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Preservation of stability under discretization of systems of ordinary differential equations

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Abstract

We study the stability preservation problem while passing from ordinary differential to difference equations. Using the method of Lyapunov functions, we determine the conditions under which the asymptotic stability of the zero solutions to systems of differential equations implies that the zero solutions to the corresponding difference systems are asymptotically stable as well. We prove a theorem on the stability of perturbed systems, estimate the duration of transition processes for some class of systems of nonlinear difference equations, and study the conditions of the stability of complex systems in nonlinear approximation.

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Author information

Correspondence to A. Yu. Aleksandrov.

Additional information

Original Russian Text Copyright © 2010 Aleksandrov A. Yu. and Zhabko A. P.

The authors were supported by the Russian Foundation for Basic Research (Grant 08-08-92208GFEN_a).

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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 51, No. 3, pp. 481–497, May–June, 2010.

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Aleksandrov, A.Y., Zhabko, A.P. Preservation of stability under discretization of systems of ordinary differential equations. Sib Math J 51, 383–395 (2010). https://doi.org/10.1007/s11202-010-0039-y

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Keywords

  • difference system
  • Lyapunov function
  • asymptotic stability
  • complex system
  • stability in nonlinear approximation