Lyapunov Spectrum Characterization of Irregular Oscillations in Climate-Coupled Dynamic Systems

Abstract

Climate-coupled dynamic systems produce irregular oscillations when fast atmospheric variability interacts nonlinearly with slower oceanic and feedback-driven processes. Such irregularity is important because waveform appearance alone does not show whether the system remains weakly predictable or has entered an unstable regime. Nonlinear climate studies increasingly use Lyapunov-based methods to diagnose instability and predictability loss, but the full Lyapunov spectrum is still used less often than leading-error indicators or event-specific measures. This creates a gap because the spectrum can distinguish bounded modulation, weak irregularity, and chaotic variability more clearly than ordinary time-series analysis. This article develops a reduced slow-fast climate-coupled nonlinear model and characterizes its oscillatory regimes through Lyapunov-spectrum analysis and trajectory-based interpretation. The results show that increasing coupling strength and slow-memory feedback move the system from regular oscillation through a weakly irregular regime and then into chaotic variability marked by a positive leading Lyapunov exponent. Overall, the study shows that irregular climate oscillations are best understood as spectral transitions in stability structure.