TY - GEN

T1 - Secants of trajectories in dimension three

AU - Alonso-González, C.

AU - Cano, F.

AU - Rosas, R.

PY - 2013

Y1 - 2013

N2 - In this paper we give a description of the sets of accumulation of secants for orbits of real analytic vector fields in dimension three with the origin as only ω-limit point. It is an infinitesimal version of the Poincaré-Bendixson problem in dimension three. These sets have structure of cyclic graph when the singularities are isolated under one blow-up. If the reduction of singularities is hyperbolic, under conditions of Morse-Smale type, we prove that the accumulation set is a single point or homeomorphic to S1.

AB - In this paper we give a description of the sets of accumulation of secants for orbits of real analytic vector fields in dimension three with the origin as only ω-limit point. It is an infinitesimal version of the Poincaré-Bendixson problem in dimension three. These sets have structure of cyclic graph when the singularities are isolated under one blow-up. If the reduction of singularities is hyperbolic, under conditions of Morse-Smale type, we prove that the accumulation set is a single point or homeomorphic to S1.

UR - http://www.scopus.com/inward/record.url?scp=84893309176&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-38830-9_1

DO - 10.1007/978-3-642-38830-9_1

M3 - Conference contribution

AN - SCOPUS:84893309176

SN - 9783642388293

T3 - Springer Proceedings in Mathematics and Statistics

SP - 1

EP - 6

BT - Progress and Challenges in Dynamical Systems - Proceedings of the International Conference Dynamical Systems

T2 - International conference "Dynamical Systems: 100 years after Poincare"

Y2 - 3 September 2012 through 7 September 2012

ER -