Local geometry of equilibria and a Poincaré-Bendixson-type theorem for holomorphic flows
Keywords:
Holomorphic dynamical system, local geometry of equilibria, definite directions, finite elliptic decomposition, elliptic sectors, Poincaré-BendixsonAbstract
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 [Differential equations and dynamical systems, Lawrence Perko (1990)] under the additional condition of holomorphy. This enables the geometric construction of a finite elliptic decomposition. We derive a holomorphic Poincaré-Bendixson-type theorem, leading to the conclusion that bounded non-periodic orbits are always homoclinic or heteroclinic.
References
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
Kevin A. Broughan, Holomorphic flows on simply connected regions have no limit cycles, Meccanica 38 (2003), no. 6, 699-709.
Kevin A. Broughan, The structure of sectors of zeros of entire flows, Topology Proc. 27 (2003), no. 2, 379-394.
Kevin A. Broughan, The holomorphic flow of Riemanns function $\xi(z)$, Nonlinearity 18 (2004), 1269-1294.
Kevin A. Broughan and Antony Ross Barnett, The holomorphic flow of the Riemann zeta function, Mathematics of Computation 73 (2004), 987-1004.
Freddy Dumortier, Jaume Llibre and Joan Artés, Differential equations and dynamical systems, Springer, 2006
Allen Hatcher, Algebraic topology, Tsinghua University Press Ltd., 2005
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
Marek Izydorek, Slawomir Rybicki and Zbigniew Szafraniec, A note on the Poincaré-Bendixson index theorem, Department of Mathematics, Tokyo Institute of Technology, Kodai Mathematical Journal 19 (1996), no. 2, 145-156.
Camille Jordan, Cours d'analyse de l'École polytechnique, Gauthier-Villars et fils, 1893
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/nicolaskainz/#c963531
Dirk Lebiedz, Holomorphic Hamiltonian $\xi$-Flow and Riemann Zeros, https://doi.org/10.48550/arXiv.2006.09165
James R. Munkres, Topology. Prentice Hall Upper Saddle River, 2000
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.
Lawrence Perko, Differential equations and dynamical systems, Springer, 1990
Jan Prüss and Mathias Wilke, Gewöhnliche Differentialgleichungen und dynamische Systeme, Springer, 2010
Wolfgang P. Schleich, Ida Bezděková, 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.
