MATLAB Implementation of Fractional-Order Chaotic Systems

Fractional-order chaotic systems are nonlinear dynamic systems based on fractional calculus, exhibiting more complex behavior and richer chaotic characteristics compared to integer-order systems. This article introduces how to implement fractional-order chaotic systems in MATLAB and plot their chaotic attractor phase diagrams. The core of fractional-order chaotic systems lies in the numerical solution of fractional differential equations. Common methods include Grünwald-Letnikov discretization, and approximation algorithms under Caputo or Riemann-Liouville definitions. In MATLAB, implementations can utilize the Fractional-order Modeling and Control (FOMCON) toolbox or custom-written solvers. Fractional Calculus Modeling: First, define the fractional differential equation form of the system, such as classical fractional-order Chen or Lorenz systems. These equations typically contain fractional derivative terms that require discretization methods to transform them into recursive forms. In MATLAB code, this involves defining system parameters, initial conditions, and fractional orders using variables like `alpha = 0.9` for the derivative order. Numerical Solution Methods: When using the Grünwald-Letnikov definition, the short memory principle can reduce computational load. MATLAB implementation involves constructing difference recurrence formulas through loop iterations or matrix operations for efficient computation of state variable time series. For example, a typical implementation would use a `for` loop to update system states with a truncated sum of previous states weighted by binomial coefficients, achieving O(n) computational complexity. Chaotic Attractor Phase Diagram Plotting: After computing time series of system state variables (e.g., x, y, z), phase diagrams of attractors can be plotted using 3D plotting functions like `plot3` or `scatter3`. Code enhancements include adjusting viewing angles with `view(3)` and applying colormaps with `colormap(jet)` to visualize chaotic trajectories' self-similar structures. Adding grid lines with `grid on` and axis labels with `xlabel('X')` improves readability. Extension: Fractional-order chaotic systems have wide applications in secure communication and image encryption. Variations in fractional orders affect Lyapunov exponents and bifurcation behaviors, warranting further investigation through parameter sweep simulations and stability analysis algorithms.