Hopf Bifurcation Analysis and Chaos Control of a Chaotic System without <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3155003pt" id="M1" height="15.4095pt" version="1.1" viewBox="-0.0657574 -15.094 8.82982 15.4095" width="8.82982pt"><g transform="matrix(.017,0,0,-0.017,8.001,-3.832)"><path id="g117-4" d="M-70 617L-95 631C-109 602 -124 588 -144 588C-163 588 -180 594 -202 604C-233 618 -248 623 -268 623C-294 623 -326 604 -353 541L-329 524C-313 552 -298 568 -278 568C-267 568 -248 562 -231 554C-203 541 -180 532 -157 532C-128 532 -93 550 -70 617Z"/></g><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g187-84" d="M442 495C434 557 427 602 423 639C405 645 348 665 284 665C142 665 55 579 55 469C55 373 127 319 229 267C304 227 348 190 348 131C348 67 309 25 248 25C152 25 93 138 73 202L38 190C45 130 58 46 68 21C81 15 100 7 128 -1C158 -10 193 -15 229 -15C375 -15 472 74 472 188C472 287 400 343 295 394C207 439 171 470 171 525C171 583 212 624 268 624C349 624 387 550 407 487L442 495Z"/></g></svg>ilnikov Orbits

<div>Bifurcation diagram of system (<a href="https://static.hindawi.com/articles/ddns/volume-2015/912798/figures/#EEq1.2">2</a>) with <svg height="11.5564pt" id="M260" style="vertical-align:-2.26807pt" version="1.1" viewbox="-0.0498162 -9.28833 90.7969 11.5564" width="90.7969pt" 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="M471 456L444 459C426 433 414 430 388 430C324 430 270 434 216 434C103 434 51 374 23 338L43 317C96 366 146 380 221 375L154 109C149 86 147 68 147 52C147 4 168 -12 197 -12C240 -12 291 25 334 71L320 96C295 75 268 58 252 58C238 58 227 79 238 138C251 211 272 296 292 372C310 372 332 368 350 368C391 368 421 369 434 371C444 388 455 413 471 456Z" id="g113-241"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="483" vert-adv-y="483"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,9.914,0)"><path d="M448 1V51H364C248 51 153 129 140 230H448V280H140C153 381 248 459 364 459H448V509H365C208 509 80 395 80 255S208 1 365 1H448Z" id="g117-173"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="528" vert-adv-y="528"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,20.41,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,24.908,0)"><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" id="g113-49"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,31.148,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z" id="g113-47"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,34.112,0)"><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" id="g113-49"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,40.352,0)"><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" id="g113-49"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,46.592,0)"><path d="M241 635C89 635 35 457 35 312C35 153 89 -12 240 -12C390 -12 443 166 443 312C443 466 390 635 241 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602 354 454 354 312C354 172 330 22 240 22C152 22 124 173 124 313S148 602 238 602Z" id="g113-49"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,70.455,0)"><path d="M113 -12C146 -12 170 11 170 46C170 78 146 103 114 103S58 78 58 46C58 11 82 -12 113 -12Z" id="g113-47"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="228" vert-adv-y="228"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,73.419,0)"><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" id="g113-50"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,79.659,0)"><path d="M153 550H386L412 615L406 623H120L82 318C104 327 142 338 184 338C294 338 347 275 347 187C347 112 305 39 221 39C160 39 119 71 97 89C88 97 80 96 71 90C59 80 50 67 49 57C48 45 52 36 66 23C80 9 123 -12 169 -12C221 -11 288 15 342 59C403 109 431 165 431 225C431 308 366 395 238 395C212 395 165 379 127 364L153 550Z" id="g113-54"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="480" vert-adv-y="480"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,86.12,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="346" vert-adv-y="346"></glyph.data></g></svg>.</div>

Discrete Dynamics in Nature and Society

fig2

Figure 2

Figure 2: Hopf Bifurcation Analysis and Chaos Control of a Chaotic System without <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.3155003pt" id="M1" height="15.4095pt" version="1.1" viewBox="-0.0657574 -15.094 8.82982 15.4095" width="8.82982pt"><g transform="matrix(.017,0,0,-0.017,8.001,-3.832)"><path id="g117-4" d="M-70 617L-95 631C-109 602 -124 588 -144 588C-163 588 -180 594 -202 604C-233 618 -248 623 -268 623C-294 623 -326 604 -353 541L-329 524C-313 552 -298 568 -278 568C-267 568 -248 562 -231 554C-203 541 -180 532 -157 532C-128 532 -93 550 -70 617Z"/></g><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g187-84" d="M442 495C434 557 427 602 423 639C405 645 348 665 284 665C142 665 55 579 55 469C55 373 127 319 229 267C304 227 348 190 348 131C348 67 309 25 248 25C152 25 93 138 73 202L38 190C45 130 58 46 68 21C81 15 100 7 128 -1C158 -10 193 -15 229 -15C375 -15 472 74 472 188C472 287 400 343 295 394C207 439 171 470 171 525C171 583 212 624 268 624C349 624 387 550 407 487L442 495Z"/></g></svg>ilnikov Orbits 

Discrete Dynamics in Nature and Society

Hopf Bifurcation Analysis and Chaos Control of a Chaotic System without ilnikov Orbits

Figure 2