Perturbed Lorenz Attractors: Effects of Time-Dependent Disturbances on Chaotic Dynamics

This paper explores the time-dependent external disturbances and its effect on the Lorenz attractor, a fundamental system in chaos theory. The Lorenz system is renowned for its sensitivity to initial conditions and its chaotic behaviour, governed by three nonlinear differential equations. This work introduces a novel approach by modifying the Rayleigh number (ρ) in the y-equation, a parameter of both physical and theoretical significance. The study implements four different types of perturbation, sinusoidal, stochastic, impulsive, and quasiperiodic, to model realistic environmental oscillations and noise. These perturbations are embedded into ρ(t), creating a non-autonomous system model, diverging from conventional approaches in the literature. The methodology employs a suite of analytical techniques, including 3D phase portraits, time series analysis, Poincar´e sections, Lyapunov exponent estimation and bifurcation diagrams, to assess the effects of these perturbations. The results demonstrate that each perturbation method induces distinct alterations to the attractor’s geometric structure, stability characteristics and chaotic intensity. Sinusoidal and quasiperiodic perturbations produce periodic structural changes; stochastic perturbations increase trajectory diffusion and sensitivity; and impulsive perturbations induce abrupt localised destabilisation. This study reveals that both the form and strength of perturbations significantly influence the system’s evolution, as highlighted through Lyapunov exponents and bifurcation structures. Overall, the research extends current understanding of chaotic system responses to external influences and underscores the potential of ρ-modulated perturbations for advancing chaos control and synchronisation strategies in systems exposed to fluctuating environments.