ISSN 0869-6632 (Print)
ISSN 2542-1905 (Online)


For citation:

Zhusubaliev Z. T., Yanochkina O. O. Formation and breakdown of a multilayered closed curve in noninvertible maps. Izvestiya VUZ. Applied Nonlinear Dynamics, 2010, vol. 18, iss. 1, pp. 51-60. DOI: 10.18500/0869-6632-2010-18-1-51-60

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Full text PDF(Ru):
(downloads: 144)
Language: 
Russian
Article type: 
Article
UDC: 
517.9

Formation and breakdown of a multilayered closed curve in noninvertible maps

Autors: 
Zhusubaliev Zhanybaj Tursunbaevich, Federal state budgetary educational institution of higher professional education "South-West state University"
Yanochkina Olga Olegovna, Federal state budgetary educational institution of higher professional education "South-West state University"
Abstract: 

The paper describes the mechanism for the formation of closed invariant curves that are formed as layered structures of several sets of interlacing manifolds each with their associated stable or unstable resonance modes. Such invariant curves can arise, for instance, if the saddle cycle on a «simple resonance curves» undergoes period-doubling or pitchfork bifurcations transversely to the circumference of the closed curve.

Reference: 
  1. Afraimovich VS, Shilnikov LP. Invariant Two-Dimensional Tori, Their Breakdown and Stochasticity. Methods of qualitative theory of differential equations. Gorky: Gorky State University. 1983:325.
  2. Arnol'd VI, Afraimovich VS, Ilyashenko YuS, Shilnikov LP. Bifurcation theory. Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Fund. Napr. Moscow: VINITI. 1986:5:5–218 (in Russian).
  3. Anishchenko VS, Astakhov VV, Neiman AB, Vadivasova TE, Schimansky-Geier L. Nonlinear Dynamics of Chaotic and Stochastic Systems. Tutorial and Modern Development. Berlin: Springer; 2007.
  4. Maistrenko V, Maistrenko Yu, Mosekilde E. Torus breakdown in noninvertible maps. Phys. Rev. E. 2003;67:046215. DOI: 10.1103/PhysRevE.67.046215.
  5. Frouzakis CE, Kevrekidis IG, Peckhman BB. A route to computational chaos revisited: noninvertibility and breakup of an invariant circle. Physica D. 2003;177:101121. DOI: 10.1016/S0167-2789(02)00751-0.
  6. Zhusubaliyev ZhT, Mosekilde E. Bifurcations and Chaos in Piecewise- Smooth Dynamical Systems. Singapore: World Scientific; 2003.
  7. Zhusubaliyev ZhT, Mosekilde E, De S, Banerjee S. Transition from phase-locked dynamics to chaos in a piecewise-linear map. Phys. Rev. E. 2008;77(2):026206. DOI: 10.1103/PhysRevE.77.026206.
  8. Lorenz EN. Computational chaos – a prelude to computational instability. Physica D. 1989;35(3):299317. DOI: 10.1016/0167-2789(89)90072-9.
  9. Krauskopf B, Osinga HM, Peckham BB. Unfolding the cusp-cusp bifurcation of planar endomorphisms. SIAM J. Applied Dynamical Systems. 2007;6(2):403440. DOI: 10.1137/060672753.
  10. England JP, Krauskof B, Osinga HM. Bifurcations of stable sets in noninvertible planar maps. Int. J. Bifurcat. Chaos. 2005;15(3):891904. DOI: 10.1142/S0218127405012466.
  11. Zhusubaliyev ZhT, Mosekilde E. Birth of bilayered torus and torus breakdown in a piecewise-smooth dynamical system. Phys. Lett. A. 2006;351(3):167174. DOI: 10.1016/j.physleta.2005.10.080.
  12. Zhusubaliyev ZhT, Mosekilde E. Formation and destruction of multilayered tori in coupled map systems. Chaos. 2008;18(3):037124. DOI: 10.1063/1.2959141.
  13. Zhusubaliyev ZhT, Mosekilde E. Multilayered tori in a system of two coupled logistic maps. Phys. Lett. A. 2009;373(10):946951. DOI: 10.1016/j.physleta.2009.01.014.
  14. Zhusubaliyev ZhT, Mosekilde E. Novel routes to chaos through torus breakdown in noninvertible maps. Physica D. 2009;283(5):589602. DOI: 10.1016/j.physd.2008.12.012.
  15. Mira C, Gardini L, Barugola A, Cathala JC. Chaotic Dynamics in Two-Dimensional Noninvertible Maps. Singapore: World Scientific; 1996.
  16. Frouzakis CE, Gardini L, Kevrekidis IG, Millerioux G, Mira C. On some properties of invariant sets of two-dimensional noninvertible maps. Int. J. Bifurcat. Chaos. 1997;7:11671194. DOI: 10.1142/S0218127497000972.
Received: 
15.03.2009
Accepted: 
04.06.2009
Published: 
31.03.2010
Short text (in English):
(downloads: 80)