The book is concerned with the concepts of chaos and fractals, which are within the scopes of dynamical systems, geometry, measure theory, topology, and numerical analysis during the last several decades. It is revealed that a special kind of Poisson stable point, which we call an unpredictable point, gives rise to the existence of chaos in the quasi-minimal set. This is the first time in the literature that the description of chaos is initiated from a single motion. Chaos is now placed on the line of oscillations, and therefore, it is a subject of study in the framework of the theories of dynamical systems and differential equations, as in this book. The techniques introduced in the book make it possible to develop continuous and discrete dynamics which admit fractals as points of trajectories as well as orbits themselves. To provide strong arguments for the genericity of chaos in the real and abstract universe, the concept of abstract similarity is suggested.
Chapter
1. Introduction.
Chapter
2. The Unpredictable Point and Poincare Chaos.
Chapter
3. Unpredictability in Bebutov Dynamics.
Chapter
4. Non-linear Unpredictable Perturbations.
Chapter
5. Unpredictability in Topological Dynamics.
Chapter
6. Unpredictable Solutions of Hyperbolic Linear Equations.
Chapter
7. Strongly Unpredictable Solutions.
Chapter
8. Li-Yorke Chaos in Hybrid Systems on a Time Scale.
Chapter
9. Homoclinic and Heteroclinic Motions in Economic Models.
Chapter
10. Global Weather and Climate in the light of El Nino-Southern Oscillation.
Chapter
11. Fractals: Dynamics in the Geometry.
Chapter
12. Abstract Similarity, Fractals and Chaos.
Marat Akhmet
Dr. Marat Akhmet is currently a Professor at Department of Mathematics, Middle East Technical University, Ankara, Turkey. He got his B.S. degree in mathematics at Aktobe State University, Kazakhstan, and Ph.D. degree in differential equations and mathematical physics at Kiev State University, Ukraine. Currently Dr. Marat Akhmets researches focus on the dynamical models and differential equations. He has published six books and more than a hundred and fifty scientific papers. In the last several years, he has been investigating dynamics of neural networks, periodic and almost periodic motions, stability, chaos and fractals.
Mehmet Onur Fen
Dr. Mehmet Onur Fen received his B.S. degree in mathematics from Middle East Technical University, Ankara, Turkey, with a double major in physics. He graduated from the mathematics Ph.D. program of the same university in September 2013. Currently he is working as an Associate Professor at Department of Mathematics, TED University, Ankara, Turkey. Dr. Mehmet Onur Fens research interests include chaotic dynamical systems, mathematical models of neural networks, and several types of differential equations.
Ejaily Milad Alejaily
Dr. Ejaily Milad Alejaily is currently a lecturer at the College of Engineering Technology, Houn, Libya. He received his B.S. degree in mechanical engineering from Sirte University, Libya in 1997. Later in 2009, Dr. Alejaily obtained his M.S. in mathematics from Universiti Teknologi Malaysia. He graduated from the mathematics Ph.D. program of Middle East Technical University, Ankara, Turkey in August 2019.
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