Taming Chaos by Linear Regulation with Bound Estimation

<div>The evolution of <svg height="11.4781pt" id="M97" style="vertical-align:-2.18977pt" version="1.1" viewbox="-0.0498162 -9.28833 32.3561 11.4781" width="32.3561pt" 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="M167 -163V704H103V-163H167ZM317 -163V704H253V-163H317Z" id="g50-10"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="420" vert-adv-y="420"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,5.46,0)"><path d="M545 403C545 425 527 451 498 451C448 451 400 404 312 286L290 341C262 412 246 451 219 451C182 451 138 404 92 345L112 323C152 368 169 378 181 378C192 378 201 359 216 321L257 214C184 115 142 69 113 69C102 69 93 73 88 79S78 89 68 89C47 89 24 61 24 40C24 8 49 -12 75 -12C125 -12 175 31 271 175L312 64C330 15 357 -12 382 -12C421 -12 475 35 515 98L496 121C466 88 439 62 420 62C402 62 384 92 363 147L325 247C349 280 374 309 397 333C422 361 446 381 462 381C471 381 481 374 488 366C493 360 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Also shown is theoretical bound of <svg height="11.4781pt" id="M99" style="vertical-align:-2.18977pt" version="1.1" viewbox="-0.0498162 -9.28833 32.3561 11.4781" width="32.3561pt" 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="M167 -163V704H103V-163H167ZM317 -163V704H253V-163H317Z" id="g50-10"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="420" vert-adv-y="420"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,5.46,0)"><path d="M545 403C545 425 527 451 498 451C448 451 400 404 312 286L290 341C262 412 246 451 219 451C182 451 138 404 92 345L112 323C152 368 169 378 181 378C192 378 201 359 216 321L257 214C184 115 142 69 113 69C102 69 93 73 88 79S78 89 68 89C47 89 24 61 24 40C24 8 49 -12 75 -12C125 -12 175 31 271 175L312 64C330 15 357 -12 382 -12C421 -12 475 35 515 98L496 121C466 88 439 62 420 62C402 62 384 92 363 147L325 247C349 280 374 309 397 333C422 361 446 381 462 381C471 381 481 374 488 366C493 360 499 357 505 357C521 357 545 380 545 403Z" id="g50-121"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="569" vert-adv-y="569"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,12.857,0)"><path d="M309 -142C211 -61 151 99 151 271S210 601 309 683L288 710C141 611 75 451 75 272V271C75 90 141 -72 288 -169L309 -142Z" id="g50-41"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="358" vert-adv-y="358"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,17.511,0)"><path d="M329 433H203L239 587L230 596L147 534L123 433H57L30 395L34 388H115L61 129C37 16 59 -12 85 -12C147 -12 222 58 260 98L241 125C212 95 160 62 144 62C132 62 127 71 138 126L192 386L305 394L329 433Z" id="g50-117"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="347" vert-adv-y="347"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,22.022,0)"><path d="M283 271C283 451 220 610 70 710L48 683C146 603 207 443 207 271S149 -59 48 -142L70 -169C220 -67 283 90 283 271Z" id="g50-42"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="358" vert-adv-y="358"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,26.676,0)"><path d="M167 -163V704H103V-163H167ZM317 -163V704H253V-163H317Z" id="g50-10"></path><glyph.data ascent="3443" descent="-2856" horiz-adv-x="420" vert-adv-y="420"></glyph.data></g></svg> as in (<a href="https://static.hindawi.com/articles/jam/volume-2015/357930/figures/#EEq7">7</a>).</div>

Journal of Applied Mathematics

Taming Chaos by Linear Regulation with Bound Estimation

Figure 2