Tracking Control Method for Double Compound-Combination Synchronization of Fractional Chaotic Systems and Its Application in Secure Communication

<div>The numerical simulations of the sent and recovered messages using the double compound-combination synchronization between the four drive systems (<a href="https://static-preview.hindawi.com/articles/mpe/volume-2022/5301689/figures/#EEq2">2</a>)–(<a href="https://static-preview.hindawi.com/articles/mpe/volume-2022/5301689/figures/#EEq22">22</a>) and the four response systems (<a href="https://static-preview.hindawi.com/articles/mpe/volume-2022/5301689/figures/#EEq23">23</a>)–(<a href="https://static-preview.hindawi.com/articles/mpe/volume-2022/5301689/figures/#EEq23">26</a>): (a) The original message <span class="nowrap"><svg height="11.5564pt" id="M180" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 87.4347 11.5564" width="87.4347pt" 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="M766 88L752 113C719 83 690 64 681 64C674 64 672 74 679 103L724 292C758 436 724 448 701 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15 54 -12 77 -12C137 -12 214 57 250 95L233 119C208 92 155 59 138 59C126 59 120 70 131 125L186 390L298 394L324 430Z"></path></g><g transform="matrix(.013,0,0,-0.013,82.618,0)"><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>,</span> (b) the encrypted message <span class="nowrap"><svg height="12.4698pt" id="M181" style="vertical-align:-3.18147pt" version="1.1" viewbox="-0.0498162 -9.28833 100.744 12.4698" width="100.744pt" 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="M352 391C352 416 319 448 267 448C236 448 173 423 147 400C107 364 96 332 96 304C96 248 143 210 193 181C241 153 258 124 258 100C258 72 232 38 184 38C151 38 107 66 81 108C77 114 64 116 55 111C34 99 23 84 23 65C23 29 81 -12 134 -12C220 -12 325 61 325 141C325 184 297 215 234 256C194 282 161 309 161 346C161 380 188 401 217 401C255 401 279 380 301 353C308 344 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252H439L447 289C476 423 450 448 419 448C398 448 379 441 355 427C307 400 234 344 162 249H160L180 324C203 409 197 448 170 448C144 448 82 413 23 349L35 321C57 343 96 374 108 374C115 374 117 371 111 341C87 227 57 112 24 -6L32 -12C53 -4 81 4 108 6C119 68 134 128 149 171C177 229 309 383 364 383C387 383 388 355 373 282C354 190 330 92 303 -6L309 -12C332 -4 356 3 386 6C396 63 411 122 424 171C458 236 590 383 642 383C658 383 664 369 652 315L603 91C587 20 593 -12 619 -12C642 -12 708 23 766 88Z"></path></g><g transform="matrix(.013,0,0,-0.013,47.424,0)"><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,-0.013,51.922,0)"><path d="M324 430H196L233 583L223 592L145 529L120 430H54L29 396L31 388H111L56 126C33 15 54 -12 77 -12C137 -12 214 57 250 95L233 119C208 92 155 59 138 59C126 59 120 70 131 125L186 390L298 394L324 430Z"></path></g><g transform="matrix(.013,0,0,-0.013,56.355,0)"><path 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Mathematical Problems in Engineering

fig7

Figure 7

Figure 7: Tracking Control Method for Double Compound-Combination Synchronization of Fractional Chaotic Systems and Its Application in Secure Communication 

Figure 7 | Tracking Control Method for Double Compound-Combination Synchronization of Fractional Chaotic Systems and Its Application in Secure Communication 