@article{Romanenko_Sharkovsky_2022, title={Asymptotic Properties of the Semigroup Generated by a Continuous Interval Map}, volume={1}, url={https://www.scipublications.com/journal/index.php/ijmebac/article/view/468}, abstractNote={<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>}, number={2}, journal={International Journal of Mathematical, Engineering, Biological and Applied Computing}, author={Romanenko, O. Yu. and Sharkovsky, O. M.}, year={2022}, month={Oct.}, pages={77–94} }