Abstract
The paper deals with the design of methodological backgrounds for the study of time-delayed piecewise-linear dynamical systems. These backgrounds allow us to define dynamical system responses to the internal and external disturbances in an analytical way. We define these responses as the piecewise functions of relative system operation time which is used by us to simplify the system model. According to the well-known method of solution of time-delayed differential equations, we split all operation time into equal slices, which are specified by the value of time delay. Due to operating with piecewise-linear right-hand expression in the differential equation, we split each slice into several stages if the output variable in the previous slice reaches the fracture point. Contrary to known methods of analytical solution of time-delayed differential equations the proposed one makes it possible to determine system motions as a function of time delay value and times where piecewise function is fractured. We formalize our approach and propose the algorithm to determine the considered system motions. We use this algorithm to study the Mackey-Glass equation with the constant parameters and piecewise-linear function in the right-hand expression. A comparison of the analytical and numerical solutions of this equation shows that the error does not exceed a step of numerical integration and gives us the possibility to claim the correctness of given formulas. Analysis of our formulas shows that the equation of the modified Mackey-Glass system can be solved analytically in advance and thus its motion can be predicted. This fact requires to use of chaotic systems with piecewise nonlinearity with big caution.
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Voliansky, R., Volianska, N., Sinkevych, O., Serhiienko, S., Kuznetsov, V. (2023). Analytical Solution of Modified Mackey-Glass Equation. In: Arsenyeva, O., Romanova, T., Sukhonos, M., Tsegelnyk, Y. (eds) Smart Technologies in Urban Engineering. STUE 2022. Lecture Notes in Networks and Systems, vol 536. Springer, Cham. https://doi.org/10.1007/978-3-031-20141-7_14
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