Computing Time Series of Lorenz Chaotic System Using Fourth-Order Runge-Kutta Method

The fourth-order Runge-Kutta method (RK4) is a high-precision numerical approach for solving systems of ordinary differential equations, particularly suitable for nonlinear differential equations like the Lorenz system. As a classical model in chaos theory, the Lorenz system's three coupled differential equations describe chaotic phenomena in atmospheric convection. The implementation process consists of three main components: Parameter initialization requires setting the three typical parameters of the Lorenz equations (σ, ρ, β) and determining the initial state vector and time step size. Classical parameter combinations generate the famous butterfly-shaped strange attractor. The core RK4 algorithm advances the state through weighted averaging of four slopes: First, compute the slope k1 at the current point Use k1 for half-step advancement to obtain an intermediate point, then compute k2 Use k2 for another half-step advancement to calculate k3 Use k3 for full-step advancement to determine k4 Finally, combine the four slopes using 1:2:2:1 weighting When storing results, special attention is needed since chaotic systems exhibit extreme sensitivity to initial conditions. Even with double-precision floating-point numbers, cumulative errors must be controlled. The typical approach involves saving three-dimensional state coordinates at each time step, which can later be used to plot 3D phase space trajectories. For numerical solving, step size selection requires balancing precision and computational load, typically ranging between 0.01 and 0.001. Although the system is dissipative and total energy gradually decays during long-term simulations, the phase space trajectory always remains near the strange attractor.