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

For citation:

Kashchenko I. S., Kashchenko S. A. Local dynamics of difference and difference-differential equations. Izvestiya VUZ. Applied Nonlinear Dynamics, 2014, vol. 22, iss. 1, pp. 71-92. DOI: 10.18500/0869-6632-2014-22-1-71-92

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: 180)
Article type: 

Local dynamics of difference and difference-differential equations

Kashchenko I. S., P. G. Demidov Yaroslavl State University
Kashchenko Sergej Aleksandrovich, P. G. Demidov Yaroslavl State University

We study local dynamics of difference and singular perturbed difference-differential systems in the neighborhood of zero equilibrium state. All critical cases in this problem have infinite dimension. We construct special nonlinear equations that play the role of normal form. Their nonlocal dynamics describes the behavior of solution of initial system.

  1. Maistrenko YL, Romanenko EY, Sharkovsky AN. Difference Equations and Their Applications. Kiev: Naukova Dumka; 1986.
  2. Maistrenko YL, Maistrenko VL, Chua LO. Cycles of chaotic intervals in a time-delayed Chua’s circuit. International Journal of Bifurcation and Chaos. 1993;3(6):1557—1572.
  3. Kaschenko DS. Dynamics of the Simplest Piecewise Linear Discontinuous Mappings. Model. Anal. Inform. Sist. 2012;19(3):73—81.
  4. Kulenovic MRS, Ladas G. Dynamics of second order rational difference equations with open problems and conjectures. Chapman and Hall/CRC; 2001. 232 p.
  5. Shnol' ÈÈ. On the stability of fixed points of two-dimensional mappings. Differ. Uravn. 1994;30(7):1156—1167.
  6. Kuramoto Y, Tsuzudi T. On the formation of dissipative structures in reaction-diffusion systems. Progr. Theor. Phys. 1975;54(3):687—699.
  7. Hartman P. Ordinary differential equations. Moscow: Mir; 1970.
  8. Bruno AD. Local method of nonlinear analysis of differential equations. Moscow: Nauka; 1979.
  9. Kashchenko SA. Application of the normalization method to the study of the dynamics of a differential-difference equation with a small factor multiplying the derivative. Differ. Uravn. 1989;25(8):1448—1451.
  10. Kaschenko SA. Normalization in the systems with small diffusion. Int. J. of Bifurcations and Chaos. 1996;6(7):1093—1109.
  11. Kashchenko IS. Asymptotic analysis of the behavior of solutions to equations with large delay. Dokl. Math. 2008;78:570—573.
  12. Kolmogorov AN, Petrovsky IG, Piskunov NS. Investigation of the Equation of Diffusion Combined with Increasing of the Substance and Its Application to a Biology Problem. Bulletin of Moscow State University Series A: Mathematics and Mechanics. 1937;1:1—25.
  13. Kashchenko SA. The Ginzburg–Landau equation as a normal form for a second-order difference-differential equation with a large delay. Zh. Vychisl. Mat. Mat. Fiz. 1998;38(3):457—465.
  14. Bogolyubov NN, Mitropolskii YA. Asymptotic Methods in the Theory of Nonlinear Oscillations. Moscow: Fizmatgiz;1974.
  15. Akhromeeva TS, Kurdyumov SP, Malinetskii GG, Samarskii AA. On the classification of the solutions of a system of nonlinear diffusion equations in a neighborhood of a bifurcation point. Itogi Nauki i Tekhniki. Ser. Sovrem. Probl. Mat. Nov. Dostizh. Moscow: VINITI. 1986;28:207–313.
Short text (in English):
(downloads: 64)