International Journal of Science, Engineering and TechnologyAuthors: Audu Abdulmalik, Onubedo, Nura A.Y Musa, Youngkumaphew Iliyasu
Abstract: This study presents a numerical investigation of nonlinear motion, recurrence, and predictability in the classical Lorenz system using a compact three-dimensional convection model. The analysis combines equilibrium considerations, local stability concepts, temporal integration, phase-space visualization, recurrence mapping, and sensitivity to nearby initial conditions. Numerical simulations show that the state variables remain bounded while developing irregular oscillations after an initial transient, with repeated reversals reflecting transitions between distinct regions of state space. The reconstructed three-dimensional trajectory exhibits the characteristic two-lobed attractor, while its planar projections reveal repeated circulation around the two non-zero equilibrium regions without convergence to a stationary state or stable periodic orbit. Successive local maxima form an organized nonlinear return relation, demonstrating that the apparently irregular oscillations retain an underlying deterministic structure. Sensitivity analysis further shows that initially close trajectories separate substantially with time, limiting reliable long-term prediction despite deterministic governing equations. The combined temporal, geometric, recurrence, and sensitivity analyses therefore provide a consistent computational characterization of bounded aperiodic motion in the Lorenz system. The results reinforce the importance of low-dimensional nonlinear models for understanding deterministic unpredictability and recurrent chaotic dynamics in mathematical physics.
