Roman Voliansky, Nina Volianska, Aji Prasetya Wibawa
Our paper is devoted to the study and design of novel chaotic systems to use in various applications. In our paper, we offer to use coordinate transformations to design a chaotic system and define its motions using algebraic-differential state space equations. We consider the known chaotic systems and apply some coordinate transformations to them. In this case, the system differential equations are used to define the known system, and observability algebraic equations depend on the used coordinate transformation. Our paper considers the transformation from cartesian coordinates into bipolar ones and vice versa. The direct transformation from cartesian coordinates in bipolar is based on using two lengths from the representative point, which define a system motion to some different base points. This transformation can be used when one interprets chaotic system state variables as coordinates in the orthogonal axes. This transformation is defined by quadratic polynomials, which usage is relatively trivial. On the contrary, the inversed transformation from bipolar to cartesian coordinates is complex enough, and its implementation can require a lot of computational resources. This drawback can be avoided by using interval methods, which allow us to define transformation equations using piecewise linear functions. In this case, one can consider the observability equations in the simplest linear-like form, which can be easily used to solve both direct and inverse transformation problems. We show the use of our approach by considering a well-known Mackey-Glass system and transforming it into bipolar coordinates by using exact and interval solutions of transformation equations. The performed research shows the similarity of the obtained results, which proves the correctness of the used approach and methods. © 2025 Copyright for this paper by its authors.
Igor Sikorsky Kyiv Polytechnic Institute, 37 Polytechnichna Str, Kyiv, 03056, Ukraine; Taras Shevchenko National University of Kyiv, 60 Volodymirska Str, Kyiv, 01033, Ukraine; State University of Malang, Str. Jl. Cakrawala, Malang, 65145, Indonesia
