Keywords: ENSO dynamics, ocean–atmosphere interaction, Hopf bifurcation, climate variability, tropical cyclones Introduction The El Niño–Southern Oscillation (ENSO) is the dominant mode of interannual climate variability in the Earth system and arises from coupled interactions between the tropical Pacific Ocean and atmosphere. It is characterized by alternating warm (El Niño) and cold (La Niña) phases, associated with anomalous sea surface temperatures, thermocline displacements, and changes in large-scale atmospheric circulation. These coupled anomalies reorganize tropical convection and modify global atmospheric teleconnections, making ENSO a primary driver of year-to-year variability in weather and climate extremes worldwide. Because of its far-reaching influence, ENSO is not only a fundamental scientific problem in climate dynamics but also a major factor in disaster risk, particularly for droughts, floods, heatwaves, and tropical cyclone activity. One of the most significant impacts of ENSO is its strong modulation of tropical cyclone activity, especially in the Atlantic basin. During El Niño events, enhanced convection over the central and eastern Pacific strengthens upper-level westerly winds over the Caribbean and tropical Atlantic, increasing vertical wind shear and suppressing hurricane formation. This dynamical environment inhibits cyclone development and reduces the intensity and frequency of Atlantic hurricanes. In contrast, La Niña conditions weaken vertical wind shear and promote warmer Atlantic sea surface temperatures, creating favorable conditions for enhanced hurricane activity. As a result, ENSO phases are strongly correlated with seasonal hurricane variability and are widely used in operational seasonal hurricane forecasts. This relationship highlights the importance of understanding ENSO dynamics not only for climate theory but also for practical forecasting and risk management. Beyond the Atlantic basin, ENSO significantly influences tropical cyclone behavior across the Pacific and Indian Oceans. El Niño events tend to shift cyclone activity eastward into the central and eastern Pacific, increasing storm frequency in regions that are typically less active, while simultaneously suppressing cyclone formation in the western Pacific. This redistribution is linked to large-scale changes in the Walker circulation and shifts in the location of maximum sea surface temperature anomalies. Consequently, ENSO acts as a global organizer of tropical cyclone climatology, altering storm genesis regions, tracks, and intensities across multiple ocean basins. These teleconnections underscore the interconnected nature of the climate system, where perturbations in one region can have far-reaching impacts on weather extremes worldwide. ENSO also plays a critical role in shaping global precipitation patterns and hydrological extremes. El Niño events are often associated with increased rainfall and flooding in parts of western South America, while simultaneously inducing drought conditions in Australia, Indonesia, and parts of Southeast Asia. La Niña events typically produce opposite anomalies, reinforcing the role of ENSO as a driver of alternating wet and dry climate extremes. These precipitation anomalies are mediated through atmospheric wave propagation, changes in convection, and shifts in jet stream positioning, all of which are directly linked to tropical Pacific sea surface temperature anomalies. The resulting impacts on agriculture, water resources, and infrastructure make ENSO one of the most economically and socially significant climate phenomena. From a dynamical systems perspective, ENSO is often interpreted as a self-sustained oscillatory phenomenon arising from nonlinear ocean–atmosphere feedbacks. Key processes include positive feedback between sea surface temperature and atmospheric circulation, delayed negative feedback through thermocline adjustment, and nonlinear saturation mechanisms that limit amplitude growth. These interactions can generate oscillatory behavior even in the absence of external periodic forcing. In simplified conceptual models, these mechanisms are often represented using low-order dynamical systems that capture the essential structure of ENSO variability. Such models are particularly valuable because they allow analytical investigation of stability, oscillation onset, and sensitivity to parameter changes. In this context, the transition from stable climate equilibrium to oscillatory ENSO behavior can be understood as a qualitative change in system dynamics driven by variations in ocean–atmosphere coupling strength. Near critical thresholds, small perturbations in system parameters can lead to large-amplitude oscillations, indicating heightened climate sensitivity. This behavior is consistent with the presence of nonlinear instabilities in the coupled system, which can give rise to sustained periodic variability in the form of El