Some remarks on inverse Jacobi multipliers around Hopf singularities

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• 作者：Adriana Buic膬^a ; ^{abuica@math.ubbcluj.ro" class="auth_mail} ; Isaac A. Garcí ; a^b ; ^{garcia@matematica.udl.cat" class="auth_mail} ; Susanna Maza^b ; ^{smaza@matematica.udl.cat" class="auth_mail}
• 关键词：Jacobi last multipliers ; Generalized Hopf bifurcation ; Poincaré ; map ; Limit cycle
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：15 October 2014
• 年：2014
• 卷：418
• 期：2
• 页码：1074-1083
• 全文大小：293 K

文摘

We study several properties of inverse Jacobi multipliers V   around Hopf singularities of analytic vector fields 2247X14003850&_mathId=si1.gif&_user=111111111&_pii=S0022247X14003850&_rdoc=1&_issn=0022247X&md5=7f1e4cd0f4cceba6a94813f65a9ba450" title="Click to view the MathML source">X$\mathcal{X}$ in 2247X14003850&_mathId=si2.gif&_user=111111111&_pii=S0022247X14003850&_rdoc=1&_issn=0022247X&md5=1b20331dc73d8aa1e306e0d362417b8f" title="Click to view the MathML source">Rⁿ${\mathbb{R}}^n$ which are relevant to the study of the local bifurcation of periodic orbits. When 2247X14003850&_mathId=si3.gif&_user=111111111&_pii=S0022247X14003850&_rdoc=1&_issn=0022247X&md5=d7c28b7eee69317635d859002c0c591a" title="Click to view the MathML source">n=3$n=3$ and the singularity is a saddle-focus we show that: (i) any two locally smooth and non-flat linearly independent inverse Jacobi multipliers have the same Taylor expansion; (ii) any smooth and non-flat V   has associated exactly one smooth center manifold 2247X14003850&_mathId=si4.gif&_user=111111111&_pii=S0022247X14003850&_rdoc=1&_issn=0022247X&md5=491056ec778638b9b0f92d42285fbc45" title="Click to view the MathML source">W^c$W^c$ of 2247X14003850&_mathId=si1.gif&_user=111111111&_pii=S0022247X14003850&_rdoc=1&_issn=0022247X&md5=7f1e4cd0f4cceba6a94813f65a9ba450" title="Click to view the MathML source">X$\mathcal{X}$ such that 2247X14003850&_mathId=si5.gif&_user=111111111&_pii=S0022247X14003850&_rdoc=1&_issn=0022247X&md5=9304d93f02dafb9e609fbae6d80c1bdb" title="Click to view the MathML source">W^c⊂V⁻¹(0)$W^c\subset V^{-1}(0)$. We also study whether the properties of the vanishing set 2247X14003850&_mathId=si6.gif&_user=111111111&_pii=S0022247X14003850&_rdoc=1&_issn=0022247X&md5=41013f516067deb1712e5ea3cf78300c" title="Click to view the MathML source">V⁻¹(0)$V^{-1}(0)$ proved in the 3-dimensional case remain valid when 2247X14003850&_mathId=si7.gif&_user=111111111&_pii=S0022247X14003850&_rdoc=1&_issn=0022247X&md5=4513c28fadcb384981a92d96ff5c1fc0" title="Click to view the MathML source">n≥4$n\ge 4$.