About some theorems by L. P. Šil'nikov
Annales de l'I.H.P. Physique théorique, Volume 40 (1984) no. 4, p. 441-461
@article{AIHPA_1984__40_4_441_0,
     author = {Tresser, C.},
     title = {About some theorems by L. P. \v Sil'nikov},
     journal = {Annales de l'I.H.P. Physique th\'eorique},
     publisher = {Gauthier-Villars},
     volume = {40},
     number = {4},
     year = {1984},
     pages = {441-461},
     zbl = {0556.58025},
     mrnumber = {757766},
     language = {en},
     url = {http://www.numdam.org/item/AIHPA_1984__40_4_441_0}
}
Tresser, C. About some theorems by L. P. Šil'nikov. Annales de l'I.H.P. Physique théorique, Volume 40 (1984) no. 4, pp. 441-461. http://www.numdam.org/item/AIHPA_1984__40_4_441_0/

[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] 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. | MR 667795 | Zbl 0489.92017

[3] A. Arnéodo, P. Coullet, E. Spiegel, C. Tresser, Asymptotic Chaos. To appear in Physica D. | Zbl 0595.58030

[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. | MR 590579

[5] A. Arnéodo, P. Coullet, C. Tresser, Possible new strange attractors with spiral structure. Commun. Math. Phys., t. 79, 1981, p. 573-579. | MR 623967 | Zbl 0485.58013

[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. | MR 598769

[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. | MR 656874 | Zbl 0522.58033

[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. | Zbl 0293.39005

[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. | MR 654883 | Zbl 0504.58031

[10] H. Broer, G. Vegter, Subordinate Šil'nikov bifurcations near some singularities of vector fields having low codimension. Preprint Groningen, 1982. | MR 779710

[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. | MR 589812

[12] R. Devaney, Homoclinic orbits in Hamiltonian systems, J. Diff. Equ., t. 21, 1976, p. 431-438. | MR 442990 | Zbl 0343.58005

[13] R. Devaney, Blue sky catastrophies in reversible and Hamiltonian systems, Indiana Univ. Math. J., t. 26, 1977, p. 247-263. | MR 431274 | Zbl 0362.58006

[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. | MR 650218 | Zbl 0512.92006

[15] J.A. Feroe, Existence and stability multiple impulse solutions of a nerve equation. SIAM J. Appl. Math., t. 42, 1982, p. 235-246. | MR 650219 | Zbl 0502.92002

[16] J.A. Feroe, Travelling waves of infinitely many pulses in nerve equations. Preprint. | MR 627953

[17] P. Gaspard, G. Nicolis, What can we learn from homoclinic orbits in chaotic dynamics. To appear in J. Stat. Phys. | MR 711488 | Zbl 0587.58035

[18] P. Glendinning, C. Sparrow, Local and global behaviour near homoclinic orbits. Preprint, Cambridge, 1983. | MR 749842

[19] J. Guckenheimer, Bifurcations of Dynamical Systems. Lecture given at C. I. M. E., July 1978 (Liguori Ed. 1980) (Reprinted Birkhauser). | MR 589591 | Zbl 0451.58025

[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. | MR 654886 | Zbl 0482.58006

[21] P. Hartman, On local homeomorphisms of Euclidean spaces, Bol. Soc. Math. Mexicana, t. 5, 1960, p. 220-241. | MR 141856 | Zbl 0127.30202

[22] S.P. Hastings, Single and multiple pulse waves for the Fitzburgh-Nagumo Equations. SIAM J. Appl. Math., t. 42, 1982, p. 247-260. | MR 650220 | Zbl 0503.92009

[23] M.W. Hirsh, C. Pugh, Stable manifolds and hyperbolic sets, in Global Analysis, Proc. A. M. S., Symposium of Pure Math., 1970. | MR 271991 | Zbl 0215.53001

[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. | MR 589999 | Zbl 0421.58016

[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] A. Libchaber, Lecture given at Les Houches Summer School on non-linear dynamics, July, 1981.

[27] J. Llibre, C. Simó, On the Hénon-Heiles potential. Actao Congresso Eq. Diff. y Applications, Santiago di Compostella, 1980.

[28] A.J. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci., t. 20, 1963, p. 130-141.

[29] A.J. Lotka, Elements of Physical Biology (Dover Publications, N. Y.), 1925.

[30] D.N. Moore, E.A. Spiegel, A thermally excited non-linear oscillator, Astrophys. J., t. 143, 1966, p. 871-887. | MR 192911

[31] J. Moser, Stable and Random Motion in Dynamical Systems (Princeton University Press, Princeton, N. J.), 1973. | MR 442980 | Zbl 0271.70009

[32] Ju. Neimark, L.P. Šil'Nikov, A case of generation of periodic motions. Soviet Math. Dokl., t. 6, 1964, p. 305-309. | MR 201754 | Zbl 0136.08203

[33] E. Nelson, Topics in Dynamics I: Flows (Princeton University Press), 1960. | MR 282379 | Zbl 0197.10702

[34] S. Newhouse, J. Palis, Bifurcations of Morse-Smale dynamical systems in Dynamical Systems (Ed. M. M. Peixoto, Academic Press, New York), 1973. | MR 334281 | Zbl 0279.58011

[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] M.I. Rabinovich, Stochastic self-oscillations and turbulence, Sov. Phys. Usp., t. 21, 1978, p. 443-469. | MR 545759

[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] T. Rikitake, Oscillations of a system of disk dynamos, Proc. Cambridge Philos. Soc., t. 54, 1958, p. 89-105. | MR 92682 | Zbl 0087.23703