TY - JOUR
AU - Romanenko, O. Yu.
AU - Sharkovsky, O. M.
PY - 2022/10/27
Y2 - 2023/02/04
TI - Asymptotic Properties of the Semigroup Generated by a Continuous Interval Map
JF - International Journal of Mathematical, Engineering, Biological and Applied Computing
JA - IJMEBAC
VL - 1
IS - 2
SE - Articles
DO -
UR - https://www.scipublications.com/journal/index.php/ijmebac/article/view/468
SP - 77-94
AB - <p>The article's purpose is twofold. First, we wish to draw attention to the insufficiently known field of continuous-time difference equations. These equations are paradigmatic for modeling complexity and chaos. Even the simplest equation , easily leads to complex dynamics, its solutions are perfectly suited to simulate strong nonlinear phenomena such as large-to-small cascades of structures, intermixing, formation of fractals, etc. Second, in the main body of the article we present a small but very important part of the theory behind the above equation marked by . Just as the discrete-time analog of this equation induces the one-dimensional dynamical system on some interval , so the equation induces the infinite-dimensional dynamical system on the space of functions . In the latter case, not only are the long-term behaviours of solutions critically dependent on the limit behaviour of the sequence (as in the discrete case) but also on the internal structure of as . Assuming to be continuous, we consider the iterations of as the semigroup generated by on the space of continuous maps, and introduce the notion of a limit semigroup for in a wider map space in order to investigate asymptotic properties of . We construct a limit semigroup in the space of upper semicontinuous maps. This enables us to describe both of the aforementioned aspects of our interest around the iterations of .</p>
ER -