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.
Author(s): Marat Akhmet, Mehmet Onur Fen, Ejaily Milad Alejaily
Series: Nonlinear Systems and Complexity
Publisher: Springer
Year: 2020
Language: English
Pages: 233
Front Matter ....Pages i-xiii
Introduction (Marat Akhmet, Mehmet Onur Fen, Ejaily Milad Alejaily)....Pages 1-14
The Unpredictable Point and Poincaré Chaos (Marat Akhmet, Mehmet Onur Fen, Ejaily Milad Alejaily)....Pages 15-23
Unpredictability in Bebutov Dynamics (Marat Akhmet, Mehmet Onur Fen, Ejaily Milad Alejaily)....Pages 25-40
Nonlinear Unpredictable Perturbations (Marat Akhmet, Mehmet Onur Fen, Ejaily Milad Alejaily)....Pages 41-56
Unpredictability in Topological Dynamics (Marat Akhmet, Mehmet Onur Fen, Ejaily Milad Alejaily)....Pages 57-79
Unpredictable Solutions of Hyperbolic Linear Equations (Marat Akhmet, Mehmet Onur Fen, Ejaily Milad Alejaily)....Pages 81-95
Strongly Unpredictable Solutions (Marat Akhmet, Mehmet Onur Fen, Ejaily Milad Alejaily)....Pages 97-108
Li–Yorke Chaos in Hybrid Systems on a Time Scale (Marat Akhmet, Mehmet Onur Fen, Ejaily Milad Alejaily)....Pages 109-124
Homoclinic and Heteroclinic Motions in Economic Models (Marat Akhmet, Mehmet Onur Fen, Ejaily Milad Alejaily)....Pages 125-137
Global Weather and Climate in the Light of El Niño-Southern Oscillation (Marat Akhmet, Mehmet Onur Fen, Ejaily Milad Alejaily)....Pages 139-172
Fractals: Dynamics in the Geometry (Marat Akhmet, Mehmet Onur Fen, Ejaily Milad Alejaily)....Pages 173-202
Abstract Similarity, Fractals, and Chaos (Marat Akhmet, Mehmet Onur Fen, Ejaily Milad Alejaily)....Pages 203-221
Back Matter ....Pages 223-226
