National Academy of Sciences of Ukraine Institute of Mathematics International Conference DYNAMICAL SYSTEMS AND THEIR APPLICATIONS Kyiv, Ukraine June 22 – 26, 2015 Abstracts Kyiv – 2015 DYNAMICAL SYSTEMS AND THEIR APPLICATIONS Kyiv, Ukraine June 22 – 26, 2015 International Conference on Dynamical Systems and Their Appli- cations is organized by the Institute of Mathematics, National Acad- emy of Sciences of Ukraine. Dynamical systems theory, being one of the rapidly developing ar- eas of modern mathematics, provides powerful theoretical base for exploring a variety of models that arise in natural and social sciences, engineering and technology. The combination of the internal wealth and beauty of results with the exceptional practical importance moti- vates a growing number of specialists to study into dynamical systems. Beginning with the 60s, the Institute of Mathematics held con- ferences and schools on various fields of mathematics, in particular, on dynamical systems. This has had a profound effect on the devel- opment not only of dynamical systems theory but also of the overall nonlinear dynamics. Not so long, the Institute of Mathematics de- cided to arrange International Conference “Dynamical Systems and Their Applications” (ICDSA), aimed to promote transnational coop- eration and share good practice in the field of dynamical systems theory. The first conference hosted in Kyiv in 2012. The second edi- tion of ICDSA takes place in Kyiv again, it considers a wide range of issues of the modern theory of dynamical systems. Organizing Committee Institute of Mathematics National Academy of Sciences of Ukraine A. A. Akbergenov O. M. Baranovsky B. G. Feshchenko S. I. Maksymenko (Chair) O. V. Marunkevych A. A. Panchuk O. V. Rybak A. G. Sivak N. A. Vasilenko Contents Plenary Speakers 6 Francisco Balibrea. Dynamics of the Thue-Morse System of Difference Equations 7 Valery Gaiko. On Limit Cycle Bifurcations 8 Rostyslav Grigorchuk. Absolutely Non Free Actions, Random Subgroups, and Factor Representations 9 Yulij Ilyashenko. Towards the Global Bifurcation Theory on the Plane 10 Sergey Kaschenko. Regular and Chaotic Oscillations in Singularly Perturbed Systems with Delay 11 Anatole Katok. On Flexibility of Entropies and Lyapunov Exponents 12 Mykola Pratsiovytyi. Topologically-Metrical and Fractal Analysis of Local Structure of Continuous Functions 13 Oleksandr Sharkovsky. !-Attractors and Their Basins 14 Andrii Sivak. σ-Attractors, µ-Attractors, and Their Basins 15 Grygorii M. Torbin. Fractals, Singular Probability Measures, and Dynamical Systems 16 Igor Vlasenko. New Invariants of Topological Conjugacy of Non-Invertible Inner Mappings 17 Abdyvali Akbergenov. On a Class of Differential-Functional Equations 18 Bianca-Elena Aram˘a. The Cost of Approximate Controllability and a Unique Continuation Result at Initial Time for the Ginzburg-Landau Equation 19 Maxim Bebiya. Stabilization of Systems with Multiple Power Nonlinearities 21 Mykola M. Bilotskyi. Application of (c)-Properties for Cesaro Summability Methods (C; 1) of Ergodic Theorem 22 Gennadii Burlak. Nonlinear Dynamics of Random Laser Generation in 3D Percolating Clusters 23 Syamal Kumar Dana. Chimera States in a Network of Oscillators under Cross-Global Coupling 24 Mahaveer Gadiya, K. D. Masalkar. The Homotopy Method to Solve Dynamical Systems and Boundary Value Problems for Differential Equations 25 Olena Karpel. Infinite Orbit Equivalence Class for a Minimal Substitution Dynamical System 26 Oleksandr V. Karpenko. Probability Measures and Dynamical Systems Generated by Continuous Nowhere Monotonic Functions 27 Svetlana Katok. Reduction Theory, Coding of Geodesics, and Continued Fractions 28 Karsten Keller. Aspects of the Permutation Entropy 29 Natalya Kizilova. A Novel Model for Explanation the Regular and Chaotic Dynamics in Arterial Blood Flow 30 Volodymyr Koshmanenko. On Dynamical System of Conflict with Fair Redistribution of Vital Resources 31 Mykhailo Kuznietsov. Analogue of Topological Entropy for Some Infinite-Dimensional Systems 32 M. Howard Lee. Solving for the Fixed Points of 3-Cycle in the Logistic Map and Toward Realizing Chaos by the Theorems of Sharkovskii and Li-Yorke 33 Xiaoyan Liu. Numerical Methods for Solving Systems of Nonlinear Volterra Integral Equations with Cardinal Splines 34 Marina Lupain. On DP-Approach to Fractal Properties of Random Variables with Independent Identically Distributed LML-Symbols 35 Taras V. Lyutyy, Stanislav I. Denisov, Vladislav V. Reva. Forced Rotation of a Ferromagnetic Fine Particle in a Viscous Carrier: the Stationary Probability Density 36 Alexander Makarenko, Sergey Lazarenko, Anton Popov. Systems with Memory, Nonlocality and Anticipation. Some New Examples and New Research Problems 37 Vitalii Marchenko. One Special Construction in the Spectral Theory of C0-Semigroups 38 Roderick Melnik. Dynamical Systems Approach for Multidimensional Phase Transformation Models and Their Applications 39 Roman Nikiforov. On Phenomena Connected with Infinite IFS 40 Leonid Nizhnik, Irina Nizhnik. Deterministic Diffusion 41 4 Olurotimi Ojoniyi. Studies on Chaos and Hyperchaos in Four-Dimensional Sprott Systems 42 George Osipenko. Invariant Measures in Projective Bundle 44 Makar Plakhotnyk. Topologically Conjugated Unimodal Interval Maps 45 Mykola Pratsiovytyi, Yanina Goncharenko, Oleksii Panasenko, Natalya Vasylenko. One-Dimensional Dynamical System Generated by One Continuous Twisted Function 46 Tetyana Revina. Robust Feedback Synthesis for a Disturbed Canonical System 47 Olena Romanenko, Oleksandr Sharkovsky. Difference Equations and Infinite-Dimensional Dynamical Systems 48 Oleksandr Rybak. Sensitivity of the Induced Systems on an Interval 49 Tetiana Rybalkina. Topological Classification of Linear Mappings 50 Vladimir M. Savchin. On Potentiality of Some Evolutionary Equations with Deviating Arguments 51 Lilia Sinelnyk. On Singularity of Distribution of Random Variables with Independent Symbols of Oppenheim Expansions 52 Oleksandr Slutskyi. About the Faithfulness of Cantor Series Expansion Cylinders Family for the Packing Dimension Calculation 53 Vasyl Yu. Slyusarchuk. Nonlinear Almost Periodic Difference Equations 54 Olena Smiian. On Some Problems of Faithful Covering Family of Cylinders Generated by the Q1-Expansion 55 Alexey Teplinsky. A Smoothly Linearizable Circle Diffeomorphism with Breaks 56 Victor Urmanchev. Four Dimensional and Singular Perturbation Systems of Differential Equations and Two Dimensional Dynamical System with Impulse 57 Inga Verygina. Comparison of Two Strategies in the Problem of “Conquest” of Territory 58 Irina Garko. On fractal properties of probability measures with independent x − Q1 -digits 59 List of Participants 60 Author Index 64 5 Plenary Speakers Francisco Balibrea (Spain) “Dynamics of the Thue-Morse system of difference equations” Valery Gaiko (Belarus) “On limit cycle bifurcations” Rostislav Grigorchuk (USA) “Absolutely non free actions, random subgroups, and factor representations” Yulij Ilyashenko (Russia) “Towards the global bifurcation theory on the plane” Anatole Katok (USA) “On Flexibility of Entropies and Lyapunov Exponents” Sergey Kaschenko (Russia) “Regular and chaotic oscillations in singularly perturbed systems with delay” Mykola Pratsiovytyi (Ukraine) “Topologically-metrical and fractal analysis of local structure of continuous functions” Oleksandr Sharkovsky (Ukraine) “!-Attractors and their basins” Andrii Sivak (Ukraine) “σ-Attractors, µ-attractors, and their basins” Grygorii Torbin (Ukraine) “Fractals, singular probability measures and dynamical systems” Igor Vlasenko (Ukraine) “New invariants of topological conjugacy of non-invertible inner mappings” 6 Dynamical Systems and Their Applications June 22 – 26, 2015, Kyiv, Ukraine dynamics of the thue-morse system of difference equations francisco balibrea Departamento de Matemticas, University of Murcia, 30100 Murcia, Spain e-mail: [email protected] The Thue–Morse system of difference equations was introduced in [1] as a model to un- derstand the electric behavior (conductor or insulator) of an array of electrical punctual positive charges occupying positions following a one dimensional distribution of points called a Thue–Morse chain which it is connected to the sequence t = (0110100110010 :::) called also the Thue–Morse sequence. Unfolding the system of difference equations, we obtain the two-dimensional dynamical system in the plane given by F (x; y) = (x(4 − x − y); xy): The interest of such system was stated by A. Sharkovsk˘ıias an open problem and proposing some questions. The most interesting dynamics of the system is developed inside an invariant plane tri- angle, where hyperbolic periodic points of almost all period appear, there are subsets of transitivity and invariant curves of spiral form around the unique inside fixed point. In this talk we will present some results concerning the behavior of all points outside the triangle, completing the known dynamics of the system. In fact we have obtained that outside the triangle, the orbits of all points are unbounded. Some of them go to infinite in an oscillating way occupying the second and third quadrant of the plane and others are going in a monotone way to infinite. Outside the triangle there are no periodic points. Such new results has an interesting interpretation in terms of the physics of the problem. Additionally we will answer some of the questions stated by Sharkovsk˘ıi concerning the inside of the mentioned triangle. We will also present graphycal analysis of such evolutions and also the visualization