Bubbling transition as a mechanism of destruction of synchronous oscillations of identical microbubble contrast agents

The paper proposes and numerically supports a bubbling-transition route in a symmetric two-bubble model: a synchronous chaotic attractor in the synchronization manifold Fix(S) with λtr<0 (e.g., at d/R10=12.01), emergence of asynchronous bursts via subcritical pitchforks involving (3,1) orbits in Fix(S) and (2,2) orbits from an off-manifold hyperchaotic saddle set SH(2,2), subsequent crossing of the largest transversal Lyapunov exponent λtr through zero leading to weakly asynchronous (hyperchaotic) dynamics, and finally a boundary crisis that destroys the attractor before a full blowout occurs; robustness under small symmetry breaking (ε≈1±0.01) is also shown. All of these elements appear explicitly in the paper’s phenomenological scenario and tables/figures (e.g., Fix(S) and λtr use, SH(2,2) hyperchaos, subcritical pitchforks, crisis before blowout, and ε-robustness) and match the candidate solution’s mechanism and sequence closely . Minor differences are present: the paper does not rigorously verify the subcritical pitchfork via multiplier computations nor the crisis scaling, while the model outlines how to certify these with validated numerics. Nonetheless, the conceptual structure and the route’s stages agree, so both are correct and substantially the same in proof strategy (phenomenological/equivariant bubbling picture corroborated by λtr and Lyapunov data).