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Topological Dynamics and Continuum Theory

Dynamics/CT Session #2 #2

Subevent of Dynamics/CT Session #2

Central Time (US & Canada)

Starts at: 2025-08-12 09:00AM

Ends at: 2025-08-12 09:25AM

Elliptic sectors and heteroclinic regions in real time holomorphic flows

Nicolas Kainz ⟨nicolas.kainz@uni-ulm.de⟩

Abstract:

The geometric description of the phase space of holomorphic dynamical systems with real time is a crucial research field. In this context, the globalization of local structures is of particular interest. For example, the local structure of an equilibrium of order mN1 has already been sufficiently investigated and characterized. Locally, there exist 2m2 elliptic sectors, all consisting of homoclinic trajectories tending to the equilibrium in both time directions. Now the question arises how this local structure can be globalized using analytical tools, as is the case, for example, with the basin of attraction of nodes and foci. I present a method to define a global elliptic sector based on so-called “sector-forming orbits” and show some topological properties for it: The global elliptic sector is open, flow-invariant, path-connected, and simply connected. Moreover, all orbits are nested inside each other, consistent with the intuitive notion of an elliptic sector. Furthermore, for the case m3, it coincides with the naively defined global elliptic sector, which merely contains all homoclinic trajectories with adjacent definite directions. This gives us an analytic precise definition of a globalization of a locally defined elliptic sector, together with important and useful topological properties. Moreover, it is possible to investigate the geometrical structure that can occur between two global elliptic sectors with no common boundary near the equilibrium. In this context, the question also arises as to how many so-called “heteroclinic regions” can appear between two such sectors.

Notes:

[1]: Nicolas Kainz and Dirk Lebiedz, “Local geometry of Equilibria and a Poincar´e-Bendixson-type Theorem for Holomorphic Flows”, Topology Proc. 65 (2025), 99–116.

[2]: Kevin A. Broughan: “The structure of sectors of zeros of entire flows”, Topology Proc. 27, no. 2 (2003), 379–394.

[3]: Freddy Dumortier, Jaume Llibre, Joan C. Artés, “Qualitative theory of planar differential systems”, Springer, 2006.

[4]: Alvarez–Parrilla, A., and Mucino–Raymundo, J.. “Dynamics of singular complex analytic vector fields with essential singularities I”. Conformal Geometry and Dynamics of the American Mathematical Society 21.7 (2017), 126-224.

[5]: Antonio Garijo, Armengol Gasull, and Xavier Jarque, “Local and global phase portrait of equation z˙=f(z)”, Discrete and Continuous Dynamical Systems 17.2 (2006), 309-329.

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