Abstract
The first objective of this paper was to study the Darboux integrability of the polynomial differential system
and the second one is to show that for \(a>0\) sufficiently small this model exhibits two small amplitude periodic solutions that bifurcate from a zero-Hopf equilibrium point localized at the origin of coordinates when \(a=0\). We note that this polynomial differential system introduced by Chen and Wang (Nonlinear Dyn 71:429–436, 2013) is relevant in the sense that it is the first system in \(\mathbb {R}^3\) exhibiting chaotic motion without having equilibria.
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Acknowledgments
The first author is partially supported by a MINECO/ FEDER Grant MTM2008-03437, and MTM2013-40998-P, an AGAUR Grant Number 2014SGR-568, an ICREA Academia, the Grants FP7-PEOPLE-2012-IRSES 318999 and 316338, FEDER-UNAB-10-4E-378. The first two authors are also supported by the joint projects FP7-PEOPLE-2012-IRSES numbers 316338 and a CAPES Grant Number 88881.030454/2013-01 from the program CSF-PVE. The third author is supported by Portuguese National Funds through FCT - Fundação para a Ciência e a Tecnologia within the project PEst-OE/EEI/ LA0009/2013 (CAMGSD).
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Appendix: Roots of a cubic polynomial
Appendix: Roots of a cubic polynomial
We recall that the discriminant \(\Delta \) of the polynomial \(ax^3+bx^2+cx+d\) is
It is known that
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If \(\Delta > 0\), then the equation has three distinct real roots.
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If \(\Delta = 0\), then the equation has a root of multiplicity \(2\) and all its roots are real.
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If \(\Delta < 0\), then the equation has one real root and two non–real complex conjugate roots.
For more details see [1].
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Llibre, J., Oliveira, R.D.S. & Valls, C. On the integrability and the zero-Hopf bifurcation of a Chen–Wang differential system. Nonlinear Dyn 80, 353–361 (2015). https://doi.org/10.1007/s11071-014-1873-4
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DOI: https://doi.org/10.1007/s11071-014-1873-4