Skip to main content
SpringerLink
Log in

Global Dynamics and Bifurcation of Periodic Orbits in a Modified Nosé-Hoover Oscillator

  • Published:
Journal of Dynamical and Control Systems Aims and scope Submit manuscript

Abstract

We perform a global dynamical analysis of a modified Nosé-Hoover oscillator, obtained as the perturbation of an integrable differential system. Using this new approach for studying such an oscillator, in the integrable cases, we give a complete description of the solutions in the phase space, including the dynamics at infinity via the Poincaré compactification. Then using the averaging theory, we prove analytically the existence of a linearly stable periodic orbit which bifurcates from one of the infinite periodic orbits which exist in the integrable cases. Moreover, by a detailed numerical study, we show the existence of nested invariant tori around the bifurcating periodic orbit. Finally, starting with the integrable cases and increasing the parameter values, we show that chaotic dynamics may occur, due to the break of such an invariant tori, leading to the creation of chaotic seas surrounding regular regions in the phase space.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Cima A, Llibre J. Bounded polynomial vector fields. Trans Amer Math Soc 1990;318:557–579.

    Article  MathSciNet  Google Scholar 

  2. Guckenheimer J, Holmes P. Nonlinear oscillations, dynamical systems and bifurcations of vectors fields. New York: Springer; 1983.

    Book  Google Scholar 

  3. Holian BL, Voter AF. Thermostatted molecular dynamics: How to avoid the Toda demon hidden in Nosé-Hoover dynamics. Phys Rev E 1995;52:2338–2347.

    Article  Google Scholar 

  4. Hoover WG. Canonical dynamics: Equilibrium phase-space distributions. Phys Rev A 1985;31:1695–1697.

    Article  Google Scholar 

  5. Jafari S, Sprott JC, Dehghan S. Categories of conservative flows. Int J Bifurcat Chaos 2019;29:1950021 (16 pages).

    MathSciNet  MATH  Google Scholar 

  6. Llibre J, Messias M. Global dynamics of the Rikitake system. Physica D 2009; 238:241–252.

    Article  MathSciNet  Google Scholar 

  7. Llibre J, Messias M, da Silva PR. Global dynamics in the Poincaré ball of the Chen system having invariant algebraic surfaces. Int J Bifurcat Chaos 2012;22: 1250154 (17 pages).

    MATH  Google Scholar 

  8. Mahdi A, Valls C. Integrability of the Nosé–Hoover equation. J Geom Phys 2011;61:1348–1352.

    Article  MathSciNet  Google Scholar 

  9. Messias M. Dynamics at infinity and the existence of singularly degenerate heteroclinic cycles in the Lorenz system. J Phys A: Math Theor 2009;42:115101(18 pages).

    Article  MathSciNet  Google Scholar 

  10. Messias M, Reinol AC. On the existence of periodic orbits and KAM tori in the Sprott A system: a special case of the Nosé-Hoover oscillator. Nonlinear Dynam 2018; 92:1287–1297.

    Article  Google Scholar 

  11. Nosé S. A unified formulation of the constant temperature molecular-dynamics methods. J Chem Phys 1984;81:511–519.

    Article  Google Scholar 

  12. Nosé S. A molecular dynamics method for simulations in the canonical ensemble. Molecular Phys 1984;52:255–268.

    Article  Google Scholar 

  13. Posch HA, Hoover WG, Vesely FJ. Canonical dynamics of the Nosé oscillator: Stability, order, and chaos. Phys Rev A 1986;33:4253–4265.

    Article  MathSciNet  Google Scholar 

  14. Rech PC. Quasiperiodicity and chaos in a generalized Nosé-Hoover Oscillator. Int J Bifurcat Chaos 2016;26:16501701 (7 pages).

    Article  Google Scholar 

  15. Sprott JC, Hoover WG, Hoover CG. Heat conduction, and the lack thereof, in time reversible dynamical systems: Generalized Nosé-Hoover oscillators with a temperature gradient. Phys Rev E 2014;89:042914.

    Article  Google Scholar 

  16. Sprott JC. A dynamical system with a strange attractor and invariant tori. Phys Lett A 2014;378:1361–1363.

    Article  MathSciNet  Google Scholar 

  17. Sprott JC. Strange attractors with various equilibrium types. Eur Phys J Special Topics 2015;224:1409–1419.

    Article  Google Scholar 

  18. Swinnerton-Dyer P, Wagenknecht T. Some third-order ordinary differential equations. Bull London Math Soc 2008;40:725–748.

    Article  MathSciNet  Google Scholar 

  19. Vehrulst F. Nonlinear differential equations and dynamical systems. Universitext. Berlin: Springer; 1996.

    Book  Google Scholar 

  20. Wang L, Yang X-S. The invariant tori of knot type and the interlinked invariant tori in the Nosé-Hoover oscillator. Eur Phys J B 2015;88:78 (5 pages).

    Google Scholar 

  21. Wolf A, Swift JB, Swinney HL, Vastano JA. Determining Lyapunov exponents from a time series. Physica D 1985;16:285–317.

    Article  MathSciNet  Google Scholar 

Download references

Funding

The first author was partially supported by the Ministerio de Ciencia, Innovación y Universidades, Agencia Estatal de Investigación grant MTM2016-77278-P (FEDER), the Agència de Gestió d’Ajuts Universitaris i de Recerca grant 2017SGR1617, and the H2020 European Research Council grant MSCA-RISE-2017-777911. The second author was partially supported by CNPq-Brazil under the grant 311355/2018-8 and by São Paulo Research Foundation (FAPESP) grant number 2019/10269-3.

Author information

Authors and Affiliations

  1. Departament de Matemàtiques, Universitat Autònoma de Barcelona (UAB), 08193, Bellaterra, Barcelona, CT, Spain

    Jaume Llibre

  2. Departamento de Matemática e Computação, Universidade Estadual Paulista (UNESP), 19060–900, Presidente Prudente, SP, Brazil

    Marcelo Messias

  3. Departamento Acadêmico de Matemática, Universidade Tecnológica Federal do Paraná, UTFPR, 86812-460, Apucarana, Paraná, Brazil

    Alisson C. Reinol

Authors
  1. Jaume Llibre
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marcelo Messias
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alisson C. Reinol
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marcelo Messias.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Llibre, J., Messias, M. & Reinol, A.C. Global Dynamics and Bifurcation of Periodic Orbits in a Modified Nosé-Hoover Oscillator. J Dyn Control Syst 27, 491–506 (2021). https://doi.org/10.1007/s10883-020-09491-5

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10883-020-09491-5

Keywords

Mathematics Subject Classification (2010)

Search

Navigation