Homoclinic Bifurcation and Chaos in a Noise-Induced <svg style="vertical-align:-0.0pt;width:25.174999px;" id="M1" height="20.9375" version="1.1" viewBox="0 0 25.174999 20.9375" width="25.174999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.022,-0,0,-.022,.062,20.875)"><path id="x3A6" d="M488 658v-28q-47 -3 -58.5 -12t-11.5 -39q49 -2 97 -15t93.5 -39.5t73.5 -75t28 -112.5q0 -53 -19.5 -96.5t-50 -71t-70 -47t-77.5 -27.5t-75 -9q0 -36 11.5 -45.5t58.5 -12.5v-28h-223v28q47 3 58.5 12.5t11.5 45.5q-47 1 -94.5 14t-93 39.5t-74.5 76t-29 114.5
q0 51 19 93.5t49 70t69.5 47t77.5 28t76 10.5q0 30 -11 39t-54 12v28h218zM335 118v429q-34 -2 -67 -13.5t-63.5 -35t-49.5 -65.5t-19 -96q0 -56 18.5 -99t49 -67.5t63.5 -37.5t68 -15zM418 547v-429q35 2 67.5 14t63.5 35.5t50 66.5t19 99t-19 99t-50 66.5t-64 35t-67 13.5
z" /></g><g transform="matrix(.016,-0,0,-.016,16.962,10.125)"><path id="x36" 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" /></g></svg> Potential

<table>Chaos for the noise system with <svg height="17.262501" id="M86" style="vertical-align:-4.0404pt" version="1.1" viewbox="0 0 75.550003 17.262501" width="75.550003" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
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<g transform="matrix(.012,-0,0,-.012,10.975,17.2)"><path d="M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z" id="x31"></path></g>
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</svg> (a) time series, (b) phase portraits, (c) poincare maps, (d) spectrum.</table>

Mathematical Problems in Engineering

fig5

Figure 5

Figure 5: Homoclinic Bifurcation and Chaos in a Noise-Induced <svg style="vertical-align:-0.0pt;width:25.174999px;" id="M1" height="20.9375" version="1.1" viewBox="0 0 25.174999 20.9375" width="25.174999" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"><g transform="matrix(.022,-0,0,-.022,.062,20.875)"><path id="x3A6" d="M488 658v-28q-47 -3 -58.5 -12t-11.5 -39q49 -2 97 -15t93.5 -39.5t73.5 -75t28 -112.5q0 -53 -19.5 -96.5t-50 -71t-70 -47t-77.5 -27.5t-75 -9q0 -36 11.5 -45.5t58.5 -12.5v-28h-223v28q47 3 58.5 12.5t11.5 45.5q-47 1 -94.5 14t-93 39.5t-74.5 76t-29 114.5
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z" /></g><g transform="matrix(.016,-0,0,-.016,16.962,10.125)"><path id="x36" 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" /></g></svg> Potential 

Mathematical Problems in Engineering

Homoclinic Bifurcation and Chaos in a Noise-Induced Potential

Figure 5