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A degenerate singularity generating geometric Lorenz attractors

Published online by Cambridge University Press:  14 October 2010

Freddy Dumortier ,
Hiroshi Kokubu  and
Hiroe Oka
Freddy Dumortier
Affiliation:
Departement Wiskunde, Limburgs Universitair Centrum, Universitaire Campus, B-3590 Diepenbeek, Belgium
Hiroshi Kokubu
Affiliation:
Department of Mathematics, Faculty of Science, Kyoto University, Kyoto 606-01, Japan
Hiroe Oka
Affiliation:
Department of Applied Mathematics and Informatics, Faculty of Science and Technology, Ryukoku University Seta, Otsu 520-21, Japan
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Abstract

A degenerate vector field singularity in R3 can generate a geometric Lorenz attractor in an arbitrarily small unfolding of it. This enables us to detect Lorenz-like chaos in some families of vector fields, merely by performing normal form calculations of order 3.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 5 , October 1995 , pp. 833 - 856
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

[1] Afraimovich, V. S., Bykov, V. V. and L. P. , Shil'nikov. On structurally stable attracting limit sets of Lorenz attractor type. Trans. Moscow Math. Soc. 44 (1983), 153–216.Google Scholar
[2] Afraimovich, V. S. and Pesin, Ya. B.. Dimension of Lorenz type attractors. Sov. Sci. Rev., C: Malh/Phys. 6 (1987), 169–241.Google Scholar
[3] Chow, S. N., Deng, B. and Fiedler, B.. Homoclinic bifurcations at resonant eigenvalues. J. Dyn. Diff. Eq. 2 (1990), 177–244.CrossRefGoogle Scholar
[4] Carr, J.. Applications of Centre Manifold Theory. Springer, Berlin, 1981.CrossRefGoogle Scholar
[5] Dumortier, F.. Local study of planar vector fields: singularities and their unfoldings. In Structures in Dynamics. Broer, H. etal., eds, pp. 161-241. North-Holland, Amsterdam, 1991.Google Scholar
[6] Guckenheimer, J.. A strange strange attractor. In Hopf Bifurcation and its Applications. Marsden, J. E. and McCracken, M., eds, pp. 368–381. Springer, Berlin, 1976.CrossRefGoogle Scholar
[7] Guckenheimer, J. and Holmes, P.. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. 3rd edn. Springer, Berlin, 1989.Google Scholar
[8] Guckenheimer, J. and Williams, R. F.. Structural stability of Lorenz attractors. Publ. Math. IHES 50 (1979), 59–72.CrossRefGoogle Scholar
[9] Gruendler, J.. The existence of homoclinic orbits and the method of Melnikov for systems in Ro. SIAM J. Math. Anal. 16 (1985), 907–931.CrossRefGoogle Scholar
[10] Hirsch, M., Pugh, C. and Shub, M.. Invariant manifolds. Springer Lecture Notes in Mathematics 583. Springer, Berlin, 1977.CrossRefGoogle Scholar
[11] Horozov, E. I.. Versal deformations of equivariant vector fields in the case of symmetry of order 2 and 3. Trudy Sem. I. C. Petrovskava 5 (1979), 163–192 (in Russian).Google Scholar
[12] Keller, G.. Generalized bounded variation and applications to piecewise monotonic transformations. Z Wahrsch. Verw. Gebiete 69 (1985), 461–478.CrossRefGoogle Scholar
[13] Kisaka, M., Kokubu, H. and Oka, H.. Supplement to homoclinic doubling bifurcation in vector fields. In Dynamical Systems, Santiago 1990, Bamon, R. et al eds, pp. 92–1 16. Pitman Research Notes Vol. 285. Longman, London, 1993.Google Scholar
[14] Kisaka, M., Kokubu, H. and Oka, H.. Bifurcations to N-homoclinic and W-periodic orbits in vector fields. J. Dyn. Dig. Eq. 5 (1993), 305–357.CrossRefGoogle Scholar
[IS] Kokubu, H.. Homoclinic and heteroclinic bifurcations of vector fields. Japan J. Appl. Math. 5 (1988), 455–501.CrossRefGoogle Scholar
[16] Kokubu, H. and Oka, H.. Bifurcation of geometric Lorenz attractors from orbit-flip double homoclinic loops. In preparation.Google Scholar
[17] Lorenz, E. N.. Deterministic nonperiodic flow. J. Atmos. Sci. 20 (1963), 130–141.2.0.CO;2>CrossRefGoogle Scholar
[18] Oka, H.. Generation of geometric Lorenz attractors from degenerate vector field singularities. Proceedings of the 7th Toyota Conference ‘Towards Harnessing of Chaos’, pp. 389–392, Elsevier, Amsterdam, 1994.Google Scholar
[19] Palis, J. Jr and Melo, W. de. Geometric Theory of Dynamical Systems: An Introduction. Springer, Berlin, 1981.Google Scholar
[20] Rand, D.. The topological classiciation of Lorenz attractors. Math. Proc. Camb. Phil. Soc. 83 (1978), 451–460.CrossRefGoogle Scholar
[21] Robinson, C.. Differentiability of the stable foliation of the model Lorenz equations. In Dynamical Systems and Turbulence. Rand, D. A. and Young, L. S., eds, pp. 302–315. Springer Lecture Notes in Mathematics 898. Springer, Berlin, 1981.Google Scholar
[22] Robinson, C.. Transitivity and invariant measures for the geometric model of the Lorenz equations. Ergod. Th. & Dynam. Sys. 4 (1984), 605–611; Errata, Math. Proc. Camb. Phil. Soc., 6 (1986), 323.CrossRefGoogle Scholar
[23] Robinson, C.. Homoclinic bifurcation to a transitive attractor of Lorenz type. Nonlinearity 2 (1989), 495–518.CrossRefGoogle Scholar
[24] Robinson, C.. Homoclinic bifurcation to a transitive attractor of Lorenz type, II. SIAM J. Math. Anal. 23 (1992), 1255–1268.CrossRefGoogle Scholar
[25] Rychlik, M.. Lorenz attractors through Silnikov-type bifurcation: Part I. Ergod. Th. & Dynam. Sys. 10 (1990), 793–821.CrossRefGoogle Scholar
[26] Sandstede, B.. Verzweigungstheorie homokliner Verdopplungen, Thesis, University of Stuttgart, 1993.Google Scholar
[27] Ushiki, S., Oka, H. and Kokubu, H.. Existence d'attracteurs etranges dans le deploieinent d'une singularity degeneree d'un champ de vecteurs invariant par translation. C. R. Acad. Sci. Paris Serie I 298 (1984), 39–42.Google Scholar
[28] Vanderbauwhede, A.. Center manifolds, normal forms and elementary bifurcations. In Dynamics Reported, Vol. 2. Kirchgraber, U. and Walther, H. O., eds, pp. 89–169. John Wiley, Chichester, 1989.CrossRefGoogle Scholar
[29] Williams, R. F.. The structure of Lorenz attractors. Publ. Math. IHES 50 (1979), 321–347.CrossRefGoogle Scholar
[30] Yanagida, E.. Branching of double pulse solutions from single pulse solutions in nerve axon equations. J. Dig. Eq. 66 (1987), 243–262.CrossRefGoogle Scholar