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Original Articles

From wild Lorenz-like to wild Rovella-like dynamics

, &
Pages 525-542 | Received 10 Feb 2015, Accepted 06 Aug 2015, Published online: 11 Sep 2015

References

  • Bamón R, Kiwi J, Rivera-Letelier J. Wild Lorenz-like attractors. arXiv 0508045. 2006. Available from: http://arxiv.org/abs/math/0508045.
  • Guckenheimer J. A strange, strange attractor. In: Marsden JE, McCracken M, editors. The Hopf bifurcation theorem and its applications. New York: Springer-Verlag; 1976. p. 368–381.
  • Guckenheimer J, Williams RF. Structural stability of Lorenz attractors. Inst Hautes Études Sci Publ Math. 1979;50(50):59–72.
  • Afraimovich VS, Bykov VV, Shilnikov LP. The origin and structure of the Lorenz attractor. Sov Phys Dokl. 1977;22:253–255.
  • Afraimovich VS, Bykov VV, Shilnikov LP. On structurally unstable attracting limit sets of Lorenz attractor type. Trans Mosc Math Soc. 1983;44:153–216.
  • Lorenz E. Deterministic nonperiodic flow. J Atmos Sci. 1963;20:130–141.
  • Asaoka M. Hyperbolic sets exhibiting C1-persistent homoclinic tangency for higher dimensions. Proc Am Math Soc. 2008;136(2):677–686.
  • Asaoka M. Erratum to “Hyperbolic sets exhibiting C1-persistent homoclinic tangency for higher dimensions”. Proc Am Math Soc. 2010;138(4):1533.
  • Gonchenko SV, Ovsyannikov II, Simó C, et al. Three-dimensional Hénon-like maps and wild Lorenz-like attractors. Int J Bifurc Chaos Appl Sci Eng. 2005;15(11):3493–3508.
  • Gonchenko SV, Shilnikov LP, Turaev DV. On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors. Regul Chaot Dyn. 2009;14(1):137–147.
  • Turaev DV, Shilnikov LP. An example of a wild strange attractor. Mat Sb. 1998;189(1–2):291–314.
  • Hittmeyer S, Krauskopf B, Osinga HM. Interacting global invariant sets in a planar map model of wild chaos. SIAM J Appl Dyn Syst. 2013;12(3):1280–1329.
  • Osinga HM, Krauskopf B, Hittmeyer S. Chaos and wild chaos in Lorenz-type systems. In: AlSharawi Z, Cushing J, Elaydi S, editors. Theory and applications of difference equations and discrete dynamical systems. Berlin: Springer-Verlag; 2014. p. 75–98.
  • Rovella A. The dynamics of perturbations of the contracting Lorenz attractor. Bol Soc Brasil Mat. 1993;24(2):233–259. doi:10.1007/BF01237679.
  • Keller G, StPierre M. Topological and measurable dynamics of Lorenz maps. In: Fiedler B, editor. Ergodic theory, analysis, and efficient simulation of dynamical systems. Berlin: Springer-Verlag; 2001. p. 333–361.
  • Araújo V, Castro A, Pacifico MJ, et al. Multidimensional Rovella-like attractors. J Differ Equat. 2011;251(11):3163–3201.
  • Hittmeyer S, Krauskopf B, Osinga HM. Interactions of the Julia set with critical and (un)stable sets in an angle-doubling map on . Int J Bifurc Chaos. 2015;25(4):1530013.
  • Bunimovich LA, Sinai JG. Stochasticity of the attractor in the Lorenz model. In: Gaponov-Grekhov AV, editor. Proceedings of the winter school on nonlinear waves. Moskow: Nauka Press; 1979. p. 212–226.
  • Sinai JG, Vul EB. Hyperbolicity conditions for the Lorenz model. Phys D. 1981;2(1):3–7.
  • Tucker W. The Lorenz attractor exists. C R Acad Sci Paris Sér I Math. 1999;328(12):1197–1202.
  • Palis J, de Melo W. Geometric theory of dynamical systems. New York: Springer-Verlag; 1982.
  • Mira C, Gardini L, Barugola A, et al. Chaotic dynamics in two-dimensional noninvertible maps. Vol. 20 of World Scientific series on nonlinear science, Series A: monograph and treatises. Singapore: World Scientific; 1996.
  • Dhooge A, Govaerts W, Kuznetsov YuA. MatCont: a Matlab package for numerical bifurcation analysis of ODEs. ACM Trans Math Software. 2003;29(2):141–164. Available from: http://sourceforge.net/projects/matcont.
  • Ghaziani RK, Govaerts W, Kuznetsov YuA, et al. Numerical continuation of connecting orbits of maps in Matlab. J Differ Equat Appl. 2009;15(8–9):849–875.
  • Govaerts W, Kuznetsov YuA, Khoshsiar Ghaziani R, et al. Cl_MatContM: a toolbox for continuation and bifurcation of cycles of maps. 2008; software. Available from: http://sourceforge.net/projects/matcont.
  • Aronson DG, Chory MA, Hall GR, et al. Bifurcations from an invariant circle for two-parameter families of maps of the plane: a computer-assisted study. Commun Math Phys. 1982;83(3):303–354.
  • Schilder F, Peckham BB. Computing Arnol′d tongue scenarios. J Comput Phys. 2007;220(2):932–951.
  • Grebogi C, Ott E, Yorke JA. Fractal basin boundaries, long-lived chaotic transients, and unstable-unstable pair bifurcation. Phys Rev Lett. 1983;50(13):935–938.
  • Doedel EJ, Krauskopf B, Osinga HM. Global invariant manifolds in the transition to preturbulence in the Lorenz system. Indag Math. 2011;22(3–4):222–240.
  • Doedel EJ, Krauskopf B, Osinga HM. Global organization of phase space in the transition to chaos in the Lorenz system. Nonlinearity. 2015. (to appear).
  • Kostelich EJ, Kan I, Grebogi C, et al. Unstable dimension variability: a source of nonhyperbolicity in chaotic systems. Phys D. 1997;109(1–2):81–90.
  • Bonatti C, Díaz L. Robust heterodimensional cycles and C1-generic dynamics. J Inst Math Jussieu. 2008;7(3):469–525.
  • Gonchenko SV, Shilnikov LP, Turaev DV. On dynamical properties of multidimensional diffeomorphisms from Newhouse regions. I. Nonlinearity. 2008;21(5):923–972.
  • Shinohara K. An example of C1-generically wild homoclinic classes with index deficiency. Nonlinearity. 2011;24(7):1961–1974.
  • Shinohara K. On the index problem of C1-generic wild homoclinic classes in dimension three. Discrete Contin Dyn Syst Ser A. 2011;31(3):913–940.
  • Bonatti C, Díaz LJ, Viana M. Dynamics beyond uniform hyperbolicity. a global geometric and probabilistic perspective. Vol. 102 of Encyclopaedia of Mathematical Sciences. Berlin: Springer-Verlag; 2005.
  • Zhang W, Krauskopf B, Kirk V. How to find a codimension-one heteroclinic cycle between two periodic orbits. Discrete Contin Dyn Syst Ser A. 2012;32(8):2825–2851.

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