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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email:

On the localization of invariant tori in a family of generalized standard mappings and its applications to scaling in a chaotic sea

Journal of Applied Nonlinear Dynamics 7(2) (2018) 123--129 | DOI:10.5890/JAND.2018.06.002

Diogo Ricardo da Costa$^{1}$; Iberê L. Caldas$^{2}$; Denis G. Ladeira$^{3}$; Edson D. Leonel$^{4}$

$^{1}$ Departamento de Física, UNESP - Univ Estadual Paulista, Av.24A, 1515, 13506-900, Rio Claro, SP - Brazil

$^{2}$ Instituto de Física da USP, Rua do Matão, Travessa R, 187 - Cidade Universitária, 05314-970, São Paulo, SP - Brazil

$^{3}$ Universidade Federal de São João Del-Rei, Departamento de Física e Matemática. Rodovia MG 443, Km 07, 36420-000, Ouro Branco, MG - Brazil

$^{4}$ Abdus Salam International Center for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy

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The localization of the last invariant spanning curve – also known as the last invariant tori – in a family of generalized standard mappings is discussed. The position of the curve dictates the size of the chaotic sea hence influencing the scaling properties observed for such region. The mapping is area preserving and is constructed such its dynamical variables are the action, J, and the angle θ . The action is controlled by a parameter ε , controlling the intensity of a generic nonlinear function, which defines a transition from integrable for ε = 0 to non integrable for ε≠ 0. The angle is dependent on a parameter γ. If γ > 0, the angle has the property that it diverges in the limit of vanishingly action and is added, by a finite function dependent on a free parameter γ , when the action is larger than zero. The case γ = −1 reproduces the expression of the angle for the traditional standard mapping. The phase space is mixed and shows, for certain ranges of control parameters, a set of periodic islands, chaotic seas and invariant spanning curves. Statistical properties for an ensemble of noninteracting particles starting in the chaotic sea with very low action is considered and we show: (i) the saturation of chaotic orbits grows with εα ; (ii) the regime of growth scales with nβ ; and (iii) the regime that marks the changeover from the diffusive dynamics to the stationary state scales with ε z. The exponents α and z depend on γ and are independent of the nonlinear function f while β is universal. To illustrate the theory here proposed, we obtain an estimation for the critical parameter Kc for a generalized standard mapping considering three different periodic functions. We also find α, β and z for different nonlinear functions.


DRC acknowledges Brazilian agencies PNPD/Capes and FAPESP (2013/22764-2). EDL thanks to CNPq (303707/2015-1) and FAPESP (2012/23688-5), Brazilian agencies. This research was supported by resources supplied by the Center for Scientific Computing (NCC/GridUNESP) of the São Paulo State University (UNESP).


  1. [1]  Moser, J.K. (1962), Nachr. Akad. Wiss. G¨ottingen Math.-Phys. Kl II, 1962, 1.
  2. [2]  Slater, N.B. (1950), Mathematical Proceedings of the Cambridge Philosophical Society, 46, 525.
  3. [3]  Abud, C.V., Caldas, I.L., (2015), Physica D, 308, 34.
  4. [4]  Altmann, E.G., Cristadoro, G. and Pazó, D. (2006), Phys. Rev. E, 73, 056201.
  5. [5]  Canadell, M. and de la Llave, R. (2015), Physica D, 310, 104.
  6. [6]  Laakso, T., Mikko, K. (2016), Physica D, 315, 72.
  7. [7]  Wang, L. and Dyn. J. (2015), Diff. Equat., 27, 283.
  8. [8]  Gentili, G. (2015), Nonlinearity, 28, 2555.
  9. [9]  Kaloshin, V. and Zhang, K. (2015), Nonlinearity, 28, 2699.
  10. [10]  Lichtenberg, A.J. and Lieberman, M.A. (1992), Regular and Chaotic Dynamics, Appl. Math. Sci., 38, Springer, New York.
  11. [11]  Chirikov, B.V. (1979), Phys. Rep., 52, 263.
  12. [12]  Lieberman, M.A. and Lichtenberg, A.J. (1971), Phys. Rev. A, 5, 1852.
  13. [13]  Karlis, A.K., et. al (2006), Phys. Rev. Lett., 97, 194102.
  14. [14]  Pustylnikov, L.D. (1978), Trans. Mosc. Math. Soc., 2, 1.
  15. [15]  Howard, J.E. and Humpherys, J. (1995), Physica D, 80, 256.
  16. [16]  Leonel, E.D. and McClintock, P.V.E. (2005), J. Phys. A, 38, 823.
  17. [17]  Petrosky, T. Y. (1986), Phys. Lett. A, 117, 328.
  18. [18]  Leonel, E.D., de Oliveira, J.A., and Saif, F. (2011), Phys. A: Math. Theor., 44, 302001.
  19. [19]  MacKay, R.S. (1992), Nonlinearity, 5, 161.