Transformation of the generalized chaotic system into the discrete-time complex domain

Abstract

The paper deals with the development of the backgrounds for the design and implementation of secured communications by using systems with chaotic dynamics. Such backgrounds allow us to perform the stable transformation of a nonlinear object into a simpler form and formulate the nonlinearities simplification optimization algorithm. This algorithm is based on the optimization problem's solution, which makes it possible to define polynomial order, approximation terms, and breakpoints. Usage of proposed algorithms is one of the ways to simplify known chaotic system models without neglecting their unique properties and features. We prove our approach by considering simplifying the Mackey-Glass system and transforming it into a discrete-time complex domain. This example shows that the transformed system produces chaotic oscillations with twice-increased highest Lyapunov exponent. This fact can be considered as improving the unpredictability of the transformed system, and thus it makes background to make highly non-intercepted and undecoded transmission channels.

Keywords

• chaotic dynamic
• secured communications
• complex and hypercomplex numbers
• lyapunov exponent
• tustin transformation