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Izv. RAN. Ser. Mat., 1992, Volume 56, Issue 6, Pages 1165–1197 (Mi izv902)  

This article is cited in 24 scientific papers (total in 24 papers)

On moduli of systems with a structurally unstable homoclinic Poincare curve

S. V. Gonchenko, L. P. Shilnikov


Abstract: For higher-dimensional dynamical systems with a structurally unstable homoclinic Poincare curve we find moduli of topological and $\Omega$-equivalence. Using these, in the case of systems with a nontrivial structure we find sufficient conditions for $\Omega$-equivalence on small neighborhoods of homoclinic curves.

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English version:
Russian Academy of Sciences. Izvestiya Mathematics, 1993, 41:3, 417–445

Bibliographic databases:

UDC: 517.9
MSC: Primary 58F15, 58F14; Secondary 58F10
Received: 24.05.1990
Revised: 10.01.1992

Citation: S. V. Gonchenko, L. P. Shilnikov, “On moduli of systems with a structurally unstable homoclinic Poincare curve”, Izv. RAN. Ser. Mat., 56:6 (1992), 1165–1197; Russian Acad. Sci. Izv. Math., 41:3 (1993), 417–445

Citation in format AMSBIB
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\by S.~V.~Gonchenko, L.~P.~Shilnikov
\paper On~moduli of systems with a structurally unstable homoclinic Poincare curve
\jour Izv. RAN. Ser. Mat.
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\vol 56
\issue 6
\pages 1165--1197
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\adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1993IzMat..41..417G}
\transl
\jour Russian Acad. Sci. Izv. Math.
\yr 1993
\vol 41
\issue 3
\pages 417--445
\crossref{https://doi.org/10.1070/IM1993v041n03ABEH002270}
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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. S. V. Gonchenko, L. P. Shil’nikov, D. V. Turaev, “Dynamical phenomena in systems with structurally unstable Poincaré́ homoclinic orbits”, Chaos, 6:1 (1996), 15  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. S. V. Gonchenko, “Moduli of $\Omega$-conjugacy of two-dimensional diffeomorphisms with a structurally unstable heteroclinic contour”, Sb. Math., 187:9 (1996), 1261–1281  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. S Gonchenko, “Bifurcations of systems with structurally unstable homoclinic orbits and moduli of ω-equivalence”, Computers & Mathematics with Applications, 34:2-4 (1997), 111  crossref  elib
    4. S Gonchenko, “Quasiattractors and homoclinic tangencies”, Computers & Mathematics with Applications, 34:2-4 (1997), 195  crossref  elib
    5. Leonid Shilnikov, “Mathematical problems of nonlinear dynamics: A tutorial”, Journal of the Franklin Institute, 334:5-6 (1997), 793  crossref
    6. O. V. Sten'kin, L. P. Shilnikov, “Homoclinic $\Omega$-explosion and domains of hyperbolicity”, Sb. Math., 189:4 (1998), 603–622  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    7. FLAVIANO BATTELLI, CLAUDIO LAZZARI, “PERTURBING TWO-DIMENSIONAL MAPS HAVING CRITICAL HOMOCLINIC ORBITS”, Int. J. Bifurcation Chaos, 09:06 (1999), 1189  crossref
    8. Chelnokov Y.N., “The use of quaternions in the optimal control problems of motion of the center of mass of a spacecraft in a Newtonian gravitational field: I”, Cosmic Research, 39:5 (2001), 470–484  crossref  isi  elib
    9. V. S. Gonchenko, “Homoclinic Tangencies, $\Omega$-Moduli, and Bifurcations”, Proc. Steklov Inst. Math., 236 (2002), 94–109  mathnet  mathscinet  zmath
    10. J. Math. Sci. (N. Y.), 128:2 (2005), 2767–2773  mathnet  crossref  mathscinet  zmath
    11. J. Math. Sci. (N. Y.), 128:2 (2005), 2774–2777  mathnet  crossref  mathscinet  zmath
    12. Jeroen S W Lamb, Oleg V Stenkin, “Newhouse regions for reversible systems with infinitely many stable, unstable and elliptic periodic orbits”, Nonlinearity, 17:4 (2004), 1217  crossref  mathscinet  zmath  isi  elib
    13. S. V. Gonchenko, V. S. Gonchenko, “On Bifurcations of Birth of Closed Invariant Curves in the Case of Two-Dimensional Diffeomorphisms with Homoclinic Tangencies”, Proc. Steklov Inst. Math., 244 (2004), 80–105  mathnet  mathscinet  zmath
    14. S. V. Gonchenko, D. V. Turaev, L. P. Shilnikov, “Existence of Infinitely Many Elliptic Periodic Orbits in Four-Dimensional Symplectic Maps with a Homoclinic Tangency”, Proc. Steklov Inst. Math., 244 (2004), 106–131  mathnet  mathscinet  zmath
    15. Sergey Gonchenko, Dmitry Turaev, Leonid Shilnikov, “Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps”, Nonlinearity, 20:2 (2007), 241  crossref  mathscinet  zmath  isi  elib
    16. S V Gonchenko, L P Shilnikov, D V Turaev, “On dynamical properties of multidimensional diffeomorphisms from Newhouse regions: I”, Nonlinearity, 21:5 (2008), 923  crossref  mathscinet  zmath  isi  elib
    17. T. M. Mitryakova, O. V. Pochinka, “K voprosu o klassifikatsii diffeomorfizmov poverkhnostei s konechnym chislom modulei topologicheskoi sopryazhennosti”, Nelineinaya dinam., 6:1 (2010), 91–105  mathnet  elib
    18. S. V. Gonchenko, O. V. Stenkin, “Gomoklinicheskii $\Omega$-vzryv: intervaly giperbolichnosti i ikh granitsy”, Nelineinaya dinam., 7:1 (2011), 3–24  mathnet  elib
    19. S.V. Gonchenko, I.I. Ovsyannikov, D. Turaev, “On the effect of invisibility of stable periodic orbits at homoclinic bifurcations”, Physica D: Nonlinear Phenomena, 2012  crossref
    20. SHIN KIRIKI, TERUHIKO SOMA, “Existence of generic cubic homoclinic tangencies for Hénon maps”, Ergod. Th. Dynam. Sys, 2012, 1  crossref
    21. A Delshams, S V Gonchenko, V S Gonchenko, J T Lázaro, O Sten'kin, “Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps”, Nonlinearity, 26:1 (2013), 1  crossref
    22. Sergey V. Gonchenko, Olga V. Gordeeva, Valery I. Lukyanov, Ivan I. Ovsyannikov, “On Bifurcations of Multidimensional Diffeomorphisms Having a Homoclinic Tangency to a Saddle-node”, Regul. Chaotic Dyn., 19:4 (2014), 461–473  mathnet  crossref  mathscinet  zmath
    23. Sergey V. Gonchenko, Ivan I. Ovsyannikov, Joan C. Tatjer, “Birth of Discrete Lorenz Attractors at the Bifurcations of 3D Maps with Homoclinic Tangencies to Saddle Points”, Regul. Chaotic Dyn., 19:4 (2014), 495–505  mathnet  crossref  mathscinet  zmath
    24. S. V. Gonchenko, M. S. Gonchenko, I. O. Sinitsky, “On mixed dynamics of two-dimensional reversible diffeomorphisms with symmetric non-transversal heteroclinic cycles”, Izv. Math., 84:1 (2020), 23–51  mathnet  crossref  crossref  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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