Nonlinear Dynamics

, Volume 92, Issue 3, pp 1287–1297 | Cite as

On the existence of periodic orbits and KAM tori in the Sprott A system: a special case of the Nosé–Hoover oscillator

  • Marcelo Messias
  • Alisson C. Reinol
Original Paper


We consider the well-known Sprott A system, which is a special case of the widely studied Nosé–Hoover oscillator. The system depends on a single real parameter a, and for suitable choices of the parameter value, it is shown to present chaotic behavior, even in the absence of an equilibrium point. In this paper, we prove that, for \(a\ne 0,\) the Sprott A system has neither invariant algebraic surfaces nor polynomial first integrals. For \(a>0\) small, by using the averaging method we prove the existence of a linearly stable periodic orbit, which bifurcates from a non-isolated zero-Hopf equilibrium point located at the origin. Moreover, we show numerically the existence of nested invariant tori surrounding this periodic orbit. Thus, we observe that these dynamical elements and their perturbation play an important role in the occurrence of chaotic behavior in the Sprott A system.


Sprott A system Nosé–Hoover oscillator Averaging method Periodic orbits Nested invariant tori Invariant algebraic surfaces Chaotic behavior 



The first author is supported by FAPESP Grant Number 2013/24541–0, by CNPq Grant Number 308159/2015–2 and by CAPES - Program CSF-PVE, Grant Number 88881.030454/2013. The second author is supported by FAPESP Grant Number 2013/26602-7. They are also grateful to the anonymous referee who pointed out the equivalence between the Sprott A system and the Nosé–Hoover oscillator, which enabled them to enrich the presentation of the results in this paper, by comparing both systems.


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Authors and Affiliations

  1. 1.Departamento de Matemática e Computação, Faculdade de Ciências e TecnologiaUniversidade Estadual Paulista (UNESP)Presidente PrudenteBrazil
  2. 2.Departamento de Matemática, Instituto de Biociências, Letras e Ciências ExatasUniversidade Estadual Paulista (UNESP)São José do Rio PretoBrazil

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