Bifurcation Analysis for a Kind of Nonlinear Finance System with Delayed Feedback and Its Application to Control of Chaos

<table>When <svg height="13.1625" id="M274" style="vertical-align:-1.59705pt" version="1.1" viewbox="0 0 111.075 13.1625" width="111.075" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,11.112)"><use xlink:href="#x1D43E"></use></g><g transform="matrix(.017,-0,0,-.017,17.962,11.112)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,32.666,11.112)"><use xlink:href="#x2212"></use></g><g transform="matrix(.017,-0,0,-.017,42.644,11.112)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,50.804,11.112)"><use xlink:href="#x2C"></use></g><g transform="matrix(.017,-0,0,-.017,63.179,11.112)"><use xlink:href="#x1D70F"></use></g><g transform="matrix(.017,-0,0,-.017,76.115,11.112)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,90.819,11.112)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,98.978,11.112)"><use xlink:href="#x2E"></use></g><g transform="matrix(.017,-0,0,-.017,102.854,11.112)"><path d="M249 635q70 0 116 -43t46 -105q0 -46 -28 -80q-22 -25 -80 -64q62 -35 97 -75t35 -99q0 -81 -63 -131t-135 -50q-83 0 -137.5 45.5t-54.5 123.5q0 52 45 95q29 28 89 64q-109 62 -109 155q0 66 50.5 115t128.5 49zM238 603q-42 0 -67.5 -31t-25.5 -72q0 -50 32.5 -79.5
t98.5 -62.5q61 48 61 124q0 59 -29.5 90t-69.5 31zM248 20q46 0 76.5 33.5t30.5 89.5q0 50 -39 85.5t-110 71.5q-81 -54 -81 -137q0 -67 35.5 -105t87.5 -38z" id="x38"></path></g>
</svg>, chaos vanishes, and <svg height="15.5375" id="M275" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 14.6125 15.5375" width="14.6125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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l-31 22q-32 23 -53.5 41.5t-45 57t-23.5 77.5q0 82 58 132.5t156 50.5q46 0 101 -17l18.5 -6t17 -6t8.5 -3q-4 -55 0 -147z" id="x1D446"></path></g>
<g transform="matrix(.012,-0,0,-.012,8.225,15.475)"><use xlink:href="#x31"></use></g>
</svg> becomes local stable. Here initial value is <svg height="15.0875" id="M276" style="vertical-align:-2.3205pt" version="1.1" viewbox="0 0 107.2 15.0875" width="107.2" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path d="M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z" id="x28"></path></g><g transform="matrix(.017,-0,0,-.017,5.944,12.138)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,14.103,12.138)"><use xlink:href="#x2E"></use></g><g transform="matrix(.017,-0,0,-.017,17.979,12.138)"><use xlink:href="#x35"></use></g><g transform="matrix(.017,-0,0,-.017,26.139,12.138)"><use xlink:href="#x2C"></use></g><g transform="matrix(.017,-0,0,-.017,38.497,12.138)"><use xlink:href="#x2212"></use></g><g transform="matrix(.017,-0,0,-.017,48.475,12.138)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,56.634,12.138)"><use xlink:href="#x2E"></use></g><g transform="matrix(.017,-0,0,-.017,60.51,12.138)"><use xlink:href="#x35"></use></g><g transform="matrix(.017,-0,0,-.017,68.669,12.138)"><use xlink:href="#x2C"></use></g><g transform="matrix(.017,-0,0,-.017,81.045,12.138)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,89.204,12.138)"><use xlink:href="#x2E"></use></g><g transform="matrix(.017,-0,0,-.017,93.08,12.138)"><path d="M137 343l67 33q37 17 63 17q79 0 129.5 -53t50.5 -131q0 -92 -58 -156.5t-147 -64.5t-147 68t-58 182q0 63 17 119t43 95.5t61.5 72t69 52t67.5 31.5q62 22 128 33l6 -32q-56 -11 -108 -35q-149 -71 -184 -231zM227 337q-47 0 -95 -27q-6 -23 -6 -70q0 -93 36 -155.5
t96 -62.5q53 0 78 45.5t25 105.5q0 68 -35 116t-99 48z" id="x36"></path></g><g transform="matrix(.017,-0,0,-.017,101.239,12.138)"><path d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" id="x29"></path></g>
</svg>.</table>

Journal of Applied Mathematics

fig4

Figure 4

Figure 4: Bifurcation Analysis for a Kind of Nonlinear Finance System with Delayed Feedback and Its Application to Control of Chaos 