2019/05/05 by Marat Akhmet, Akhmet, Marat, Ejaily Milad Alejaily +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #Chaos control and synchronization #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals
paper · pdf · doi:10.48550/arxiv.1905.02198
openalex publication_date 2019/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
To prove presence of chaos for fractals, a new mathematical concept of abstract similarity is introduced. As an example, the space of symbolic strings on a finite number of symbols is proved to possess the property. Moreover, Sierpinski fractals, Koch curve as well as Cantor set satisfy the definition. A similarity map is introduced and the problem of chaos presence for the sets is solved by considering the dynamics of the map. This is true for Poincare, Li-Yorke and Devaney chaos, even in multi-dimensional cases. Original numerical simulations which illustrate the results are delivered.