General solutions of weakly delayed discrete systems in 3D

Abstract Discrete systems <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>x</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mi>A</m:mi> <m:mi>x</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>k</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>+</m:mo> <m:mi>B</m:mi> <m:mi>x</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mo>−</m:mo> <m:mi>m</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> x\left(k+1)=Ax\left(k)+Bx\left(k-m) , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo form="prefix">,</m:mo> <m:mn>1</m:mn> <m:mo form="prefix">,</m:mo> <m:mrow> <m:mo>…</m:mo> </m:mrow> <m:mspace width="0.33em"/> </m:math> k=0,1,\ldots \hspace{0.33em} are analyzed, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>m</m:mi> </m:math> m is a fixed positive integer, <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>A</m:mi> </m:math> A , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>B</m:mi> </m:math> B are constant 3 by 3 matrices and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>x</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mo>{</m:mo> <m:mrow> <m:mo>−</m:mo> <m:mi>m</m:mi> <m:mo>,</m:mo> <m:mo>−</m:mo> <m:mi>m</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mrow> <m:mo>…</m:mo> </m:mrow> </m:mrow> <m:mo>}</m:mo> </m:mrow> <m:mo>→</m:mo> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>3</m:mn> </m:mrow> </m:msup> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>A</m:mi> </m:math> A . It is shown that, for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> <m:mo>≥</m:mo> <m:mn>3</m:mn> <m:mi>m</m:mi> </m:math> k\ge 3m or for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> <m:mi>m</m:mi> </m:math> 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.