A polynomial class of Markus–Yamabe counterexamples and examples in ℝ³

Abstract

We give a polynomial family of counterexamples to the Markus–Yamabe conjecture which contains and generalizes that of Cima–Gasull–Mañosas. Furthermore, we construct a new class of polynomial vector fields in ℝ³ having the origin as a global attractor which satisfy the Markus–Yamabe hypotheses.

Keywords:

• polynomial vector fields
• global attractors

AMS Subject Classifications::

• 37C10
• 37C70
• 37C75

Acknowledgements

The authors were supported in part by FONDECYT Grant #1080172, by CONICYT Grant PBCT ADI 17, and by MATH–AMSUD DySET. The second author was also supported by FONDECYT Postdoctoral Grant #3100082.