Non-Perturbative Analysis of Primary Resonance and Chaos in Fractal Mems Photoacoustic Transducers

The present study proposes a cohesive analytical framework for the primary resonance and chaotic dynamics of fractal-structured MEMS photoacoustic transducers. The non-perturbative approach is extended to fractal MEMS systems, yielding closed-form equivalent linearisation that accurately captures amplitude-dependent behaviour with relative errors below 5%. The analysis demonstrates that fractal parameters significantly regulate dynamic stability. An increase in fractal dimension or a decrease in scale factor has been shown to enhance electrostatic softening, reduce effective nonlinearity, and lower the chaotic threshold through a period-doubling cascade. A fundamental trade-off is evident between sensitivity and stability. The fractal geometries that optimise photoacoustic performance are precisely those that compromise dynamic reliability. The establishment of a design criterion is imperative for ensuring stable operation within the hardening regime. The framework also offers pathways towards fractional-order damping and digital twin integration, providing both theoretical foundations and practical guidelines for robust fractal MEMS design.