Bifurcation diagram and stability for a one-parameter family of planar vector fields

@article{GarciaSaldana2013BifurcationDA,
  title={Bifurcation diagram and stability for a one-parameter family of planar vector fields},
  author={J. D. Garc'ia-Saldana and A. Gasull and H. Giacomini},
  journal={Journal of Mathematical Analysis and Applications},
  year={2013},
  volume={413},
  pages={321-342}
}
We consider the 1-parameter family of planar quintic systems, ˙x = y3−x3, y˙ = −x + my5, introduced by A. Bacciotti in 1985. It is known that it has at most one limit cycle and that it can exist only when the parameter m is in (0.36, 0.6). In this paper, using the Bendixon-Dulac theorem, we give a new unified proof of all the previous results, we shrink this to (0.547, 0.6), and we prove the hyperbolicity of the limit cycle. We also consider the question of the existence of polycycles. The main… Expand

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References

SHOWING 1-10 OF 35 REFERENCES
Limit-Cycles and Rotated Vector Fields
The disappearance of the limit cycle in a mode interaction problem with symmetry
Differential equations defined by the sum of two quasi-homogeneous vector fields
Monodromy and Stability for Nilpotent Critical Points
A global analysis of the Bogdanov-Takens system
Symétrie et forme normale des centres et foyers dégénérés
...
1
2
3
4
...