Geometry of canonical regions and elliptic sectors in holomorphic flows
Keywords:
Holomorphic dynamical system, entire vector field, center, period annulus, focus, node, basin of attraction/stability, elliptic sector, canonical regionAbstract
In this follow-up paper, we investigate the global geometry and topology of dynamical systems $\dot{x}=F(x)$ with entire vector field $F$, building on and constructively extending the local structure of simple and higher-order equilibria. We provide a step-by-step analysis to reveal topological properties of the basins of centers, nodes, and foci, while excluding isolated equilibria at the boundaries of the latter two. We propose a definition of global elliptic sectors and introduce the concept of sector-forming orbits based on the geometry within a finite elliptic decomposition of multiple equilibria. Finally, we characterize the structure of heteroclinic regions connecting two equilibria.
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
Antonio Garijo, Armengol Gasull and Xavier Jarque, Normal forms for singularities of one dimensional holomorphic vector fields, Electronic Journal of Differential Equations (2004), no. 122, 1-7.
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.
Dean A. Neumann, Classification of continuous flows on 2-manifolds, Proceedings of the American Mathematical Society 48 (1975), no. 1, 73-81.
Lawrence Markus, Global structure of ordinary differential equations in the plane, Transactions of the American Mathematical Society 76 (1954), no. 1, 127-148.
Lawrence Markus, Global integrals of $f z x+g z y=h$, Bulletins de l'Académie Royale de Belgique 38 (1952), no. 1, 311-332.
Dean A. Neumann and Thomas O'Brien, Global structure of continuous flows on 2-manifolds, Journal of differential equations 22 (1976), no. 1, 89-110.
Janina Kotus, Michał Krych, and Zbigniew Nitecki, Global structural stability of flows on open surfaces, Memoirs of the American Mathematical Society 261 (1982).
Louis Brickman and E. S. Thomas, Conformal equivalence of analytic flows, Journal of Differential Equations 25 (1977), no. 3, 310-324.
Antonio Garijo, Armengol Gasull, and Xavier Jarque, Local and global phase portrait of equation $\dot{z}=f(z)$, Discrete and Continuous Dynamical Systems 17 (2006), no. 2, 309-329.
Allen Hatcher, Algebraic topology, Tsinghua University Press Ltd., 2005
Freddy Dumortier, Jaume Llibre and Joan Artés, Differential equations and dynamical systems, Springer, 2006
Pascal Heiter and Dirk Lebiedz, Towards differential geometric characterization of slow invariant manifolds in extended phase space: Sectional curvature and flow invariance, SIAM Journal on Applied Dynamical Systems 17 (2018), 732-753.
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/
Nicolas Kainz and Dirk Lebiedz, Local geometry of Equilibria and a Poincaré-Bendixson-type Theorem for Holomorphic Flows, Topology Proc. 65 (2025), 99116.
Johannes Poppe and Dirk Lebiedz, Sensitivities in complex-time flows: Phase transitions, Hamiltonian structure, and differential geometry, Chaos 35 (2025), no. 3, 033114.
Dirk Lebiedz, Holomorphic Hamiltonian $\xi$-Flow and Riemann Zeros, https:// doi.org/10.48550/arXiv.2006.09165
Dirk Lebiedz, Julia Kammerer and Ulrich Brandt-Pollmann, Automatic network coupling analysis for dynamical systems based on detailed kinetic models, Physical Review E 72, 041911 (2005)
Dirk Lebiedz, Volkmar Reinhardt and Julia Kammerer, Novel Trajectory Based Concepts for Model and Complexity Reduction in (Bio)Chemical Kinetics, in Model Reduction and Coarse-Graining Approaches for Multiscale Phenomena, 343-364 (2006)
Dirk Lebiedz, Jochen Siehr and Jonas Unger, A variational principle for computing slow invariant manifolds in dissipative dynamical systems, SIAM Journal on Scientific Computing 33, 703-720 (2011)
Dirk Lebiedz and Jochen Siehr, A continuation method for the efficient solution of parametric optimization problems in kinetic model reduction, SIAM Journal on Scientific Computing 35, 1584-1603 (2013)
Dirk Lebiedz and Jonas Unger, On unifying concepts for trajectory-based slow invariant attracting manifold computation in kinetic multiscale models, Mathematical and Computer Modelling of Dynamical Systems 22, 87-112 (2016)
Dirk Lebiedz and Johannes Poppe On differential geometric formulations of slow invariant manifold computation: geodesic stretching and flow curvature, Journal of Dynamical Systems and Geometric Theories 20 1-32 (2022)
James R. Munkres, Topology. Prentice Hall Upper Saddle River, 2000
Lawrence Perko, Differential Equations and Dynamical Systems, Springer, 2006 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.
Gerald Teschl, Ordinary differential equations and dynamical systems, American Mathematical Soc., 2012
Wilson A. Sutherland, Introduction to metric and topological spaces, Oxford University Press, 2009
