Asian Research Journal of Mathematics

This study identifies critical parameter thresholds associated with climate regime shifts and extreme events in the Kenyan Arid and Semi-Arid Lands (ASALs) using a calibrated stochastic coupled Lorenz system. The analysis combines bifurcation theory, stability analysis, numerical simulation, extreme-value statistics, and critical-slowing-down indicators to investigate transitions in the model dynamics. For the classical Lorenz parameters (\(\sigma\), \(\beta\)) = (10, 8/3), the non-zero equilibria lose stability through a subcritical Hopf bifurcation at the analytically derived critical value pH = 470/19 ≈ 24.74. The homoclinic bifurcation occurs at approximately phom ≈ 13.926, marking the emergence of a non-attracting chaotic invariant set and transient chaotic dynamics, while an attracting chaotic set emerges near pA ≈ 24.06. The calibrated ASAL parameter values are then examined relative to these dynamical thresholds. The stochastic extension is further used to investigate noise-induced transitions, synchronisation extreme-value behaviour, and early-warning indicators. The results demonstrate how critical thresholds in the reduced-order model can provide a mathematical framework for analysing nonlinear climate variability and extreme-event behaviour in Kenyan ASALs.