MATLAB Implementation of Chen's Fractional-Order Chaotic System

Chen's fractional-order chaotic system is an extension of the classical Chen system within the framework of fractional calculus, exhibiting richer dynamic behaviors. Implementing this system requires combining numerical solutions for fractional differential equations with chaotic system modeling techniques.

The core implementation approach consists of three key components: Fractional Calculus Processing Utilizing Caputo definition or Grünwald-Letnikov approximation methods, converting infinite series summation to finite-term computation through short-memory principle. Common implementation methods include constructing fractional differential operators or calling fractional integration functions from toolboxes like FOMCON. In MATLAB code, this typically involves implementing recursive algorithms for fractional derivatives calculation.

System Equation Discretization After rewriting the original Chen system's integer-order differential equations into fractional-order form, discretization should be performed using predictor-corrector methods or Adams-Bashforth-Moulton algorithms. Special attention must be paid to handling coupled nonlinear terms separately to prevent numerical divergence. Code implementation requires careful iteration design for coupled differential equations.

Parameter Sensitivity Analysis Typical parameter settings are (α,β,γ)=(35,3,28), but fractional orders significantly influence system states. Bifurcation diagrams or Lyapunov exponent spectra analysis is recommended to observe period-doubling routes to chaos under different fractional orders (e.g., 0.8-1.0). MATLAB implementation involves creating parameter sweep functions and visualization code for dynamical analysis.

Critical implementation details: Step size selection requires balancing precision and computational cost, typically 0.005-0.01 When using ODE solvers, adaptive step size functionality should be disabled Phase diagrams should employ 3D projection to observe attractor structures - use MATLAB's plot3 or scatter3 functions

This model has special application value in secure communication and image encryption fields, as its fractional-order characteristics provide higher-dimensional key spaces. Extension directions could include adding time-delay terms or random perturbations to construct more complex variant systems through additional MATLAB function modules.