Dependency of the positive and negative long-time behaviors of flows on surfaces. (English) Zbl 1528.37037
In this paper, the author studies the structure of \(\omega\)-limit sets of flows defined on compact surfaces. In particular, it is shown that the \(\omega\)-limit of an orbit whose closure has non-empty interior (locally dense orbit), is either a nowhere dense subset of singular points or the closure of a non-closed locally dense recurrent trajectory (Theorem A). Further, the author observes that this is also the structure depicted by the \(\omega\)-limit sets of non-closed trajectories of non-wandering flows (Theorem B). For Hamiltonian flows, Theorem C establishes that \(\omega\)-limits of non-closed trajectories on possibly non-compact surfaces consist of singular points.
The author also introduces the notion of pre-Hamiltonian flow which is a flow whose non-singular trajectories are the connected components of the level sets of the restriction of some real-valued function to the surface obtained by removing the singular points, and comprise a codimension-one foliaton of this new surface. Theorem E states that if a flow has a finite number of singular points being pre-Hamiltonian is equivalent to being Hamiltonian.
The author also introduces the notion of pre-Hamiltonian flow which is a flow whose non-singular trajectories are the connected components of the level sets of the restriction of some real-valued function to the surface obtained by removing the singular points, and comprise a codimension-one foliaton of this new surface. Theorem E states that if a flow has a finite number of singular points being pre-Hamiltonian is equivalent to being Hamiltonian.
Reviewer: Héctor Barge (Madrid)
MSC:
| 37E35 | Flows on surfaces |
| 37C15 | Topological and differentiable equivalence, conjugacy, moduli, classification of dynamical systems |
| 37C75 | Stability theory for smooth dynamical systems |
| 37J39 | Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.) |
| 37J12 | Fixed points and periodic points of finite-dimensional Hamiltonian and Lagrangian systems |
Keywords:
Poincaré-Bendixson theorem; \( \omega \)-limit set; non-wandering flow; locally dense orbit; totally disconnectivity of singular pointsReferences:
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