Geodesic
Systems with a homoclinic curve of multidimensional saddle-focus, and spiral chaos
Sbornik. Mathematics, Tome 73 (1992) no. 2, pp. 415-443 
I. M. Ovsyannikov; L. P. Shilnikov. Systems with a homoclinic curve of multidimensional saddle-focus, and spiral chaos. Sbornik. Mathematics, Tome 73 (1992) no. 2, pp. 415-443. http://geodesic.mathdoc.fr/item/SM_1992_73_2_a6/
@article{SM_1992_73_2_a6,
     author = {I. M. Ovsyannikov and L. P. Shilnikov},
     title = {Systems with a~homoclinic curve of multidimensional saddle-focus, and spiral chaos},
     journal = {Sbornik. Mathematics},
     pages = {415--443},
     year = {1992},
     volume = {73},
     number = {2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1992_73_2_a6/}
}
TY  - JOUR
AU  - I. M. Ovsyannikov
AU  - L. P. Shilnikov
TI  - Systems with a homoclinic curve of multidimensional saddle-focus, and spiral chaos
JO  - Sbornik. Mathematics
PY  - 1992
SP  - 415
EP  - 443
VL  - 73
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/SM_1992_73_2_a6/
LA  - en
ID  - SM_1992_73_2_a6
ER  - 
%0 Journal Article
%A I. M. Ovsyannikov
%A L. P. Shilnikov
%T Systems with a homoclinic curve of multidimensional saddle-focus, and spiral chaos
%J Sbornik. Mathematics
%D 1992
%P 415-443
%V 73
%N 2
%U http://geodesic.mathdoc.fr/item/SM_1992_73_2_a6/
%G en
%F SM_1992_73_2_a6

Voir la notice de l'article provenant de la source Math-Net.Ru

Consider the space $\mathscr B^1$ of dynamical systems having an isolated equilibrium point $O$ of saddle-focus type with a one- or two-dimensional unstable manifold and a trajectory $\Gamma$ homoclinic at $O$. The following results are proved: 1. Systems with structurally unstable periodic motions are dense in $\mathscr B^1$. 2. Systems with a countable set of stable periodic motions are dense in the open subset $\mathscr B^1_s$ of $\mathscr B^1$ comprised of systems whose second saddle parameter $\sigma_2$ is negative. 3. Neither the subset $\mathscr B^1_u$ of $\mathscr B^1$ consisting of systems satisfying $\sigma_2>0$ nor any sufficiently small neighborhood of $\mathscr B^1_u$ in the space of all dynamical systems contains a system with stable periodic motions in a sufficiently small neighborhood of the contour $O\cup\Gamma$.

[1] Ovsyannikov I. M., Shilnikov L. P., “O sistemakh s gomoklinicheskoi krivoi sedlo-fokusa”, Matem. sb., 130 (172) (1986), 552–570 | MR

[2] Shil'nikov L. P., “Bifurcation theory and turbulence”, Nonlinear and Turbulent Processes in Physics, v. 3, Harwood Academic Publ., 1984, 1627–1635 | MR

[3] Shilnikov L. P., “Teoriya bifurkatsii i turbulentnost”, Metody kachestvennoi teorii differentsialnykh uravnenii, Gorkii, 1986, 150–165 | MR

[4] Afraimovich V. ., Shilnikov L. P., “Invariantnye dvumernye tory, ikh razrushenie i stokhastichnost”, Metody kachestvennoi teorii differentsialnykh uravnenii, Gorkii, 1983, 3–26 | MR | Zbl

[5] Anischenko V. S., Stokhasticheskie kolebaniya v radiofizicheskikh sistemakh, ch. 2, Izd-vo Saratovskogo un-ta, Saratov, 1986

[6] Chil M., Childess S., Topics in Geophysical Fluid Dynamics: Atmospheric Dynamics, Dynamo Theory and Climate Dynamics, Springer-Verlag, 1987 | MR

[7] Argoul F., Arneodo A., Richetti P., “Experimental evidence for homoclinic chaos in the Belousov - Zhabotinskii reaction”, Physics Letters. A, 120:6, 269–275 | DOI | MR

[8] Argoul F., Arneodo A., Richetti P., Symbolic dynamic in the the Belousov - Zhabotinskii reaction: an experimental and theoretical approach of Shil'nikov, Preprint, 1987

[9] Babenko K. I., “Ob ispolzovanii EVM pri issledovanii gidrodinamicheskoi neustoichivosti”, Issledovanie gidrodinamicheskoi ustoichivosti s pomoschyu EVM, IPM AN im. M. V. Keldysha, M., 1981, 5–79

[10] Moore D. R., Toomre J. , Knobloch E., Weiss N. O., “Period doubling and chaos in partial differential equations for thermosolutal convection”, Nature, 303 (1983), 663–667 | DOI

[11] Chia K. N., Osswald G. A., Chia V., “Simulation of self-induced unsteady motion in the near wave of a Joukowski airfoli”, Lect. Not. Eng., 24 (1986), 112–127 | MR

[12] Lankaster P., Teoriya matrits, Nauka, M., 1982 | MR