Lorenz equations part I: Existence and nonexistence of homoclinic orbits

SIAM Journal on Mathematical Analysis

Volumn 27, Issue 4, 1996, Pages 1057-1069

  Chen, Xinfu ^a  

^a UNIVERSITY OF PITTSBURGH   (United States)

Author keywords

Asymptotic behavior; Homoclinic orbits; Liapunov functions; Lorenz equations; Shooting methods

Indexed keywords

EID: 0030540448     PISSN: 00361410     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036141094264414     Document Type: Article

References (18)

1. X. CHEN, Lorenz equations, part II: Existence of rotated homoclinic orbits and chaotic orbits, Discrete Continuous Dynam. Sys., 2 (1996), pp. 121-140.
2. X. CHEN, Lorenz equations, part III: Existence of homoclinic explosion, preprint, University of Pittsburgh, Pittsburgh, PA, 1995.
3. J. GUCKENHEIMER, A strange, strange attractor, in The Hopf Bifurcation and Its Applications, J. E. Marsden and M. McCracken, eds., Springer-Verlag, Berlin, New York, 1976.
4. J. GUCKENHEIMER AND P. HOLMES, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Appl. Math. Sci. 42, Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1983.
5. J. GUCKENHEIMER AND R. F. WILLIAMS, Structural stability of the Lorenz attractor, Inst. Hautes Études Sci. Publ. Math., 50 (1979), pp. 59-72.
6. B. HASSARD AND J. ZHANG, Existence of a homoclinic orbit of the Lorenz system by precise shooting, SIAM, J. Math. Anal., 25 (1994), pp. 179-196.
7. B. HASSARD, S. HASTINGS, W. TROY, AND J. ZHANG, A computer proof that the Lorenz equations have "chaotic" solutions, Appl. Math. Lett., to appear.
8. S. P. HASTINGS AND W. C. TROY, A shooting approach to the Lorenz equations, Bull. Amer. Math. Soc., 27 (1992), pp. 298-303.
9. _, A shooting approach to chaos in the Lorenz equations, J. Differential Equations, 126 (1996), to appear.
10. _, A proof that the Lorenz equations have a homoclinic orbit, J. Differential Equations, 113 (1994), pp. 166-188.
11. E. N. LORENZ, Deterministic non-periodic flows, J. Atmos. Sci., 20 (1963), pp. 130-141.
12. K. MISCHAIKOW AND M. MROZEK, Chaos in the Lorenz equations: A computer-assisted proof, Bull. Amer. Math. Soc., 32 (1995), pp. 66-72.
13. K. A. ROBBINS, Periodic solutions and bifurcation structure at a high R in the Lorenz model, SIAM J. Appl. Math., 36 (1979), pp. 457-472.
14. C. ROBINSON, Homoclinic bifurcation to a transitive attractor of Lorenz type, Nonlinearity, 2 (1989), pp. 495-518.
15. M. R. RYCHLIK, Lorenz attractors through Sil'nikov-type bifurcation, part I, Ergodic Theory Dynamical Systems, 10 (1989), pp. 793-821.
16. L. P. SIL'NIKOV, A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle focus type, Math. USSR-Sb., 10 (1970), pp. 91-102.
17. C. SPARROW, The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Appl. Math. Sci. 41, Springer-Verlag, Berlin, New York, Heidelberg, 1982.
18. H. P. F. SWINNERTON-DYER, The method of averaging for some almost-conservative differential equations, J. London Math. Soc., 22 (1980), pp. 534-542.