Document Zbl 08013580

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Proc.</a> <a href="/?q=in%3A520144" title="Articles in this Issue">65, 99-116 (2025)</a>. </div> <div class="abstract">Summary: In this paper, we explore the local geometry of dynamical systems $\dot{x} =F(x)$ with real time parameterization, where $F$ is holomorphic on connected open subsets of $\mathbb{C}\cong \mathbb{R}^2$. We describe the geometry of first-order equilibria. For equilibria of higher orders, we establish an equivalent condition for “definite directions”, allowing us to reverse the implication in Theorem 2 of Chapter 2.10 in [<span class="zbmathjax-textit">L. Perko</span>, Differential equations and dynamical systems. New York, NY: Springer (2001; <a href="/0973.34001">Zbl 0973.34001</a>)]. This enables the geometric construction of a finite elliptic decomposition. We derive a holomorphic Poincar茅-Bendixson-type theorem, leading to the conclusion that bounded non-periodic orbits are always homoclinic or heteroclinic.</div> <div class="clear"></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A30A99" title="MSC2020">30A99</a> </td> <td class="space"> General properties of functions of one complex variable </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A30C10" title="MSC2020">30C10</a> </td> <td class="space"> Polynomials and rational functions of one complex variable </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A30C15" title="MSC2020">30C15</a> </td> <td class="space"> Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A30D30" title="MSC2020">30D30</a> </td> <td class="space"> Meromorphic functions of one complex variable (general theory) </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A32M25" title="MSC2020">32M25</a> </td> <td class="space"> Complex vector fields, holomorphic foliations, $\mathbb{C}$-actions </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37F10" title="MSC2020">37F10</a> </td> <td class="space"> Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A37F75" title="MSC2020">37F75</a> </td> <td class="space"> Dynamical aspects of holomorphic foliations and vector fields </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Aholomorphic+dynamical+system">holomorphic dynamical system</a>; <a href="/?q=ut%3Alocal+geometry+of+equilibria">local geometry of equilibria</a>; <a href="/?q=ut%3Adefinite+directions">definite directions</a>; <a href="/?q=ut%3Afinite+elliptic+decomposition">finite elliptic decomposition</a></div><div class="keywords"> <h3>Citations:</h3><a href="/0973.34001">Zbl 0973.34001</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 08013580" data-ciurl="/ci/08013580" data-biburl="/bibtex/08013580.bib" data-amsurl="/amsrefs/08013580.bib" data-xmlurl="/xml/08013580.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/08013580.pdf" title="Zbl 08013580 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://arxiv.org/abs/2402.07612"title="Note: arXiv document may differ from published version">arXiv</a> <a class="btn btn-default btn-xs" type="button" href="https://topology.nipissingu.ca/tp/restricted/v65/tp65006.pdf" title="Full Text Link">Link</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Alexander A. Andronov, Evgeniya A. Leontovich, I. I. Gordon and A. G. Maier, Qualitative Theory of Second-Order Dynamic Systems. Translated from Russian, John Wiley & Sons, 1973 · <a href="/0282.34022" class="nowrap">Zbl 0282.34022</a></td> </tr><tr> <td>[2]</td> <td class="space">Kevin A. Broughan, Holomorphic flows on simply connected regions have no limit cycles, Meccanica 38 (2003), no. 6, 699-709. · <a href="/1034.37027" class="nowrap">Zbl 1034.37027</a></td> </tr><tr> <td>[3]</td> <td class="space">The structure of sectors of zeros of entire flows, Topology Proc. 27 (2003), no. 2, 379-394. · <a href="/1152.37330" class="nowrap">Zbl 1152.37330</a></td> </tr><tr> <td>[4]</td> <td class="space">The holomorphic flow of Riemanns function 尉(z), Nonlinearity 18 (2004), 1269-1294. · <a href="/1125.11047" class="nowrap">Zbl 1125.11047</a></td> </tr><tr> <td>[5]</td> <td class="space">Kevin A. Broughan and Antony Ross Barnett, The holomorphic flow of the Rie-mann zeta function, Mathematics of Computation 73 (2004), 987-1004. · <a href="/1060.37039" class="nowrap">Zbl 1060.37039</a></td> </tr><tr> <td>[6]</td> <td class="space">Freddy Dumortier, Jaume Llibre and Joan Art茅s, Differential equations and dy-namical systems, Springer, 2006 · <a href="/1110.34002" class="nowrap">Zbl 1110.34002</a></td> </tr><tr> <td>[7]</td> <td class="space">Allen Hatcher, Algebraic topology, Tsinghua University Press Ltd., 2005</td> </tr><tr> <td>[8]</td> <td class="space">Marcus Heitel and Dirk Lebiedz, On Analytical and Topological Properties of Separatrices in 1-D Holomorphic Dynamical Systems and Complex-Time Newton Flows, https://doi.org/10.48550/arXiv.1911.10963 · <a href="https://doi.org/10.48550/arXiv.1911.10963" class="nowrap">doi:10.48550/arXiv.1911.10963</a></td> </tr><tr> <td>[9]</td> <td class="space">Marek Izydorek, Slawomir Rybicki and Zbigniew Szafraniec, A note on the Poincar茅-Bendixson index theorem, Department of Mathematics, Tokyo Institute of Technology, Kodai Mathematical Journal 19 (1996), no. 2, 145-156. · <a href="/0863.34045" class="nowrap">Zbl 0863.34045</a></td> </tr><tr> <td>[10]</td> <td class="space">Camille Jordan, Cours d’analyse de l’脡cole polytechnique, Gauthier-Villars et fils, 1893</td> </tr><tr> <td>[11]</td> <td class="space">Nicolas Kainz, Planar Analytic Dynamical Systems and their phase space structure. Master thesis at Ulm University, 2023. https://www.uni-ulm.de/ mawi/institut-fuer-numerische-mathematik/institut/mitarbeiter/nicolas-kainz/#c963531</td> </tr><tr> <td>[12]</td> <td class="space">Dirk Lebiedz, Holomorphic Hamiltonian 尉-Flow and Riemann Zeros, https:// doi.org/10.48550/arXiv.2006.09165 · <a href="https://doi.org/10.48550/arXiv.2006.09165" class="nowrap">doi:10.48550/arXiv.2006.09165</a></td> </tr><tr> <td>[13]</td> <td class="space">James R. Munkres, Topology. Prentice Hall Upper Saddle River, 2000 · <a href="/0951.54001" class="nowrap">Zbl 0951.54001</a></td> </tr><tr> <td>[14]</td> <td class="space">John W. Neuberger, Cornelia Feiler, Helmut Maier and Wolfgang P. Schleich, Newton flow of the Riemann zeta function: separatrices control the appearance of zeros, New J. Phys. 16 (2014), 103023. · <a href="/1451.11093" class="nowrap">Zbl 1451.11093</a></td> </tr><tr> <td>[15]</td> <td class="space">Lawrence Perko, Differential equations and dynamical systems, Springer, 1990 · <a href="/0685.34002" class="nowrap">Zbl 0685.34002</a></td> </tr><tr> <td>[16]</td> <td class="space">Jan Pr眉ss and Mathias Wilke, Gew枚hnliche Differentialgleichungen und dynamis-che Systeme, Springer, 2010 · <a href="/1211.34001" class="nowrap">Zbl 1211.34001</a></td> </tr><tr> <td>[17]</td> <td class="space">Wolfgang P. Schleich, Ida Bezd臎kov谩, Moochan B. Kim, Paul C. Abbott, Helmut Maier, Hugh L. Montgomery and John W. Neuberger, Equivalent formulations of the Riemann hypothesis based on lines of constant phase, Physica Scripta 93 (2018), 065201. Institute of Numerical Mathematics, Ulm University, Germany Email address, Kainz: nicolas.kainz@uni-ulm.de Email address, Lebiedz: dirk.lebiedz@uni-ulm.de</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching. </div> </div></article> </div></div> </div> </div> <div class="clearfix"></div> </div> </div> <div id="foot"><div class="copyright"> © 2025 <a target="fiz" href="https://www.fiz-karlsruhe.de/en">FIZ Karlsruhe GmbH</a> <a href="/privacy-policy/">Privacy Policy</a> <a href="/legal-notices/">Legal Notices</a> <a href="/terms-conditions/">Terms & Conditions</a> <div class="info"> <ul class="nav"> <li class="mastodon"> <a href="https://mathstodon.xyz/@zbMATH" target="_blank" class="no-new-tab-icon"> <img src="/static/mastodon.png" title="zbMATH at Mathstodon (opens in new tab)" alt="Mastodon logo"> </a> </li> </ul> </div> </div> <div class="clearfix" style="height: 0px;"></div> </div> </div> <script src="https://static.zbmath.org/contrib/jquery/1.9.1/jquery.min.js"></script> <script src="https://static.zbmath.org/contrib/jquery-caret/1.5.2/jquery.caret.min.js"></script> <script src="/static/js/jquery-ui-1.10.1.custom.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap/v3.3.7zb1/js/bootstrap.min.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-lightbox/v0.7.0/bootstrap-lightbox.min.js"></script> <script src="https://static.zbmath.org/contrib/retina/unknown/retina.js"></script> <script src="https://static.zbmath.org/contrib/bootstrap-select/v1.13.14/js/bootstrap-select.min.js"></script> <script> var SCRIPT_ROOT = ""; </script> <script src="/static/scripts.js?v=20240926"> </script> <script src="https://static.zbmath.org/contrib/mathjax/2.7.1/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script> <script type="text/x-mathjax-config"> MathJax.Hub.Config({ "HTML-CSS": { preferredFont: "TeX", availableFonts: [ "STIX", "TeX" ], linebreaks: { automatic: true }, EqnChunk: (MathJax.Hub.Browser.isMobile ? 10 : 50) }, tex2jax: { processEscapes: true, ignoreClass: "tex2jax_ignore|dno" }, TeX: { Macros: { Aut: "\\operatorname{Aut}", Hom: "\\operatorname{Hom}" }, noUndefined: { attributes: { mathcolor: "#039", //"red", mathbackground: "white", //"#FFEEEE", mathsize: "90%" } } }, messageStyle: "none" }); </script> <script type="text/javascript"> $(document).ready(function() { $("#MathInput").stop(true, true).keyup(function() { $.ajax({ url: "/mwsq/", type: "POST", data: { query : $("#MathInput").val() }, dataType: "text" }) .done(function(xml) { $("#MathPreview").html(xml); $(window).resize(); }); }); var press = jQuery.Event("keyup"); press.ctrlKey = false; press.which = 40; $("#MathInput").trigger(press); }); </script> <div id="new_tab_icon" style="display: none"> <span class="glyphicon glyphicon-new-window" aria-hidden="true"></span><span class="sr-only">(opens in new tab)</span></div> </body> </html>

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