From Order to Chaos: The Destabilizing Role of Time-Varying Mass in Fractional Pendulum Systems

This study investigates the nonlinear dynamics of a double pendulum system with exponentially decaying mass by integrating classical mechanics with fractional calculus to capture memory-dependent chaotic behavior. The primary objectives are: (1) to quantify how time-varying mass amplifies chaos relative to fixed-mass systems, and (2) to establish a fractional-order framework that incorporates non-local effects. Using the Caputo fractional derivative and dimensionless Euler–Lagrange equations, we derive governing equations incorporating mass decay μ(τ) = μ 0 e -γτ (γ = 0.1 s -1 ). Numerical simulations via fourth-order Runge–Kutta and Adams–Bashforth methods reveal that variable mass intensifies nonlinearity, increasing the maximal Lyapunov exponent from 0.1 s -1 (fixed mass) to 0.41 s -1 —a 310% rise. Key results include: angular displacements growing by 44.4% (α: 0.18 → 0.26 rad) and 106.7% (β: 0.15 → 0.31 rad); energy fluctuations reaching 10.41 J (vs. 9.82 J for fixed mass) due to asymmetric torque from mass loss; and Poincaré sections confirming chaotic attractors only in the variable-mass case. The fractional-order model reveals that memory effects synergistically amplify chaos, providing more accurate predictions for engineering systems such as adaptive robotics (e.g., arms with shifting payloads) and aerospace vehicles (e.g., fuel-depleting spacecraft), where our model predicts instability thresholds at critical ≈ −0.015 s -1 . This work bridges the gap between fractional calculus and real-world variable-mass systems, offering a robust tool for instability prediction and control design.