Numerical solutions, dynamical analysis and circuit implementation of fractional-order chaotic system with time-delay

In this research work, the time-delay problems existing in signal transmission and mechanical structures are considered, based on the fractional-order Lü chaotic system, a fractional - order Lü time-delay chaotic system is constructed. The frequency-domain method, Grunwald-Liouvile method, and Adomian decomposition method among the numerical solution methods for fractional-order systems are used to numerically solve the fractional-order time-delay system, and the numerical solutions of the fractional-order time-delay chaotic system are obtained and the corresponding phase diagrams are drawn. Through numerical solution, the bifurcation diagram, complexity, and attractor phase diagrams of the system are drawn, and the influence of parameters on the system is verified. The simulation results show that the fractional-order time-delay system has rich dynamic characteristics and differences among the three solution algorithms. Finally, the fractional-order time-delay chaotic circuit is designed and implemented based on the integer-order circuit using Laplace transform, and the 0.01- 0.99 order fractional-order Laplace transform expansion formula is given, which provides a theoretical basis for the study of more accurate fractional-order systems and circuit design.