Abstract
Lorenz type attractors are found from a codimension one bifurcation of a system on the boundary of Morse-Smale systems. Conditions of their emerging are formulated in terms of conventional Floquet exponents of homoclinic orbits—a new characteristic of homoclinic orbits at the bifurcation moment.
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References
Afraimovich, V. (1990). On Smooth Changes of Variables.Selecta Math. Sovietica. 9, (3) 205–213.
Afraimovich, V., Arnold, V., Ilyashenko, Y., and Shilnikov, L. (1986).The Theory of Bifurcations, Moscow, Nauka (in Russian). Translated inEncyclopedia of Math. Sci. Vol. 5, Springer-Verlag, 1994.
Afraimovich, V., Bykov, V., and Shilnikov, L. (1983). On Structurally Unstable Attracting Limit Sets of Lorenz Attractor Type.Tran. of Moscow Math. Soc. 44, 153–216.
Afraimovich, V., and Pesin, Ya. (1987). Dimension of Lorenz Type Attractors.Sov. Sci. Rev. C Math./Phys. 6, 169–241.
Afraimovich, V., and Shilnikov, L. (1974). On Some Global Bifurcations Connected with the Disappearance of a Fixed Point of Saddle-node Type.Doklady Akad. Nauk. SSSR 219, 1281–1285 (in Russian). English translation inSov. Math. Doklady.
Benedicks, M., and Carleson, L. (1993). The Dynamics of the Henon Map.Am. of Math.
Chow, S.-N., and Lu, K. (1988).C k center unstable manifolds.Royal Soc. Edinburgh 108(A), 303–320.
Deng, B. (1989). The Shilnikov problem, exponential expansion, strongλ-lemma,C 1-linearizations and homoclinic bifurcation.J. Differential Equations 79, 189–231.
Dumortier, F., Kokubu, H., and Oak, H. (1993). A Degenerate Singularity Denerating Geometric Lorenz Attractors. CDSNS92-110 Georgia Tech. (Preprint).
Guckenheimer, J. (1976). A Strange Attractor. In Marsden, J., and McCracken, M. (eds.),Hopf Bifurcation and its Applications, Springer-Verlag, pp. 368–381.
Guckenheimer, J., and Williams, R. (1979). Structural Stability of the Lorenz Attractors.Publ. Math. IHES 50, 59–72.
Hirsch, M., Pugh, C., and Shub, M. (1976). Invariant Manifolds.Lec. Notes in Math., No. 583, Springer-Verlag, New York.
Mischaikow, K., and Mrozek, M. Chaos in the Lorenz Equations: a Computer Assisted Proof.Bull. AMS (to appear).
Newhouse, S., Palis, J., and Takens, F. (1983). Bifurcations and Stability of Families of Diffeomorphisms.Publ. Math. IHES 57, 5–71.
Pesin, Ya. (1992). Dynamical Systems With Generalized Hyperbolic Attractors: Hyperbolic, Ergodic and Topological Properties.Ergod. Theor. Dynam. Syst. 12, 123–151.
Lorenz, E. (1963). Deterministic nonperiodic flow.J. of Atm. Sci. 20, 130–141.
Robinson, C. (1989). Homoclinic bifurcation to a transitive attractor of Lorenz type. I.Non-linearity 2, 495–518.
Rychlik, M. (1990). Lorenz attractors through Silnikov-type bifurcation: Part 1.Ergod. Theor. Dynam. Syst. 10, 793–821.
Smale, S. (1967). Differentiable Dynamical Systems.Bull. Amer. Math. Soc. 73, 747–817.
Shilnikov, A., Shilnikov, L., and Turaev, D. (1993). Normal Forms and Lorenz Attractors.International J. of Bifurcation and Chaos. 3(5), 1123–1139.
Turaev, D., and Shilnikov, L. (1986). Torus-Chaos Bifurcations of Quasi-attractors. In Mitropolsky, Yu., and Sharkovsky, A. (eds.),Math. Mech. of Turbulence, Kiev, in Russian, pp. 113–121.
Viana, M. Persistence of strange attractors when unfolding homoclinic tatigendes.IMPA. Preprint.
Williams, R. (1979). The structure of Lorenz attractors.Publ. Math. IHES 50, 321–347.
Young, T. (1993).C k-smoothness of Invariant Curves in a Global Saddle-node Bifurcation. CDSNS93-138 Georgia Tech. Preprint.
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Afraimovich, V., Chow, SN. & Liu, W. Lorenz type attractors from codimension one bifurcation. J Dyn Diff Equat 7, 375–407 (1995). https://doi.org/10.1007/BF02219362
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DOI: https://doi.org/10.1007/BF02219362