Nonlinear stability of oscillating microbubbles: Lyapunov analysis and Gaussian process regression

Encapsulated microbubbles subjected to ultrasound exhibit rich nonlinear dynamics ranging from stable periodic oscillations to chaotic behavior, which can strongly influence their performance in biomedical applications. In this study, a comprehensive stability analysis of polymer-shelled microbubbles is performed using Lyapunov exponent spectra evaluated over a four-dimensional parameter space defined by acoustic pressure (0–10 MPa), driving frequency (0–10 MHz), initial bubble radius (0.5–5.0 μm), and environmental temperature (290–390 K). The results reveal well-defined dynamical transitions across the investigated domain. Acoustic pressure acts as the primary bifurcation parameter, with period-doubling routes to chaos emerging around 3450–3500 kPa. The excitation frequency plays a resonance-like role, with a critical bifurcation frequency of 1.00 MHz separating ordered and chaotic regimes. A critical radius threshold at approximately 1.9 μm is identified, where the maximum Lyapunov exponent changes sign, marking the onset of instability. Temperature further modulates the system response, with chaotic oscillations predominantly observed in the range of 310–348 K and more stable dynamics appearing between 350 and 400 K, including abrupt transitions near 345–350 K. To extend the analysis beyond discrete parameter combinations, Gaussian process regression (GPR) with an automatic relevance determination kernel is employed to construct a probabilistic surrogate model that maps the input parameters to the maximum Lyapunov exponent. The GPR model reconstructs the nonlinear stability landscape with high predictive accuracy (R2=0.9989) while providing uncertainty-aware predictions. Overall, the proposed Lyapunov-GPR framework offers an efficient methodology for mapping stability boundaries in high-dimensional parameter spaces and provides model-based insight into the coupled mechanisms governing nonlinear microbubble dynamics.