Computation of Lyapunov Exponents and Implementation of the Lorenz Model

Computing Lyapunov exponents is a crucial method for studying chaotic behavior in dynamical systems, while the Lorenz model serves as one of the classical chaotic systems. This article explains how to implement the Lorenz model simulation using MATLAB and compute its Lyapunov exponents to analyze the system's chaotic dynamics.

The Lorenz model, introduced by meteorologist Edward Lorenz, is a simplified convection model consisting of three nonlinear differential equations involving parameters σ (Prandtl number), ρ (Rayleigh number), and β (geometric parameter). Under specific parameters, the model exhibits chaotic characteristics with extreme sensitivity to initial conditions.

MATLAB implementation typically involves these key steps: First, numerical integration using solvers like ode45 to solve the Lorenz differential equations and obtain the system's time evolution trajectory. For Lyapunov exponent calculation, tracking the exponential divergence rate of neighboring trajectories requires:

System Solution: Define the differential function for Lorenz equations and numerically solve using ode45 with syntax: [t,y] = ode45(@lorenz_eq, tspan, y0). This generates phase space trajectories by storing solutions in variable y containing [x,y,z] coordinates over time vector t.

Tangent Space Evolution: While computing the main trajectory, simultaneously track tangent vectors of the linearized system using variational equations. This quantifies local divergence behavior through Jacobian matrix multiplication at each integration step.

Exponent Extraction: Estimate Lyapunov exponents by long-term averaging of logarithmic growth rates of tangent vector magnitudes. Implement orthogonalization (via QR decomposition) and normalization routines periodically to prevent numerical error accumulation. A positive maximum Lyapunov exponent indicates chaotic behavior.

Lyapunov exponent computation requires extended numerical integration for convergence assurance. In MATLAB, use iterative loops to update tangent vectors with regular orthonormalization (qrupdate function) and normalization (vecnorm operations). Final analysis of exponent magnitudes and signs determines system stability and chaos degree.

As a classic chaos study case, the Lorenz model's MATLAB implementation not only helps understand fundamental chaotic dynamics but also extends to other complex systems. Parameter adjustment (e.g., ρ variation) enables observation of system transitions from steady states through periodic oscillations to chaotic regimes, visually demonstrating rich nonlinear system behaviors through bifurcation diagrams.