Generation of Typical Chaotic Time Series

This document presents the implementation of several benchmark chaotic time series generation algorithms, including Rossler, Chen, Logistical, and Lorenz sequences. These sequences exhibit fascinating properties such as sensitive dependence on initial conditions, fractal geometric structures, and complex dynamical behaviors. The implementation typically involves solving differential equations using numerical methods like Runge-Kutta integration (e.g., ode45 in MATLAB) for continuous systems or iterative mapping functions for discrete systems. For instance, the Lorenz system is governed by three coupled differential equations dx/dt = σ(y-x), dy/dt = x(ρ-z)-y, dz/dt = xy-βz, while the Logistical map uses the recursive equation xₙ₊₁ = rxₙ(1-xₙ). Chaotic time series find broad applications across cryptography, secure communications, financial modeling, and meteorological forecasting. Therefore, thorough investigation of chaotic time series carries significant theoretical importance and practical value in engineering applications.