 384 381 384C403 384 405 375 390 295C371 194 348 91 323 -6L327 -12C347 -6 379 3 407 6C417 61 432 125 445 170C479 232 607 384 659 384C679 384 690 370 677 322L616 88C600 26 608 -12 640 -12C667 -12 736 23 798 93Z"></path></g></svg> plane; (d) projection on the <svg height="11.927pt" id="M290" style="vertical-align:-3.291101pt" version="1.1" viewbox="-0.0498162 -8.6359 42.9745 11.927" width="42.9745pt" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M219 86C216 168 211 250 206 337C201 410 189 448 163 448C131 448 79 396 43 344L60 322C91 359 110 375 118 375S132 358 136 298C141 238 152 81 155 -12H182C242 62 331 177 390 258C435 321 451 360 451 391C450 424 432 448 408 448C390 448 372 435 366 419C362 410 362 401 366 394C373 383 376 367 376 350C376 283 262 138 221 86H219Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,5.811,3.132)"><path d="M690 629C653 643 577 667 483 667C225 667 24 528 24 292C24 92 174 -17 367 -17C448 -17 534 1 570 10C592 34 642 131 658 174C648 180 639 186 629 192C562 89 499 21 382 21C223 21 130 138 130 298C130 467 255 627 462 627C582 627 643 589 650 462L686 466C689 517 688 580 690 629Z"></path></g><g transform="matrix(.013,0,0,-0.013,15.398,0)"><path d="M535 230V280H52V230H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,25.935,0)"><path d="M244 607C244 633 228 655 200 655C166 655 146 618 146 594C146 564 166 546 191 546C221 546 244 574 244 607ZM222 91L209 114C184 94 148 66 133 66C127 66 124 73 130 96L201 370C213 416 211 448 191 448C162 448 88 407 29 352L42 328C73 354 104 371 114 371C120 371 119 365 115 345L53 92C32 5 45 -12 68 -12C103 -12 186 50 222 91Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,29.419,3.132)"><path d="M1006 650H829L492 147L357 650H174L166 618C270 609 275 601 247 502L194 315C158 189 136 127 122 96C104 56 80 37 18 32L12 0H256L263 32C181 40 176 51 183 98C188 132 197 179 226 281L299 541H301L433 6H458L809 551H811L753 141C739 41 734 39 650 32L642 0H917L923 32C836 39 827 41 839 142L887 535C896 610 901 612 997 618L1006 650Z"></path></g><g transform="matrix(.0065,0,0,-0.0065,38.475,-.065)"><path d="M488 431L463 439C445 444 397 454 366 454C349 454 328 451 316 449C267 440 195 412 147 369C79 308 25 212 25 108C25 39 56 -12 93 -12C125 -12 164 1 202 29C245 61 314 115 353 164H355L291 -147C280 -199 266 -208 213 -219L208 -220L200 -253L444 -241L448 -209L424 -207C373 -203 371 -195 380 -141L439 201C462 332 478 398 488 431ZM395 384C387 334 370 258 362 240C323 176 211 60 145 60C133 60 119 84 119 135C119 209 154 312 220 371C242 391 283 403 315 403C352 403 379 395 395 384Z"></path></g></svg> plane.</p>

Complexity

fig20

Figure 20

Figure 20: Dynamic Behaviors Analysis of a Chaotic Circuit Based on a Novel Fractional-Order Generalized Memristor 