Numerical Models of Novel Discrete-Time Chaotic Systems

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.