Analyzing Nonlinear Dynamics via Data-Driven Dynamic Mode Decomposition-Like Methods

<div>The CGLE for <span class="nowrap"><svg height="11.7782pt" id="M230" style="vertical-align:-3.42938pt" version="1.1" viewbox="-0.0498162 -8.34882 34.7918 11.7782" width="34.7918pt" 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="M538 96L524 119C492 88 454 63 446 63C439 63 434 70 440 101C463 223 491 341 518 448H508L433 422L401 276C355 192 240 56 188 56C163 56 154 89 163 133C184 233 207 338 235 443L230 448L152 424L58 17C40 -60 23 -143 23 -185C23 -241 42 -261 63 -261C92 -261 117 -241 125 -227L124 -221C109 -209 89 -165 89 -103C89 -55 96 -13 105 26H107C121 -3 134 -12 151 -12C172 -12 194 -5 221 16C279 61 330 124 384 194H386C381 159 377 141 368 105C343 3 363 -12 383 -12C416 -12 486 35 538 96Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.848,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,22.111,0)"><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,-0.013,28.351,0)"><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> plotting the amplitudes versus frequencies in the DMD expansion (left plot) for low (black circles) and high (black crosses) accuracy computations using DMD-50, and using DMD-1 (red symbols), and the color map of <svg height="11.5564pt" id="M231" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 29.7072 11.5564" width="29.7072pt" 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="M813 114C785 84 767 69 759 69C706 69 706 154 694 237C687 293 662 353 617 381V383C698 414 741 479 741 555C741 596 724 631 689 660C655 689 607 704 545 704L412 607C365 666 306 695 235 695C183 695 139 681 103 653C68 626 50 588 50 540C50 450 164 435 164 350C164 311 125 281 89 269L98 247C143 258 181 278 214 307S264 368 264 401C264 490 191 500 161 550C134 588 153 667 216 664C288 664 324 578 324 405C324 364 324 330 323 305S321 247 318 208S310 140 302 119C295 99 285 89 273 89S221 109 153 150L59 5L81 -9C92 8 107 25 125 40C137 38 176 17 243 -22C339 71 412 201 440 322L514 340C549 329 566 278 566 187C566 141 574 97 589 56C605 15 630 -11 665 -22H679C702 -9 728 10 759 34S813 78 829 94L813 114ZM444 357C447 386 448 422 448 465C448 500 441 536 428 573C468 613 492 627 531 627C560 627 583 617 599 596C616 575 624 554 624 531C624 456 582 387 477 364L444 357Z"></path></g><g transform="matrix(.013,0,0,-0.013,11.362,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,15.86,0)"><path d="M686 28C612 35 607 44 591 112C563 234 541 360 519 489L489 666L457 658L147 121C100 40 89 36 24 28L17 0H240L250 28C168 34 159 41 190 101L262 237H482C495 180 503 137 510 91C517 47 514 35 441 28L433 0H677L686 28ZM475 280H285L429 541H431L475 280Z"></path></g><g transform="matrix(.013,0,0,-0.013,24.995,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g></svg> (middle plot) and <svg height="11.439pt" id="M232" style="vertical-align:-2.15067pt" version="1.1" viewbox="-0.0498162 -9.28833 16.1365 11.439" width="16.1365pt" 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="M686 28C612 35 607 44 591 112C563 234 541 360 519 489L489 666L457 658L147 121C100 40 89 36 24 28L17 0H240L250 28C168 34 159 41 190 101L262 237H482C495 180 503 137 510 91C517 47 514 35 441 28L433 0H677L686 28ZM475 280H285L429 541H431L475 280Z"></path></g><g transform="matrix(.013,0,0,-0.013,12.554,0)"><path d="M162 -163V703H101V-163H162Z"></path></g></svg> (right plot).</div>

Complexity

fig3

Figure 3

Figure 3: Analyzing Nonlinear Dynamics via Data-Driven Dynamic Mode Decomposition-Like Methods 