On the existence of periodic orbits and KAM tori in the Sprott A system: a special case of the Nos,-Hoover oscillator

Author(s)

Messias, Marcelo ^[1] ; Reinol, Alisson C. ^[2]

Affiliation

^[1] Univ Estadual Paulista UNESP, Dept Matemat & Comp, Fac Ciencias & Tecnol, Presidente Prudente, SP - Brazil

^[2] Univ Estadual Paulista UNESP, Inst Biociencias Letras & Ciencias Exatas, Dept Matemat, Sao Jose Do Rio Preto, SP - Brazil

• Document type: Journal article
• Source: NONLINEAR DYNAMICS; v. 92, n. 3, p. 1287-1297, MAY 2018.
• Web of Science Citations: 1

Abstract

We consider the well-known Sprott A system, which is a special case of the widely studied Nos,-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 the Sprott A system has neither invariant algebraic surfaces nor polynomial first integrals. For 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. (AU)

• FAPESP's process: 13/26602-7 - Integrability and global dynamics of quadratic vector fields defined on R3 with Quadrics as invariant surfaces
• Grantee: Alisson de Carvalho Reinol
• Support type: Scholarships in Brazil - Doctorate
• FAPESP's process: 13/24541-0 - Ergodic and qualitative theory of dynamical systems
• Grantee: Claudio Aguinaldo Buzzi
• Support type: Research Projects - Thematic Grants