Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-23T02:33:29.720Z Has data issue: false hasContentIssue false
  • English
  • Français

On Global Bifurcations of Three-dimensional Diffeomorphisms Leading to Lorenz-like Attractors

Published online by Cambridge University Press:  17 September 2013

S.V. Gonchenko  and
I.I. Ovsyannikov
S.V. Gonchenko
Affiliation:
Research Institute of Applied Mathematics and Cybernetics, 10, Ulyanova Str., 603005 Nizhny Novgorod, Russia
I.I. Ovsyannikov*
Affiliation:
Research Institute of Applied Mathematics and Cybernetics, 10, Ulyanova Str., 603005 Nizhny Novgorod, Russia Imperial College, SW7 2AZ London, UK
*
⋆ Corresponding author. E-mail: [email protected]
Get access

Abstract

We study dynamics and bifurcations of three-dimensional diffeomorphisms with nontransversal heteroclinic cycles. We show that bifurcations under consideration lead to the birth of Lorenz-like attractors. They can be viewed as attractors in the Poincare map for periodically perturbed classical Lorenz attractors and hence they can allow for the existence of homoclinic tangencies and wild hyperbolic sets.

Keywords

homoclinic and heteroclinic orbitsbifurcationsstrange attractorssaddle-focus
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 8 , Issue 5: Bifurcations , 2013 , pp. 71 - 83
Copyright
© EDP Sciences, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Gonchenko, S.V., Sten’kin, O.V., Shilnikov, L.P.. On existence of infinitely many stable and unstable invariant tori for systems from newhouse regions with heteroclinic tangencies. Rus. Nonlinear Dynamics, 2 (2006), 3–25. Google Scholar
Gavrilov, N.K.. On three-dimensional dynamical systems having a structurally unstable homoclinic contour. Rus. Math. Notes, 14 (1973), 687–696. Google Scholar
Gavrilov, N.K., Shilnikov, L.P.. On three-dimensional dynamical systems close to systems with a structurally unstable homoclinic curve. I. Math. USSR Sbornik, 17 (1972), 467–485; II. Math. USSR Sbornik, 19 (1973), 139–156. CrossRefGoogle Scholar
Gonchenko, S.V., Shilnikov, L.P., Turaev, D.V.. On Newhouse regions of two-dimensional diffeomorphisms close to a diffeomorphism with a nontransversal heteroclinic cycle. Proc. Steklov Inst. Math. 216 (1997), 70–118. Google Scholar
S.V. Gonchenko, L.P. Shilnikov, O.V. Stenkin. On Newhouse regions with infinitely many stable and unstable invariant tori. Proceedings of the Int. Conf. “Progress in Nonlinear Science” dedicated to 100th Anniversary of A.A. Andronov, July 2-6; v. 1 “Mathematical Problems of Nonlinear Dynamics”, Nizhny Novgorod (2002) 80–102.
Gonchenko, S., Shilnikov, L., Turaev, D.. Homoclinic tangencies of arbitrarily high orders in conservative and dissipative two-dimensional maps. Nonlinearity, 20 (2007), 241–275. CrossRefGoogle Scholar
Gonchenko, S.V., Shilnikov, L.P., Turaev, D.V.. On dynamical properties of multidimensional diffeomorphisms from Newhouse regions. I. Nonlinearity, 21 (2008), 923–972. CrossRefGoogle Scholar
Lamb, J.S.W., Sten’kin, O.V.. Newhouse regions for reversible systems with infinitely many stable, unstable and elliptic periodic orbits. Nonlinearity, 17 (2004), 1217–1244. CrossRefGoogle Scholar
Delshams, A., Gonchenko, S.V., Gonchenko, V.S., Lazaro, J.T., Sten’kin, O.. Abundance of attracting, repelling and elliptic periodic orbits in two-dimensional reversible maps. Nonlinearity, 26 (2013), 1–33. CrossRefGoogle Scholar
Gonchenko, S.V., Gonchenko, A.S., Kazakov, A.O.. On new aspects of chaotic dynamics of “celtic stone”. Rus. Nonlinear Dynamics, 8 (2013), 507–518. Google Scholar
Turaev, D.V.. On dimension of non-local bifurcational problems. Bifurcation and Chaos, 6 (1996), 919–948. CrossRefGoogle Scholar
Gonchenko, S.V., Shilnikov, L., Turaev, D.. On global bifurcations in three-dimensional diffeomorphisms leading to wild Lorenz-like attractors. Regul. Chaotic Dyn., 14 (2009), 137–147. CrossRefGoogle Scholar
Gonchenko, S.V., Ovsyannikov, I.I., Simó, C., Turaev, D.. Three-dimensional Hénon-like maps and wild Lorenz-like attractors. Bifurc. Chaos, 15 (2005), 3493–3508. CrossRefGoogle Scholar
Gonchenko, S.V., Meiss, J.D., Ovsyannikov, I.I.. Chaotic dynamics of three-dimensional Hénon maps that originate from a homoclinic bifurcation. Regular Chaotic Dyn., 11 (2006), 191–212. CrossRefGoogle Scholar
Turaev, D.V., Shilnikov, L.P.. An example of a wild strange attractor. Sbornik Mathematics, 189 (1998), 137–160. CrossRefGoogle Scholar
Turaev, D.V., Shilnikov, L.P.. Pseudo-hyperbolisity and the problem on periodic perturbations of Lorenz-like attractors. Russian Dokl. Math., 77 (2008), 17–21. CrossRefGoogle Scholar
Gonchenko, S.V., Gonchenko, A.S., Ovsyannikov, I.I., Turaev, D.V.. Examples of Lorenz-like attractors in Hénon-like maps. Math. Model. Nat. Phenom. 8 (2013), 32–54. CrossRefGoogle Scholar
L.P. Shilnikov, A.L. Shilnikov, D.V. Turaev, L.O. Chua. Methods of qualitative theory in nonlinear dynamics. Part I, World Scientific, 1998.
Gonchenko, S.V., Shilnikov, L.P.. Invariants of ??-conjugacy of diffeomorphisms with a nongeneric homoclinic trajectory. Ukrainian Mathematical Journal, 42 (1990), 134–140. CrossRefGoogle Scholar
Gonchenko, S.V.. Dynamics and moduli of Ω-conjugacy of 4D-diffeomorphisms with a structurally unstable homoclinic orbit to a saddle-focus fixed point. AMS Transl. Math., 200 (2000), 107–134. Google Scholar
Gonchenko, S.V., Shilnikov, L.P., Turaev, D.V.. On the existence of Newhouse regions near systems with non-rough Poincare homoclinic curve (multidimensional case). Russian Acad. Sci. Dokl. Math., 47 (1993), 268–283. Google Scholar
Shilnikov, A.L., Shilnikov, L.P., Turaev, D.V.. Normal forms and Lorenz attractors. Bifurc. Chaos, 3 (1993), 1123–1139. CrossRefGoogle Scholar
A.L. Shilnikov. Bifurcation and chaos in the Marioka-Shimizu system. Methods of qualitative theory of differential equations, Gorky (1986), 180–193 [English translation in Selecta Math. Soviet., 10 (1991), 105–117]
Shilnikov, A.L.. On bifurcations of the Lorenz attractor in the Shimuizu-Morioka model. Physica D, 62 (1993), 338–346. CrossRefGoogle Scholar
Tigan, G., Turaev, D.. Analytical search for homoclinic bifurcations in the Shimizu-Morioka model. Physica D: Nonlinear Phenomena, 240 (2011), 985–989. CrossRefGoogle Scholar
Homoclinic tangencies, edited by S.V. Gonchenko and L.P. Shilnikov, Moscow-Izhevsk, 2007.
Gonchenko, S.V., Gonchenko, V.S., Shilnikov, L.P.. On homoclinic origin of Henon-like maps. Regular and Chaotic Dynamics, 4–5 (2010), 462–481. Google Scholar
Gonchenko, S.V., Ovsyannikov, I.I., Turaev, D.V.. On the effect of invisibility of stable periodic orbits at homoclinic bifurcations. Physica D, 241 (2012), 1115–1122. CrossRefGoogle Scholar
Rom-Kedar, V., Turaev, D.. Big islands in dispersing billiard-like potential. Physica D, 130 (1999), 187–210. CrossRefGoogle Scholar
Gonchenko, S.V., Gonchenko, V.S.. On bifurcations of the birth of closed invariant curves in the case of two-dimensional diffeomorphisms with homoclinic tangencies. Proc. Steklov Inst. Math., 244 (2004), 80–105. Google Scholar
Gonchenko, S.V., Turaev, D.V., Shilnikov, L.P.. Dynamical phenomena in multi-dimensional systems with a non-rough Poincare homoclinic curve. Russ. Acad. Sci. Dokl. Math., 47 (1993), 410–415. Google Scholar
Gonchenko, S.V., Shilnikov, L.P., Turaev, D.V.. Dynamical phenomena in systems with structurally unstable Poincare homoclinic orbits. Chaos, 6 (1996), 15–31. CrossRefGoogle ScholarPubMed
Gonchenko, S.V., Shilnikov, L.P., Turaev, D.V.. On dynamical properties of diffeomorphisms with homoclinic tangencies. J. Math. Sci., 126 (2005), 1317–1343. CrossRefGoogle Scholar
Gonchenko, S.V., Gonchenko, V.S., Tatjer, J.C.. Bifurcations of three-dimensional diffeomorphisms with non-simple quadratic homoclinic tangencies and generalized Hénon maps. Regular and Chaotic Dynamics, 12 (2007), 233–266. CrossRefGoogle Scholar