# Advanced Periodic Characteristics and New Research Questions

## Abstract

In the previous chapters we explored varieties of periodic traits of linear difference equations, piece-wise difference equations, rational difference equations, and Max-Type difference equations. First of all, we observed how the periodicity nature varied from the existence of unique periodic cycles to every solution being periodic. Second, we discovered the necessary and sufficient conditions for the existence of periodic cycles. Furthermore, we perceived the disparity between patterns of the even ordered periodic cycles and odd ordered periodic cycles with uniqueness of solutions and alternating patterns. Our next aim is to broaden our developed knowledge to ascertain the periodicity essence of third and higher order difference equations, systems of difference equations, when the periodic sequence \(\{A_{n}\}_{n=0}^{\infty }\) is periodic with period-k, (*k* ≥ 3) and study of new systems of piecewise difference equations and neural networking models. How will these criteria for the existence of periodic cycles be similar or different compared to our current mastery in previous chapters? What new periodic traits can we expect that will be quite different? How will the delay *m* ≥ 3 affect the new periodic traits? Will even and odd values of delay *m* ≥ 3 influence the new periodic traits? We will pose conjectures that will lead to new theorems and results based on our previous studies and will pose new research questions.

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