Chaos meets stochasticity: A variance-based method for Lyapunov exponent estimation

The Largest Lyapunov Exponent (LLE) is a fundamental diagnostic of chaotic behavior in nonlinear dynamical systems, quantifying the exponential divergence of nearby trajectories. Classical computational approaches, such as Wolf’s algorithm, track individual particle trajectories to estimate the LLE, but these techniques face challenges related to noise sensitivity, computational efficiency, and scalability to high-dimensional systems. This work introduces a novel variance-based methodology for computing the LLE using intrusive polynomial chaos (IPC), an uncertainty quantification technique that evolves the probability distribution of initial conditions under deterministic dynamics rather than tracking discrete trajectories. The key innovation is extracting the LLE from the exponential growth rate of ensemble variance, which connects deterministic chaos with probabilistic descriptions. Validation against the classical trajectory-based algorithm is performed on three benchmark chaotic systems: the three-dimensional Lorenz and Rössler attractors, and a six-dimensional system from Al-Azzawi and Al-Obeidi, demonstrating that the IPC approach achieves comparable accuracy and convergence rates while offering the distinct advantage of directly computing the full statistical structure of ensemble dynamics. Comparison of convergence histories, probability density functions of instantaneous Lyapunov exponents, and statistical error measures confirms excellent agreement between the proposed IPC-based methodology and established algorithms. The results indicate that variance-based LLE estimation via polynomial chaos is a robust and viable alternative to trajectory-based methods.