Abstract
Using bifurcation methods and the Abelian integral, we investigate the number of the limit cycles that bifurcate from the period annulus of the singular point when we perturb the planar ordinary differential equations of the form <svg style="vertical-align:-3.56265pt;width:98.675003px;" id="M1" height="16.625" version="1.1" viewBox="0 0 98.675003 16.625" width="98.675003" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,9.262,11.95)"><path id="x307" d="M-161 571q0 -21 -15 -36t-36 -15q-20 0 -35 15.5t-15 35.5q0 21 15 36.5t35 15.5q22 0 36.5 -15t14.5 -37z" /></g><g transform="matrix(.017,-0,0,-.017,.062,12.138)"><path id="x1D465" d="M536 404q0 -17 -13.5 -31.5t-26.5 -14.5q-8 0 -15 10q-11 14 -25 14q-22 0 -67 -50q-47 -52 -68 -82l37 -102q31 -88 55 -88t78 59l16 -23q-32 -48 -68.5 -78t-65.5 -30q-19 0 -37.5 20t-29.5 53l-41 116q-72 -106 -114.5 -147.5t-79.5 -41.5q-21 0 -34.5 14t-13.5 37
q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 25 -10q22 0 57.5 36t89.5 111l-40 108q-22 58 -36 58q-21 0 -67 -57l-19 20q81 107 125 107q17 0 30 -22t39 -88l22 -55q68 92 108.5 128.5t74.5 36.5q20 0 32.5 -14t12.5 -30z" /></g><g transform="matrix(.017,-0,0,-.017,14.273,12.138)"><path id="x3D" d="M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,28.977,12.138)"><path id="x2212" d="M535 230h-483v50h483v-50z" /></g><g transform="matrix(.017,-0,0,-.017,38.956,12.138)"><path id="x1D466" d="M556 393q0 -39 -36 -106q-42 -78 -185 -279q-47 -66 -81 -108t-117 -135l-112 -26l-8 22q150 90 251 219q-6 136 -39 340q-8 53 -21 53q-6 0 -27 -19.5t-38 -42.5l-16 26q80 111 127 111q23 0 35 -28t20 -90q18 -137 27 -263h2q142 200 142 279q0 24 -14 48q-4 7 5 26
q13 28 43 28q18 0 30 -15.5t12 -39.5z" /></g><g transform="matrix(.017,-0,0,-.017,48.798,12.138)"><path id="x1D436" d="M682 629q-1 -16 -1.5 -72t-2.5 -86l-31 -4q-5 92 -51 129t-139 37q-100 0 -177 -49t-116 -125t-39 -162q0 -122 66 -201t182 -79q83 0 137 42.5t112 128.5l26 -15q-12 -31 -42.5 -88t-45.5 -75q-139 -27 -199 -27q-148 0 -243 81.5t-95 226.5q0 173 129.5 274.5
t325.5 101.5q114 0 204 -38z" /></g><g transform="matrix(.017,-0,0,-.017,60.782,12.138)"><path id="x28" 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" /></g><g transform="matrix(.017,-0,0,-.017,66.664,12.138)"><use xlink:href="#x1D465"/></g><g transform="matrix(.017,-0,0,-.017,76.166,12.138)"><path id="x2C" d="M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z" /></g><g transform="matrix(.017,-0,0,-.017,82.88,12.138)"><use xlink:href="#x1D466"/></g><g transform="matrix(.017,-0,0,-.017,92.723,12.138)"><path id="x29" d="M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z" /></g> </svg>, <svg style="vertical-align:-3.56265pt;width:88.6875px;" id="M2" height="16.625" version="1.1" viewBox="0 0 88.6875 16.625" width="88.6875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,10.113,11.95)"><use xlink:href="#x307"/></g><g transform="matrix(.017,-0,0,-.017,.063,12.138)"><use xlink:href="#x1D466"/></g><g transform="matrix(.017,-0,0,-.017,14.613,12.138)"><use xlink:href="#x3D"/></g><g transform="matrix(.017,-0,0,-.017,29.317,12.138)"><use xlink:href="#x1D465"/></g><g transform="matrix(.017,-0,0,-.017,38.82,12.138)"><use xlink:href="#x1D436"/></g><g transform="matrix(.017,-0,0,-.017,50.804,12.138)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,56.685,12.138)"><use xlink:href="#x1D465"/></g><g transform="matrix(.017,-0,0,-.017,66.188,12.138)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,72.902,12.138)"><use xlink:href="#x1D466"/></g><g transform="matrix(.017,-0,0,-.017,82.744,12.138)"><use xlink:href="#x29"/></g> </svg> with an arbitrary polynomial vector field, where <svg style="vertical-align:-3.56265pt;width:110.875px;" id="M3" height="20.275" version="1.1" viewBox="0 0 110.875 20.275" width="110.875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,15.775)"><use xlink:href="#x1D436"/></g><g transform="matrix(.017,-0,0,-.017,12.047,15.775)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,17.928,15.775)"><use xlink:href="#x1D465"/></g><g transform="matrix(.017,-0,0,-.017,27.43,15.775)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,34.128,15.775)"><use xlink:href="#x1D466"/></g><g transform="matrix(.017,-0,0,-.017,43.97,15.775)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,54.577,15.775)"><use xlink:href="#x3D"/></g><g transform="matrix(.017,-0,0,-.017,69.281,15.775)"><path id="x31" 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" /></g><g transform="matrix(.017,-0,0,-.017,81.215,15.775)"><use xlink:href="#x2212"/></g><g transform="matrix(.017,-0,0,-.017,94.967,15.775)"><use xlink:href="#x1D465"/></g> <g transform="matrix(.012,-0,0,-.012,104.475,7.613)"><path id="x33" 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" /></g> </svg> or <svg style="vertical-align:-3.56265pt;width:110.875px;" id="M4" height="20.237499" version="1.1" viewBox="0 0 110.875 20.237499" width="110.875" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg"> <g transform="matrix(.017,-0,0,-.017,.062,15.738)"><use xlink:href="#x1D436"/></g><g transform="matrix(.017,-0,0,-.017,12.047,15.738)"><use xlink:href="#x28"/></g><g transform="matrix(.017,-0,0,-.017,17.928,15.738)"><use xlink:href="#x1D465"/></g><g transform="matrix(.017,-0,0,-.017,27.43,15.738)"><use xlink:href="#x2C"/></g><g transform="matrix(.017,-0,0,-.017,34.128,15.738)"><use xlink:href="#x1D466"/></g><g transform="matrix(.017,-0,0,-.017,43.97,15.738)"><use xlink:href="#x29"/></g><g transform="matrix(.017,-0,0,-.017,54.577,15.738)"><use xlink:href="#x3D"/></g><g transform="matrix(.017,-0,0,-.017,69.281,15.738)"><use xlink:href="#x31"/></g><g transform="matrix(.017,-0,0,-.017,81.215,15.738)"><use xlink:href="#x2212"/></g><g transform="matrix(.017,-0,0,-.017,94.967,15.738)"><use xlink:href="#x1D465"/></g> <g transform="matrix(.012,-0,0,-.012,104.475,7.587)"><path id="x34" d="M456 178h-96v-72q0 -51 12.5 -62.5t72.5 -16.5v-27h-256v27q65 5 78 17t13 62v72h-260v28q182 271 300 426h40v-407h96v-47zM280 225v295h-2q-107 -148 -196 -295h198z" /></g> </svg>.
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