Dynamics with Chaos and Fractals

This book considers for the first time chaos for a single motion that implies chaotic behavior of infinite neighbors and makes the property generic for the dynamics. It also considers fractals as geometrical objects-very popular nowadays for their complexity, sophisticated structure, and ubiquity in the natural world and in modern engineering. Recent research presented herein suggests ways for how to map fractals, and explains how the mapping can easily be applied to realize discrete and continuous dynamics in fractals and for fractals. The authors also introduce the use of special functional space and new, unpredictable functions determined on the real axis, continuous and bounded. Thus, they describe an extension of oscillations theory for differential and discrete equations for unpredictable, sensitive, and irregular phenomena. Novel results for the unpredictable solutions are further obtained and basic properties and methods of verification are developed. The book is written by mathematicians and it contains results that are rigorously proven. The material explained in this volume is highly relevant for those working in applied mathematics with a strong theoretical approach. It is indisputable that chaos and fractals are very connected, which is why it is natural that these two subjects are united in this single book.