Stochastic Stability of Damped Mathieu Oscillator Parametrically Excited by a Gaussian Noise

<table>White noise excitation: <svg height="19.924999" id="M200" style="vertical-align:-3.276pt" version="1.1" viewbox="0 0 76.612503 19.924999" width="76.612503" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,15.775)"><path d="M609 650l-19 -162l-30 -2q2 69 -14 94q-9 18 -28.5 26t-74.5 8h-88q-30 0 -37 -6.5t-12 -36.5l-41 -212h103q48 0 69 5.5t32 19t26 50.5h29l-40 -198h-30q2 58 -11.5 70.5t-85.5 12.5h-99l-30 -167q-17 -87 3 -101q18 -15 111 -15q59 0 89 7t54 27q8 7 15.5 16t16 21.5
l14 20.5t15 23.5t13.5 21.5l29 -10q-52 -129 -71 -163h-500l6 28q66 4 83 17.5t28 73.5l77 409q11 61 -0.5 75.5t-78.5 18.5l10 28h467z" id="x1D438"></path></g><g transform="matrix(.017,-0,0,-.017,10.806,15.775)"><path d="M290 -163h-170v866h170v-28q-79 -7 -94 -19.5t-15 -72.5v-627q0 -59 14.5 -71.5t94.5 -19.5v-28z" id="x5B"></path></g><g transform="matrix(.017,-0,0,-.017,16.67,15.775)"><path d="M775 650l-6 -28q-60 -6 -81.5 -16t-61.5 -54l-175 -191l125 -243q30 -58 48.5 -71t82.5 -19l-5 -28h-275l7 28l35 4q31 4 37 12t-6 34l-108 216q-140 -165 -177 -219q-16 -22 -10.5 -30.5t41.5 -13.5l22 -3l-7 -28h-244l8 28q52 4 75 15.5t67 52.5q48 46 206 231
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<g transform="matrix(.017,-0,0,-.017,36.463,15.775)"><path d="M226 -163h-170v27q79 7 94 20t15 73v627q0 59 -15 72t-94 20v27h170v-866z" id="x5D"></path></g><g transform="matrix(.017,-0,0,-.017,47.053,15.775)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,61.74,15.775)"><path d="M475 445l-64 -69q-119 -128 -289 -280q27 6 81 -15q66 -26 102 -26q48 0 111 93l23 -19q-41 -84 -75.5 -116.5t-73.5 -32.5q-37 0 -147 53q-43 21 -63 7q-22 -15 -38 -41q-4 -7 -11 -7q-8 10 -8 26q0 38 47 83l292 273q-32 -5 -67 -1q-63 7 -105 7q-32 0 -53.5 -22
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<g transform="matrix(.012,-0,0,-.012,70.225,19.863)"><use xlink:href="#x31"></use></g>
</svg> with respect to time, <svg height="15.1375" id="M201" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 50.549999 15.1375" width="50.549999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,5.725,10.862)"><use xlink:href="#x1D714"></use></g>
<g transform="matrix(.012,-0,0,-.012,16.575,14.938)"><use xlink:href="#x30"></use></g>
<g transform="matrix(.017,-0,0,-.017,27.625,10.862)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,42.329,10.862)"><use xlink:href="#x32"></use></g>
</svg><i>π</i>, <svg height="17" id="M202" style="vertical-align:-3.39066pt" version="1.1" viewbox="0 0 67.599998 17" width="67.599998" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,5.725,12.712)"><use xlink:href="#x1D701"></use></g>
<g transform="matrix(.012,-0,0,-.012,13.425,16.787)"><use xlink:href="#x30"></use></g>
<g transform="matrix(.017,-0,0,-.017,24.488,12.712)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,39.174,12.712)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,47.334,12.712)"><use xlink:href="#x2E"></use></g><g transform="matrix(.017,-0,0,-.017,51.21,12.712)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,59.369,12.712)"><use xlink:href="#x35"></use></g>
</svg>,  <svg height="16.6625" id="M203" style="vertical-align:-3.56265pt" version="1.1" viewbox="0 0 49.612499 16.6625" width="49.612499" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,12.162)"><use xlink:href="#x1D6FD"></use></g><g transform="matrix(.017,-0,0,-.017,14.647,12.162)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,29.351,12.162)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,37.511,12.162)"><use xlink:href="#x2E"></use></g><g transform="matrix(.017,-0,0,-.017,41.386,12.162)"><use xlink:href="#x31"></use></g>
</svg>,  <svg height="11.125" id="M204" style="vertical-align:-0.1638pt" version="1.1" viewbox="0 0 34.674999 11.125" width="34.674999" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,10.862)"><use xlink:href="#x1D700"></use></g><g transform="matrix(.017,-0,0,-.017,11.741,10.862)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,26.445,10.862)"><use xlink:href="#x31"></use></g>
</svg>; (a) <svg height="20.0625" id="M205" style="vertical-align:-3.39064pt" version="1.1" viewbox="0 0 127.4125 20.0625" width="127.4125" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,15.775)"><use xlink:href="#x1D70E"></use></g>
<g transform="matrix(.012,-0,0,-.012,9.663,7.613)"><use xlink:href="#x32"></use></g>
<g transform="matrix(.017,-0,0,-.017,20.725,15.775)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,35.412,15.775)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,43.571,15.775)"><use xlink:href="#x2E"></use></g><g transform="matrix(.017,-0,0,-.017,47.447,15.775)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,55.606,15.775)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,63.766,15.775)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,71.925,15.775)"><use xlink:href="#x32"></use></g><g transform="matrix(.017,-0,0,-.017,80.085,15.775)"><path d="M285 378v-2q65 -13 102 -54.5t37 -97.5q0 -57 -30.5 -104.5t-74 -75t-85.5 -42t-72 -14.5q-31 0 -59.5 11t-40.5 23q-19 18 -16 36q1 16 23 33q13 10 24 0q58 -51 124 -51q55 0 88 40t33 112q0 64 -39 96.5t-88 32.5q-29 0 -64 -11l-6 29q77 25 118 57.5t41 84.5
q0 45 -26.5 69.5t-68.5 24.5q-67 0 -120 -79l-20 20l43 63q51 56 127 56h1q66 0 107 -37t41 -95q0 -42 -31 -71q-22 -23 -68 -54z" id="x33"></path></g><g transform="matrix(.017,-0,0,-.017,92.97,15.775)"><path d="M512 -3l-437 233v51l437 233v-58l-378 -200v-2l378 -199v-58z" id="x3C"></path></g><g transform="matrix(.017,-0,0,-.017,107.674,15.775)"><use xlink:href="#x1D70E"></use></g>
<g transform="matrix(.012,-0,0,-.012,117.275,7.613)"><use xlink:href="#x32"></use></g>
<g transform="matrix(.012,-0,0,-.012,117.275,19.863)"><use xlink:href="#x63"></use></g><g transform="matrix(.012,-0,0,-.012,122.308,19.863)"><use xlink:href="#x72"></use></g>
</svg>; (b) <svg height="20.0625" id="M206" style="vertical-align:-3.39064pt" version="1.1" viewbox="0 0 131.375 20.0625" width="131.375" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink">
<g transform="matrix(.017,-0,0,-.017,.062,15.775)"><use xlink:href="#x1D70E"></use></g>
<g transform="matrix(.012,-0,0,-.012,9.663,7.613)"><use xlink:href="#x32"></use></g>
<g transform="matrix(.017,-0,0,-.017,20.725,15.775)"><use xlink:href="#x3D"></use></g><g transform="matrix(.017,-0,0,-.017,39.373,15.775)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,47.532,15.775)"><use xlink:href="#x2E"></use></g><g transform="matrix(.017,-0,0,-.017,51.408,15.775)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,59.567,15.775)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,67.727,15.775)"><use xlink:href="#x31"></use></g><g transform="matrix(.017,-0,0,-.017,75.886,15.775)"><use xlink:href="#x33"></use></g><g transform="matrix(.017,-0,0,-.017,84.045,15.775)"><use xlink:href="#x30"></use></g><g transform="matrix(.017,-0,0,-.017,96.93,15.775)"><path d="M512 230l-437 -233v58l378 199v2l-378 200v58l437 -233v-51z" id="x3E"></path></g><g transform="matrix(.017,-0,0,-.017,111.634,15.775)"><use xlink:href="#x1D70E"></use></g>
<g transform="matrix(.012,-0,0,-.012,121.25,7.613)"><use xlink:href="#x32"></use></g>
<g transform="matrix(.012,-0,0,-.012,121.25,19.863)"><use xlink:href="#x63"></use></g><g transform="matrix(.012,-0,0,-.012,126.283,19.863)"><use xlink:href="#x72"></use></g>
</svg>.</table>

Mathematical Problems in Engineering

fig7

Figure 7

Figure 7: Stochastic Stability of Damped Mathieu Oscillator Parametrically Excited by a Gaussian Noise 