About some theorems by L. P. Šil'nikov

Tresser, C.

Annales de l'I.H.P. Physique théorique, Tome 40 (1984) no. 4, pp. 441-461.

[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 | Zbl

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

[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

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

[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

[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 | Zbl

[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

[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 | Zbl

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

[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

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

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

[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 | Zbl

[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 | Zbl

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

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

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

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

[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 | Zbl

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

[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 | Zbl

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

[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 | Zbl

[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

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

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

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

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

[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

[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 | Zbl