Abstract
From the normal form of polynomial differential systems in \(\mathbb {R}^3\) having a sphere as invariant algebraic surface, we obtain a class of quadratic systems depending on ten real parameters, which encompasses the well-known Sprott A system. For this reason, we call them generalized Sprott A systems. In this paper, we study the dynamics and bifurcations of these systems as the parameters are varied. We prove that, for certain parameter values, the z-axis is a line of equilibria, the origin is a non-isolated zero-Hopf equilibrium point, and the phase space is foliated by concentric invariant spheres. By using the averaging theory we prove that a small linearly stable periodic orbit bifurcates from the zero-Hopf equilibrium point at the origin. Finally, we numerically show the existence of nested invariant tori around the bifurcating periodic orbit.
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Acknowledgements
The first author was supported by State of So Paulo Foundation (FAPESP) grant number 2019/10269-3 and by CNPq-Brazil grant number 311355/2018-8.
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Messias, M., Reinol, A.C. (2020). The Occurrence of Zero-Hopf Bifurcation in a Generalized Sprott A System. In: Lacarbonara, W., Balachandran, B., Ma, J., Tenreiro Machado, J., Stepan, G. (eds) Nonlinear Dynamics of Structures, Systems and Devices. Springer, Cham. https://doi.org/10.1007/978-3-030-34713-0_16
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DOI: https://doi.org/10.1007/978-3-030-34713-0_16
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