Scientific heritage of L.P. Shilnikov

Regular and Chaotic Dynamics

Volumn 19, Issue 4, 2014, Pages 435-460

  Afraimovich, Valentin S ^a   Gonchenko, Sergey V ^b   Lerman, Lev M ^b   Shilnikov, Andrey L ^b,c   Turaev, Dmitry V ^d  

^a UNIVERSIDAD AUTÓNOMA DE SAN LUIS POTOSÍ   (Mexico)

^b LOBACHEVSKY STATE UNIVERSITY OF NIZHNY NOVGOROD   (Russian Federation)

^c GEORGIA STATE UNIVERSITY   (United States)

^d IMPERIAL COLLEGE LONDON   (United Kingdom)

Author keywords

Global bifurcations; Homoclinic chaos; Homoclinic loop; Hyperbolic set; Lorenz attractor; Saddle focus; Saddle node; Saddle saddle; Spiral chaos; Strange attractor

Indexed keywords

EID: 84925829871     PISSN: 15603547     EISSN: 14684845     Source Type: Journal    
DOI: 10.1134/S1560354714040017     Document Type: Article

References (85)

1. Neĭmark, Y. I. and Shil'nikov, L.P., Application of the Small-Parameter Method to a System of Differential Equations with Discontinuous Right-Hand Sides, Izv. Akad. Nauk SSSR, 1959, no. 6, pp. 51- 59 (Russian).
2. Shil'nikov, L.P., Some Cases of Generation of Period Motions from Singular Trajectories, Mat. Sb. (N. S.), 1963, vol. 61(103), no. 4, pp. 443-466 (Russian).
3. Shil'nikov, L.P., A Case of the Existence of a Countable Number of Periodic Motions, Soviet Math. Dokl., 1965, vol. 6, pp. 163-166; see also: Dokl. Akad. Nauk SSSR, 1965, vol. 169, no. 3, pp. 558-561.
4. Shilnikov, L.P., On the Generation of a Periodic Motion from a Trajectory Which Leaves and Re-Enters a Saddle-Saddle State of Equilibrium, Soviet Math. Dokl., 1966, vol. 7, pp. 1155-1158; see also: Dokl. Akad. Nauk SSSR, 1966, vol. 170, no. 1, pp. 49-52.
5. Neimark, Y. I. and Shil'nikov, L.P., A Case of Generation of Periodic Motions, Soviet Radiophys., 1965, vol. 8, pp. 234-241; see also: Izv. VUZ. Radiofizika, 1965, vol. 8, no. 2, pp. 330-340.
6. Shil'nikov, L.P., Existence of a Countable Set of Periodic Motions in a Four-Dimensional Space in an Extended Neighborhood of a Saddle-Focus, Soviet Math. Dokl., 1967, vol. 8, pp. 54-58; see also: Dokl. Akad. Nauk SSSR, 1967, vol. 172, no. 1, pp. 54-57.
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8. Shil'nikov, L.P., On the Question of the Structure of the Neighborhood of a Homoclinic Tube of an Invariant Torus, Dokl. Akad. Nauk SSSR, 1968, vol. 180, no. 2, pp. 286-289 (Russian).
9. Shil'nikov, L.P., On the Generation of a Periodic Motion from Trajectories Doubly Asymptotic to an Equilibrium State of Saddle Type, Math. USSR-Sb., 1968, vol. 6, no. 3, pp. 427-438; see also: Mat. Sb. (N. S.), 1968, vol. 77(119), no. 3, pp. 461-472.
10. Shil'nikov, L.P., A Certain New Type of Bifurcation of Multidimensional Dynamic Systems, Dokl. Akad. Nauk SSSR, 1969, vol. 189, no. 1, pp. 59-62 (Russian).
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12. Gavrilov, N. K. and Shil'nikov, L.P., On Three-Dimensional Dynamical Systems Close to Systems with a Structurally Unstable Homoclinic Curve: 1, Math. USSR-Sb., 1972, vol. 17, no. 4, pp. 467-485; see also: Mat. Sb. (N. S.), 1972, vol. 88(130), no. 4(8), pp. 475-492.
13. Gavrilov, N. K. and Shil'nikov, L.P., On Three-Dimensional Dynamical Systems Close to Systems with a Structurally Unstable Homoclinic Curve: 2, Math. USSR-Sb., 1973, vol. 19, no. 1, pp. 139-156; see also: Mat. Sb. (N. S.), 1973, vol. 90(132), no. 1, pp. 139-156.
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18. Afraimovich, V. S., Bykov, V.V., and Shilnikov, L.P., The Origin and Structure of the Lorenz Attractor, Dokl. Akad. Nauk SSSR, 1977, vol. 234, no. 2, pp. 336-339 (Russian).
19. Luk'janov, V. I. and Shil'nikov, L.P., Some Bifurcations of Dynamical Systems with Homoclinic Structures, Soviet Math. Dokl., 1978, vol. 19, pp. 1314-1318; see also: Dokl. Akad. Nauk SSSR, 1978, vol. 243, no. 1, pp. 26-29.
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28. Biragov, V. and Shilnikov, L.P., On the Bifurcation of a Saddle-Focus Separatrix Loop in a Three- Dimensional Conservative Dynamical System, in Methods of Qualitative Theory and Theory of Bifurcations, Gorky: GGU, 1989, pp. 25-34 (Russian).
29. Gonchenko, S. V., Turaev, D. V., and Shil'nikov, L.P., Models with a Structurally Unstable Homoclinic Poincaré Curve, Soviet Math. Dokl., 1992, vol. 44, no. 2, pp. 422-426; see also: Dokl. Akad. Nauk SSSR, 1991, vol. 320, no. 2, pp. 269-272.
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32. Alekseeva, S. A. and Shilnikov, L.P., On Cusp-Bifurcations of Periodic Orbits in Systems with a Saddle- Focus Homoclinic Curve, in Methods of Qualitative Theory of Differential Equations and Related Topics, Amer. Math. Soc. Transl. Ser. 2, vol. 200, Providence, R.I.: AMS, 2000, pp. 23-34.
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85. Afraimovich, V. S., Gonchenko, S.V., Lerman, L.M., Shilnikov, A. L., Turaev, D.V., Scientific Heritage of L.P. Shilnikov. Part II, Regul. Chaotic. Dyn., 2014, to appear.