Browsing by Author "Volianska, Nina"

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The Generalized Chaotic System in the Hyper-Complex Form and Its Transformations
(Springer, Cham, 2022) Voliansky, Roman; Volianska, Nina; Kuznetsov, Vitaliy V.; Tryputen, Mykola; Kuznetsova, Alisa; Tryputen, Maksym
ENG: The paper deals with the development of the mathematical backgrounds to design the novel chaotic systems by transforming existent ones. These backgrounds are based on using well-known shift, rotation, and scale transformations and we offer using hyper-complex numbers to simplify these transformations and represent the transformed chaotic system by using the one 1st order ordinary differential equation. In such form all well known, newly discovered and unknown chaotic systems have the similar mathematical models that are differs only by used nonlinear function of hyper complex variable in the right hand expression. That is why the consideration chaotic system dynamic in the hyper-complex domain allows us to simplify initial system definition as well without applying any transformations. This fact simplifies mathematical definition of chaotic systems and their modeling and simulation. The right-hand expression of the transformed equation in this case are defined as the combination of transformation hyper-complex numbers and source system nonlinearity which is given in the hyper-complex domain. We offer to use variable transformation factors to improve the performance of the considered chaotic system. Since the above-mentioned variable factors can be produced by other chaotic systems, we suggest designing the novel chaotic system by combining existed ones with the linear transformations. As an example, we consider the transformation of the well-known Lorenz system and show the differences between the source system and target one.
Lyapunov Function in the Hyper-Complex Phase Space
(Springer, Singapore, 2022) Voliansky, Roman; Volianska, Nina; Kuznetsov, Valeriy; Sadovoi, Aleksandr; Kuznetsov, Vitaliy V.; Kuznetsova, Yevheniia; Ostapchuk, Oleksandr
ENG: The paper deals with the development of background for defining Lyapunov functions for a wide range of linear dynamical objects. This background is based on assuming that the Lyapunov function is redundant energy in the considered object, and this energy is dissipated only during controlled motion. We assume the full derivative of the Lyapunov function for an autonomous motion of the control objects equals zero, and we use its summands to define linear algebraic equations. The solution of these equations allows us to find unknown terms of the Lyapunov function. The use of these terms, while the Lyapunov equation is being written down, shows that the left-hand expression in the Lyapunov equation is equal to the zero matrix. Thus, we avoid subjective assuming of quadratic form terms in the right-hand of the Lyapunov equation. We extend the proposed approach to the class dynamical system with uncertainty. This extension is performed by using interval methods, which allow defining object motions for minimal and maximal values of parameters. We show that for the control object, which parameters are not exactly known, one should consider two equations of object motions, which correspond to its trajectories on the boundaries of the intervals. Lyapunov functions are defined for these boundary trajectories. Since such an approach increases the number of the considered equations, we offer to decrease them by using hyper-complex numbers while object equations are written down.
Numerical Models of Novel Discrete-Time Chaotic Systems
(CEUR-WS Team, Aachen, Germany, 2025) Voliansky, Roman; Volianska, Nina; Yakovenko, Vitaly; Tkalenko, Oleksandr; Kuznetsov, Vitalii V.
ENG: The paper proposes a theoretical approach to designing novel dynamical systems based on known discrete maps. We show some algebraic transformations based on the derivative operators' discrete-time approximation. These operators' approximations are considered as defined by some nonlinear algebraic combinations of finite and infinite element numbers of maps. Such an approach allows us to use various differential operators, including fractional-order and complex fractional-order ones. As a result, matrix nonlinear algebraic equations are defined for the considered discrete map. The order of these equations depends on the number and order of the derivatives used to define the system dynamics. Such a formalized approach allows us to easily define the system's finite-difference equations and solve them using known numerical methods. The obtained solutions make it possible to determine the systems' motions, and we offer to add some external signals to increase the range where these motions are defined. One can consider such signals from a control theory viewpoint as control one and use known control approaches to define system equations in various state spaces. Using one of these approaches allows us to significantly increase the number of system outputs by using observability equations for defined system state variables. We show the use of our approach by designing and studying several chaotic systems based on a well-known logistic equation. Our studies prove the possibility of constructing novel systems that produce previously unknown chaotic signals significantly different from known ones. The given system equations not only allow the generation of new signals for use in various applications but also give us the possibility to improve system performance.