The limit set of trajectory in quasi-homogeneous system in ℝ³

Abstract

In this article, we study the limit set of trajectory in three-dimensional quasi-homogeneous system. For the trajectory which is close to a singular curve, we show that either it approaches a fixed point or infinity, or it oscillates. Moreover, an oscillating example is given. The behaviour of trajectory which is near a closed orbit is also studied. At the end, we classify the integral manifolds of the system with isolated singularities.

Keywords:

• limit set
• quasi-homogeneous system
• oscillating trajectory

AMS Subject Classifications::

• 37C10

Acknowledgements

The authors would like to thank the referee for his/her valuable comments and useful suggestions. This work was supported by NSF of China (No. 10871214), Program for New Century Excellent Talents in University and Ph.D. Programs Foundation of Ministry of Education of China (No. 20100171110040).