A Chimera Oscillatory State in a Globally Delay-Coupled Oscillator Network

<div>(a) The order parameter <svg height="11.439pt" id="M133" style="vertical-align:-2.15067pt" version="1.1" viewbox="-0.0498162 -9.28833 15.993 11.439" width="15.993pt" 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="M162 -163V703H101V-163H162Z"></path></g><g transform="matrix(.013,0,0,-0.013,3.419,0)"><path d="M669 636L663 650H260C230 650 221 652 208 675H186C177 621 162 548 144 481L173 479C195 534 211 560 226 578C248 604 266 615 373 615H544C460 511 114 112 23 16L31 0H561C577 34 611 135 622 170L594 182C566 125 544 88 515 65C481 38 416 35 319 35C249 35 197 37 152 41C270 184 542 501 669 636Z"></path></g><g transform="matrix(.013,0,0,-0.013,12.415,0)"><path d="M162 -163V703H101V-163H162Z"></path></g></svg> as a function of the time delay <svg height="9.49473pt" id="M134" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 6.84582 9.49473" width="6.84582pt" 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="M494 514C482 587 419 712 303 712C238 712 174 667 174 603C174 561 205 514 249 449C219 438 187 422 162 407C93 366 23 283 23 177C23 69 87 -12 190 -12C244 -12 288 5 328 33C406 87 444 170 444 249C444 329 404 391 331 475C265 550 222 605 222 627C222 647 238 657 267 657C355 657 421 585 484 499L494 514ZM359 234C359 143 319 30 219 30C172 30 114 75 114 178C114 275 163 343 195 378C212 397 241 415 269 425C305 382 359 313 359 234Z"></path></g></svg> in the small-world networks for <span class="nowrap"><svg height="11.7782pt" id="M135" style="vertical-align:-3.42938pt" version="1.1" viewbox="-0.0498162 -8.34882 38.2627 11.7782" width="38.2627pt" 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="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"></path></g><g transform="matrix(.013,0,0,-0.013,11.342,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,22.605,0)"><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,-0.013,28.845,0)"><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,-0.013,31.809,0)"><path d="M447 623H65C61 580 56 530 47 475H76C100 541 106 550 172 550H388C308 376 196 170 91 -1L98 -12L172 -2C268 204 360 408 455 611L447 623Z"></path></g></svg>,</span> 0.8, and 0.9, respectively. (b) The order parameter <svg height="11.439pt" id="M136" style="vertical-align:-2.15067pt" version="1.1" viewbox="-0.0498162 -9.28833 15.993 11.439" width="15.993pt" 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="M162 -163V703H101V-163H162Z"></path></g><g transform="matrix(.013,0,0,-0.013,3.419,0)"><path d="M669 636L663 650H260C230 650 221 652 208 675H186C177 621 162 548 144 481L173 479C195 534 211 560 226 578C248 604 266 615 373 615H544C460 511 114 112 23 16L31 0H561C577 34 611 135 622 170L594 182C566 125 544 88 515 65C481 38 416 35 319 35C249 35 197 37 152 41C270 184 542 501 669 636Z"></path></g><g transform="matrix(.013,0,0,-0.013,12.415,0)"><path d="M162 -163V703H101V-163H162Z"></path></g></svg> versus the ratio of inactive ratio <svg height="10.2124pt" id="M137" style="vertical-align:-3.42943pt" version="1.1" viewbox="-0.0498162 -6.78297 7.83752 10.2124" width="7.83752pt" 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="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"></path></g></svg> for different values of <span class="nowrap"><svg height="9.49473pt" id="M138" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 6.84582 9.49473" width="6.84582pt" 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="M494 514C482 587 419 712 303 712C238 712 174 667 174 603C174 561 205 514 249 449C219 438 187 422 162 407C93 366 23 283 23 177C23 69 87 -12 190 -12C244 -12 288 5 328 33C406 87 444 170 444 249C444 329 404 391 331 475C265 550 222 605 222 627C222 647 238 657 267 657C355 657 421 585 484 499L494 514ZM359 234C359 143 319 30 219 30C172 30 114 75 114 178C114 275 163 343 195 378C212 397 241 415 269 425C305 382 359 313 359 234Z"></path></g></svg>.</span> <span class="nowrap"><svg height="8.55521pt" id="M139" style="vertical-align:-0.2063904pt" version="1.1" viewbox="-0.0498162 -8.34882 43.404 8.55521" width="43.404pt" 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="M484 406C484 429 465 448 438 448C400 448 352 405 302 356L156 213L191 373C195 391 197 404 197 416C197 436 191 448 178 448C151 448 83 406 23 350L37 328C67 350 101 369 109 369C114 369 115 362 111 344L29 -5L34 -12C63 -5 86 -1 111 3C124 73 137 138 147 161C165 180 189 202 206 217C215 194 249 111 275 57C304 -1 319 -12 342 -12C368 -12 405 -5 471 92L454 115C427 80 395 55 384 55C374 55 364 58 341 106L262 274C282 296 367 378 414 378C427 378 440 370 444 367C449 363 454 362 462 368C476 379 484 395 484 406Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.224,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,21.487,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,27.727,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"></path></g><g transform="matrix(.013,0,0,-0.013,33.967,0)"><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,-0.013,36.931,0)"><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></svg>.</span> (c) The order parameter <svg height="11.439pt" id="M140" style="vertical-align:-2.15067pt" version="1.1" viewbox="-0.0498162 -9.28833 15.993 11.439" width="15.993pt" 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="M162 -163V703H101V-163H162Z"></path></g><g transform="matrix(.013,0,0,-0.013,3.419,0)"><path d="M669 636L663 650H260C230 650 221 652 208 675H186C177 621 162 548 144 481L173 479C195 534 211 560 226 578C248 604 266 615 373 615H544C460 511 114 112 23 16L31 0H561C577 34 611 135 622 170L594 182C566 125 544 88 515 65C481 38 416 35 319 35C249 35 197 37 152 41C270 184 542 501 669 636Z"></path></g><g transform="matrix(.013,0,0,-0.013,12.415,0)"><path d="M162 -163V703H101V-163H162Z"></path></g></svg> against <svg height="10.2124pt" id="M141" style="vertical-align:-3.42943pt" version="1.1" viewbox="-0.0498162 -6.78297 7.83752 10.2124" width="7.83752pt" 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="M570 304C570 398 525 448 414 448C385 448 343 445 312 434L329 511L321 518C297 504 262 482 244 460L233 411C195 397 159 381 128 358L135 332C160 347 189 360 224 373L111 -147C97 -210 84 -218 17 -231L13 -257L254 -247L259 -218L233 -216C183 -212 177 -202 189 -142L218 -1C238 -10 266 -12 283 -12C351 3 429 48 483 105C543 168 570 242 570 304ZM482 289C482 161 380 33 304 33C278 33 248 51 233 69L303 396C326 400 352 403 369 403C428 403 482 380 482 289Z"></path></g></svg> with different coupling strength. <span class="nowrap"><svg height="9.49473pt" id="M142" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 43.5344 9.49473" width="43.5344pt" 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="M494 514C482 587 419 712 303 712C238 712 174 667 174 603C174 561 205 514 249 449C219 438 187 422 162 407C93 366 23 283 23 177C23 69 87 -12 190 -12C244 -12 288 5 328 33C406 87 444 170 444 249C444 329 404 391 331 475C265 550 222 605 222 627C222 647 238 657 267 657C355 657 421 585 484 499L494 514ZM359 234C359 143 319 30 219 30C172 30 114 75 114 178C114 275 163 343 195 378C212 397 241 415 269 425C305 382 359 313 359 234Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.354,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,21.617,0)"><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,-0.013,27.857,0)"><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,-0.013,30.821,0)"><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,-0.013,37.061,0)"><path d="M137 343C167 482 260 545 321 574C357 591 397 603 429 609L423 641C382 634 335 622 295 608C189 570 37 457 37 238C37 84 125 -12 242 -12C362 -12 447 89 447 209C447 311 374 393 267 393C247 393 226 386 204 376L137 343ZM227 337C318 337 361 256 361 173C361 105 336 22 258 22C176 22 126 120 126 240C126 266 127 291 132 310C155 323 189 337 227 337Z"></path></g></svg>.</span></div>

Complexity

fig8

Figure 8

Figure 8: A Chimera Oscillatory State in a Globally Delay-Coupled Oscillator Network 