General solutions of weakly delayed discrete systems in 3D
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Date
2025-10-24
Authors
Diblík, Josef
Boháčková, Hana
Růžičková, Miroslava
Šafařík, Jan
0000-0001-5009-316XAdvisor
Referee
Mark
Journal Title
Journal ISSN
Volume Title
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Altmetrics
Abstract
Discrete systems x ( k + 1 ) = A x ( k ) + B x ( k - m ) x\left(k+1)=Ax\left(k)+Bx\left(k-m) , k = 0 , 1 , & mldr; k=0,1,\ldots \hspace{0.33em} are analyzed, where m m is a fixed positive integer, A A , B B are constant 3 by 3 matrices and x : { - m , - m + 1 , & mldr; } -> R 3 x:\left\{-m,-m+1,\ldots \right\}\to {{\mathbb{R}}}{3} . Assuming that the system is weakly delayed, its general solution is constructed for every case of the Jordan form of the matrix A A . It is shown that, for k >= 3 m k\ge 3m or for k >= 2 m k\ge 2m , these formulas reduce to simple forms depending on only the three independent parameters generated by the initial values. Formulas connecting these parameters with the initial ones are found. The results are illustrated by examples. Open problems for future research are discussed, and comparisons are given with the previous results.
Discrete systems x ( k + 1 ) = A x ( k ) + B x ( k - m ) x\left(k+1)=Ax\left(k)+Bx\left(k-m) , k = 0 , 1 , & mldr; k=0,1,\ldots \hspace{0.33em} are analyzed, where m m is a fixed positive integer, A A , B B are constant 3 by 3 matrices and x : { - m , - m + 1 , & mldr; } -> R 3 x:\left\{-m,-m+1,\ldots \right\}\to {{\mathbb{R}}}{3} . Assuming that the system is weakly delayed, its general solution is constructed for every case of the Jordan form of the matrix A A . It is shown that, for k >= 3 m k\ge 3m or for k >= 2 m k\ge 2m , these formulas reduce to simple forms depending on only the three independent parameters generated by the initial values. Formulas connecting these parameters with the initial ones are found. The results are illustrated by examples. Open problems for future research are discussed, and comparisons are given with the previous results.
Discrete systems x ( k + 1 ) = A x ( k ) + B x ( k - m ) x\left(k+1)=Ax\left(k)+Bx\left(k-m) , k = 0 , 1 , & mldr; k=0,1,\ldots \hspace{0.33em} are analyzed, where m m is a fixed positive integer, A A , B B are constant 3 by 3 matrices and x : { - m , - m + 1 , & mldr; } -> R 3 x:\left\{-m,-m+1,\ldots \right\}\to {{\mathbb{R}}}{3} . Assuming that the system is weakly delayed, its general solution is constructed for every case of the Jordan form of the matrix A A . It is shown that, for k >= 3 m k\ge 3m or for k >= 2 m k\ge 2m , these formulas reduce to simple forms depending on only the three independent parameters generated by the initial values. Formulas connecting these parameters with the initial ones are found. The results are illustrated by examples. Open problems for future research are discussed, and comparisons are given with the previous results.
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Citation
Advances in Nonlinear Analysis. 2025, vol. 14, issue 1, p. 1-49.
https://www.degruyterbrill.com/document/doi/10.1515/anona-2025-0121/html
https://www.degruyterbrill.com/document/doi/10.1515/anona-2025-0121/html
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Peer-reviewed
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en