Exploring Constraints, Solvability and Stability of a Circular System of Complex Riccati Type Difference Equations | Chapter 9 | Research Updates in Mathematics and Computer Science Vol. 8

In this paper, the circular system of Riccati type complex difference equations of the form

𝑢(𝑗)𝑛+1=𝑎𝑗⁢𝑢(𝑗−1)𝑛+𝑏𝑗𝑐𝑗⁢𝑢(𝑗−1)𝑛+𝑑𝑗 ,𝑛=0,1,2,⋯,𝑗=1,2,⋯,𝑘 where un(0) := un(k) for all n, is investigated. First, the forbidden set of the equation is given. Then the solvability of the system is examined and then the expression of the solutions are given in terms of their initial values. Next, the asymptotic behaviour of the solutions is studied. Finally, in case of negative Riccati real numbers

𝑅𝑗:=𝑎𝑗⁢𝑑𝑗−𝑏𝑗⁢𝑐𝑗[𝑎𝑗+𝑑𝑗]2 ,𝑗∈―――1,𝑘,it is shown that there exists a unique positive fixed point which attracts all solutions starting from positive states.

 

 

Author(s) Details

George L. Karakostas

Department of Mathematics, University of Ioannina, Ioannina 45110, Greece.

 

Please see the book here:- https://doi.org/10.9734/bpi/rumcs/v8/3703G