Uniform upper bounds for the cyclicity of the zero solution of the Abel differential equation

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作者：Dmitry Batenkovasup> ; sup>1sup> ; sup>batenkov@cs.technion.ac.il" class="auth_mail" title="E-mail the corresponding authorsup> ; Gal Binyaminibsup> ; sup>2sup> ; sup>galbin@gmail.com" class="auth_mail" title="E-mail the corresponding authorsup>
刊名：Journal of Differential Equations
出版年：2015
出版时间：5 December 2015
年：2015
卷：259
期：11
页码：5769-5781
全文大小：268 K

文摘

Given two polynomials <span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002203961500368X&_mathId=si1.gif&_user=111111111&_pii=S002203961500368X&_rdoc=1&_issn=00220396&md5=1e41c0834dc9aee569810be6faceb4e8" title="Click to view the MathML source">P,qspan>

si1.gif" overflow="

scroll">P,qspan>span>span> we consider the following question: “how large can the index of the first non-zero moment source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002203961500368X&_mathId=si2.gif&_user=111111111&_pii=S002203961500368X&_rdoc=1&_issn=00220396&md5=4fbd410de72af33210ce95b1323cae64">ss="imgLazyJSB inlineImage" height="26" width="86" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S002203961500368X-si2.gif">script>style="vertical-align:bottom" width="86" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S002203961500368X-si2.gif">script>

si2.gif" overflow="

scroll">sub>m˜ksub>=subsup>∫absubsup>sup>Pksup>qspan>span>span> be, assuming the sequence is not identically zero?” The answer K to this question is known as the moment Bautin index, and we provide the first general upper bound: <span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002203961500368X&_mathId=si3.gif&_user=111111111&_pii=S002203961500368X&_rdoc=1&_issn=00220396&md5=59fa0200e217fff7cc3c00d58c09297c" title="Click to view the MathML source">K猢?+deg鈦+3(deg鈦−1)2sup>span>

si3.gif" overflow="

scroll">K猢?/mo>2+deg鈦?/mo>q+3sup>stretchy="false">(deg鈦?/mo>P−1stretchy="false">)2sup>span>span>span>. The proof is based on qualitative analysis of linear ODEs, applied to Cauchy-type integrals of certain algebraic functions.

sp0020">The moment Bautin index plays an important role in the study of bifurcations of periodic solution in the polynomial Abel equation <span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002203961500368X&_mathId=si4.gif&_user=111111111&_pii=S002203961500368X&_rdoc=1&_issn=00220396&md5=0c708b3ff0ade5a81cdcafcf1b080410" title="Click to view the MathML source">y′sup>=py2sup>+蔚qy3sup>span> $si4.gif" overflow="$ scroll">sup>y′sup>=psup>y2sup>+蔚qsup>y3sup>span>span>span> for <span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002203961500368X&_mathId=si5.gif&_user=111111111&_pii=S002203961500368X&_rdoc=1&_issn=00220396&md5=26be93bfba7296edd5385680b567e44c" title="Click to view the MathML source">p,qspan> $si5.gif" overflow="$ scroll">p,qspan>span>span> polynomials and <span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002203961500368X&_mathId=si6.gif&_user=111111111&_pii=S002203961500368X&_rdoc=1&_issn=00220396&md5=8516f639514b58138c74c43c669dc4b3" title="Click to view the MathML source">蔚鈮?span> $si6.gif" overflow="$ scroll">蔚鈮?/mo>1span>span>span>. In particular, our result implies that for p satisfying a well-known generic condition, the number of periodic solutions near the zero solution does not exceed <span class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S002203961500368X&_mathId=si7.gif&_user=111111111&_pii=S002203961500368X&_rdoc=1&_issn=00220396&md5=0fa1e3db2d6f0e102701ce6bc5214377" title="Click to view the MathML source">5+deg鈦+3deg2sup>鈦span> $si7.gif" overflow="$ scroll">5+deg鈦?/mo>q+3sup>deg2sup>鈦?/mo>pspan>span>span>. This is the first such bound depending solely on the degrees of the Abel equation.

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