Geodesic
Analysis of local dynamics of difference and close to them differential-difference equations
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2018), pp. 29-41

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We consider the local dynamics of a class nonlinear difference equations which is important for applications. Using the perturbation theory methods we built the sets of singularly perturbed differential-difference equations close to the original difference equations to some extent. We show that the critical cases in the problem of stability of a null balance state have infinite dimension. We offer the method to set special non-linear boundary-value problems that do not contain small parameters. They play the role of normal forms. Their nonlocal dynamics describes the structure of solutions to original equations in a small neighborhood of a balance state. We show that the dynamic properties of difference and close to them differential-difference equations considerably differ.
Mots-clés : bifurcation, singular perturbation
Keywords: stability, normal form, dynamics. 
@article{IVM_2018_9_a3,
     author = {I. S. Kashchenko and S. A. Kashchenko},
     title = {Analysis of local dynamics of difference and close to them differential-difference equations},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {29--41},
     publisher = {mathdoc},
     number = {9},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2018_9_a3/}
}
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I. S. Kashchenko; S. A. Kashchenko. Analysis of local dynamics of difference and close to them differential-difference equations. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2018), pp. 29-41. http://geodesic.mathdoc.fr/item/IVM_2018_9_a3/