Lorenz attractors do not have the pseudo-orbit tracing property

Motomasa KOMURO

doi:10.2969/jmsj/03730489

Motomasa KOMURO

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JOURNAL FREE ACCESS

1985 Volume 37 Issue 3 Pages 489-514

DOI https://doi.org/10.2969/jmsj/03730489

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Published: 1985 Received: February 21, 1984 Available on J-STAGE: October 20, 2006 Accepted: - Advance online publication: - Revised: -
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Date of correction: October 20, 2006 Reason for correction: - Correction: CITATION Details: Wrong : 1) R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1-30.
2) R. Bowen and P. Walters, Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193.
3) J. E. Franke and J. F. Selgrade, Hyperbolicity and chain recurrence, J. Differential Equations, 26 (1977), 27-36.
4) J. Guckenheimer, A strange, strange attractor, in The Hopf Bifurcation Theorem and its Applications, ed. by J. E. Marsden and M. McCracken, Springer-Verlag, 1976, 368-381.
5) J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, IHES Publ. Math., 50 (1979), 59-72.
6) J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Applied Mathematical Sciences, 42, Springer, Berlin-Heidelberg-New York, 1983.
7) K. Kato, Pseudo-orbit and stabilities of flows, Mem. Fac. Sci. Kochi Univ. (Math.), 5(1984), 45-62.
8) K. Kato and A. Morimoto, Topological Ω-stability of Axiom A flows with no Ω-explosions, J. Differential Equations, 34 (1979), 464-481.
9) E. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Sciences, 20 (1963), 130-141.
10) A. Morimoto, The method of pseudo-orbit tracing and stability of dynamical systems, Seminar Note, 39, Dept. Math. Tokyo Univ., 1979, (in Japanese).
11) C. Sparrow, The Lorenz Equations, Springer-Verlag, New York, Heidelberg, Berlin, 1982.
12) R. F. Thomas, Stability properties of one-parameter flows, Proc. London Math. Soc., 45 (1982), 479-505.
13) P. Walters, On the pseudo-orbit tracing property and its relationship to stability, Lecture Notes in Math., 668 (1978), Springer, 231-244.
14) R. F. Williams, Structure of Lorenz attractors, IHES Publ. Math., 50 (1979), 73-99.
15) R.C. Robinson, Differentiability of the stable foliation for the model Lorenz equations, Lecture Notes in Math., 898 (1981), Springer, 302-315.

Right : [1] R. Bowen, Periodic orbits for hyperbolic flows, Amer. J. Math., 94 (1972), 1-30.
[2] R. Bowen and P. Walters, Expansive one-parameter flows, J. Differential Equations, 12 (1972), 180-193.
[3] J. E. Franke and J. F. Selgrade, Hyperbolicity and chain recurrence, J. Differential Equations, 26 (1977), 27-36.
[4] J. Guckenheimer, A strange, strange attractor, in The Hopf Bifurcation Theorem and its Applications, ed. by J. E. Marsden and M. McCracken, Springer-Verlag, 1976, 368-381.
[5] J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, IHES Publ. Math., 50 (1979), 59-72.
[6] J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vector fields, Applied Mathematical Sciences, 42, Springer, Berlin-Heidelberg-New York, 1983.
[7] K. Kato, Pseudo-orbit and stabilities of flows, Mem. Fac. Sci. Kochi Univ. (Math.), 5 (1984), 45-62.
[8] K. Kato and A. Morimoto, Topological Ω-stability of Axiom A flows with no Ω-explosions, J. Differential Equations, 34 (1979), 464-481.
[9] E. Lorenz, Deterministic nonperiodic flow, J. Atmospheric Sciences, 20 (1963), 130-141.
[10] A. Morimoto, The method of pseudo-orbit tracing and stability of dynamical systems, Seminar Note, 39, Dept. Math. Tokyo Univ., 1979, (in Japanese).
[11] C. Sparrow, The Lorenz Equations, Springer-Verlag, New York, Heidelberg, Berlin, 1982.
[12] R. F. Thomas, Stability properties of one-parameter flows, Proc. London Math. Soc., 45 (1982), 479-505.
[13] P. Walters, On the pseudo-orbit tracing property and its relationship to stability, Lecture Notes in Math., 668 (1978), Springer, 231-244.
[14] R. F. Williams, Structure of Lorenz attractors, IHES Publ. Math., 50 (1979), 73-99.
[15] R. C. Robinson, Differentiability of the stable foliation for the model Lorenz equations, Lecture Notes in Math., 898 (1981), Springer, 302-315.

Date of correction: October 20, 2006 Reason for correction: - Correction: PDF FILE Details: -

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