Chaotic-Saddle-Organized Hidden Bursting Oscillations in a 4D Slow–Fast System with No Equilibria

Bursting oscillations, characterized by alternating quiescent and spiking states, are a fundamental class of nonlinear dynamics that play a crucial role in diverse scientific fields. Therefore, understanding the mechanisms that generate bursting is of both theoretical and practical importance. Bursting is typically studied as a class of self-excited or forced oscillations in classical slow–fast dynamics, with analysis relying on stable invariant sets and their bifurcations. In contrast to these cases, we extend the analysis to hidden bursting in a 4D slow–fast system with no equilibria using classical slow–fast decomposition. Specifically, we show how chaotic saddle dynamics in the fast subsystem organize the bursting mechanism. Bifurcation and basin analyses identify two symmetric chaotic saddle ranges between interior crises and the folds of limit cycles. When quiescent-to-spiking transitions fall within these ranges, chaotic saddles render the full system highly sensitive to parameters, acting as the organizing center for hidden bursting. By tuning the slow adaptation strength, we uncover three transition-path modes: lift-captured double-reversal (LCDR), lift-escape single-reversal (LESR), and direct-captured no-reversal (DCNR). These modes and their combinations classify the observed bursting patterns. These findings show that non-attracting invariant sets can affect bursting dynamics, offering a new perspective on bursting dynamics.