About some theorems by L. P. Šil'nikov

C. Tresser

Annales de l'I.H.P. Physique théorique (1984)

  • Volume: 40, Issue: 4, page 441-461
  • ISSN: 0246-0211

How to cite

top

Tresser, C.. "About some theorems by L. P. Šil'nikov." Annales de l'I.H.P. Physique théorique 40.4 (1984): 441-461. <http://eudml.org/doc/76245>.

@article{Tresser1984,
author = {Tresser, C.},
journal = {Annales de l'I.H.P. Physique théorique},
keywords = {homoclinic orbit; hyperbolic periodic points; Lotka-Volterra equations},
language = {eng},
number = {4},
pages = {441-461},
publisher = {Gauthier-Villars},
title = {About some theorems by L. P. Šil'nikov},
url = {http://eudml.org/doc/76245},
volume = {40},
year = {1984},
}

TY - JOUR
AU - Tresser, C.
TI - About some theorems by L. P. Šil'nikov
JO - Annales de l'I.H.P. Physique théorique
PY - 1984
PB - Gauthier-Villars
VL - 40
IS - 4
SP - 441
EP - 461
LA - eng
KW - homoclinic orbit; hyperbolic periodic points; Lotka-Volterra equations
UR - http://eudml.org/doc/76245
ER -

References

top
  1. [1] A.A. Andronov, E. Leontovich, I. Gordon, A. Maier, Theory of bifurcation of dynamical systems in the plane. Israel Program of Scientific Translations, 1971. 
  2. [2] A. Arnéodo, P. Coullet, J. Peyraud, C. Tresser, Strange attractors in Volterra equations for species in competition. J. Math. Biology, t. 14, 1982, p. 153-157. Zbl0489.92017MR667795
  3. [3] A. Arnéodo, P. Coullet, E. Spiegel, C. Tresser, Asymptotic Chaos. To appear in Physica D. Zbl0595.58030
  4. [4] A. Arnéodo, P. Coullet, C. Tresser, Occurence of strange attractors in three–dimensional Volterra equations, Phys. Lett., t. 79 A, 1980, p. 259-263. MR590579
  5. [5] A. Arnéodo, P. Coullet, C. Tresser, Possible new strange attractors with spiral structure. Commun. Math. Phys., t. 79, 1981, p. 573-579. Zbl0485.58013MR623967
  6. [6] A. Arnéodo, P. Coullet, C. Tresser, A possible new mechanism for the onset of turbulence, Phys. Lett., t. 81 A, 1981, p. 197-201. MR598769
  7. [7] A. Arnéodo, P. Coullet, C. Tresser, Oscillations with chaotic behavior: an illustration of a theorem by Šhil'nikov. J. Stat. Phys., t. 27, 1982, p. 171-182. Zbl0522.58033MR656874
  8. [8] G.R. Belitskii, Functional equations and conjugacy of local diffeomorphisms of a finite smoothness class. Functional Anal. Appl., t. 7, 1973, p. 268-277. Zbl0293.39005
  9. [9] H. Broer, Formal normal form theorems for vector fields and some consequences for bifurcations in the volume preserving case, in Dynamical Systems and Turbulence (D. A. Rand and L. S. Young, ed.), Springer Lecture notes in Maths., n° 898, 1981. Zbl0504.58031MR654883
  10. [10] H. Broer, G. Vegter, Subordinate Šil'nikov bifurcations near some singularities of vector fields having low codimension. Preprint Groningen, 1982. Zbl0553.58024MR779710
  11. [11] P. Coullet, C. Tresser, A. Arnéodo, Transition to stochasticity for a class of forced oscillators, Phys. Lett., t. 72 A, 1979, p. 268-270. MR589812
  12. [12] R. Devaney, Homoclinic orbits in Hamiltonian systems, J. Diff. Equ., t. 21, 1976, p. 431-438. Zbl0343.58005MR442990
  13. [13] R. Devaney, Blue sky catastrophies in reversible and Hamiltonian systems, Indiana Univ. Math. J., t. 26, 1977, p. 247-263. Zbl0362.58006MR431274
  14. [14] J.W. Evans, N. Fenichel, J.A. Feroe, Double impulse solution in nerve axon equations. SIAM J. Appl. Math., t. 42, 1982, p. 219-234. Zbl0512.92006MR650218
  15. [15] J.A. Feroe, Existence and stability multiple impulse solutions of a nerve equation. SIAM J. Appl. Math., t. 42, 1982, p. 235-246. Zbl0502.92002MR650219
  16. [16] J.A. Feroe, Travelling waves of infinitely many pulses in nerve equations. Preprint. Zbl0476.92008MR627953
  17. [17] P. Gaspard, G. Nicolis, What can we learn from homoclinic orbits in chaotic dynamics. To appear in J. Stat. Phys. Zbl0587.58035MR711488
  18. [18] P. Glendinning, C. Sparrow, Local and global behaviour near homoclinic orbits. Preprint, Cambridge, 1983. Zbl0588.58041MR749842
  19. [19] J. Guckenheimer, Bifurcations of Dynamical Systems. Lecture given at C. I. M. E., July 1978 (Liguori Ed. 1980) (Reprinted Birkhauser). Zbl0451.58025MR589591
  20. [20] J. Guckenheimer, On a codimension two bifurcation in Dynamical Systems and Turbulence. Warwick, 1980 (D. A. Rand and L. S. Young Ed.), Springer Lecture Notes in Math., n° 898, 1981. Zbl0482.58006MR654886
  21. [21] P. Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Math. Mexicana, t. 5, 1960, p. 220-241. Zbl0127.30202MR141856
  22. [22] S.P. Hastings, Single and multiple pulse waves for the Fitzburgh-Nagumo Equations. SIAM J. Appl. Math., t. 42, 1982, p. 247-260. Zbl0503.92009MR650220
  23. [23] M.W. Hirsh, C. Pugh, Stable manifolds and hyperbolic sets, in Global Analysis, Proc. A. M. S., Symposium of Pure Math., 1970. Zbl0215.53001MR271991
  24. [24] P.J. Holmes, A strange family of three-dimensional vector fields near a degenerate singularity. J. Diff. Equ., t. 37, 1980, p. 382-403. Zbl0421.58016MR589999
  25. [25] N. Koppel, L.N. Howard, Target patterns and horseshoes from a perturbed central force problem : some temporally periodic solutions to reaction diffusion equations. Preprint, 1979. 
  26. [26] A. Libchaber, Lecture given at Les Houches Summer School on non-linear dynamics, July, 1981. 
  27. [27] J. Llibre, C. Simó, On the Hénon-Heiles potential. Actao Congresso Eq. Diff. y Applications, Santiago di Compostella, 1980. 
  28. [28] A.J. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci., t. 20, 1963, p. 130-141. 
  29. [29] A.J. Lotka, Elements of Physical Biology (Dover Publications, N. Y.), 1925. Zbl51.0416.06
  30. [30] D.N. Moore, E.A. Spiegel, A thermally excited non-linear oscillator, Astrophys. J., t. 143, 1966, p. 871-887. MR192911
  31. [31] J. Moser, Stable and Random Motion in Dynamical Systems (Princeton University Press, Princeton, N. J.), 1973. Zbl0271.70009MR442980
  32. [32] Ju. Neimark, L.P. ŠIL'NIKOV, A case of generation of periodic motions. Soviet Math. Dokl., t. 6, 1964, p. 305-309. Zbl0136.08203MR201754
  33. [33] E. Nelson, Topics in Dynamics I: Flows (Princeton University Press), 1960. Zbl0197.10702MR282379
  34. [34] S. Newhouse, J. Palis, Bifurcations of Morse-Smale dynamical systems in Dynamical Systems (Ed. M. M. Peixoto, Academic Press, New York), 1973. Zbl0279.58011MR334281
  35. [35] A.S. Pikovskii, M.I. Rabinovich, V. Yu. TRAKHTENGERTS, Onset of stochasticity in decay confinement of parametric instability, Soc. Phys. J. E. T. P., t. 47, 1979, p. 715-719. 
  36. [36] M.I. Rabinovich, Stochastic self-oscillations and turbulence, Sov. Phys. Usp., t. 21, 1978, p. 443-469. MR545759
  37. [37] M.I. Rabinovich, A.L. Fabrikant, Stochastic self-modulation of waves in non–equilibrium media, Sov. Phys. J. E. T. P., t. 50, 1979, p. 311-323. 
  38. [38] T. Rikitake, Oscillations of a system of disk dynamos, Proc. Cambridge Philos. Soc., t. 54, 1958, p. 89-105. Zbl0087.23703MR92682

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.