ОБЧИСЛЮВАЛЬНІ СКЛАДНОСТІ МОДЕЛЮВАННЯ ДИНАМІЧНИХ СИСТЕМ З АНТИЦИПАЦІЄЮ

• Дата: 2025
• Автори: Lazarenko, S.V., Makarenko, A.S.
• Формат: Стаття
• Мова: English
• Опубліковано: V.M. Glushkov Institute of Cybernetics of NAS of Ukraine 2025
• Теми: обчислювальна складність; дискретні динамічні системи; оператор Хатчинсона; періодична траєкторія
• Онлайн доступ: https://jais.net.ua/index.php/files/article/view/638
• Назва журналу: Problems of Control and Informatics

The concept of anticipation stands for the dependence of future states not only on the past, but also on themselves. One of the main reasons for the relevance of the study of the anticipatory systems is the over-complexity of modeling of the systems with multiple possible scenarios, since anticipatory systems often imply multiple solutions. A non-prevalence of this field of computer science is also often caused by not well-posing of the problem due to non-uniqueness of the solutions. Thus, systems with anticipation represent a new direction in cybernetics and the models based on anticipation can more formally describe a large number of existing systems and processes, compared with classical models with delay. In current paper, we consider such nonlinear discrete dynamical systems with strong anticipation, in which future states can be represented by an explicit dependence on the past with the help of the Hutchinson operator. The evolution of such dynamical systems is carried out in a Hausdorff metric space. The article focuses on the fundamental problem of modeling such systems – the amount of use of computing resources. A number of definitions have been introduced to study the dynamics of anticipatory systems. The necessary concepts of the theory of computational complexity are presented. An important tool for studying the dynamics of systems is the Atlas of Charts of dynamic regimes. The construction of which requires the adaptation of procedures for finding periodic trajectories for systems of this type. The article proposes and describes in detail the procedures for searching for periodic trajectories of dynamical systems with anticipation. Time and spatial complexities are obtained for the construction of states, trajectories, and these procedures in general. In order to minimize the time complexity during the simulation of anticipatory systems, the presentation of the states of the corresponding dynamical systems with the help of multisets is justified. In order to optimize further computational costs, one should take into account the structure of the phase space of a dynamical system with an anticipation, thereby combining the proposed procedures.