Numerical Methods for mapping band-type resonance in insect flight

Abstract

Insect flight is a highly complex and energy-intensive process. Flapping-wing insects employ a unique muscle contraction mechanism that enables high-frequency wing beats, with metabolic rates reaching several times those at rest. Their remarkable endurance during flight highlights the importance of understanding the energy optimization involved. This thesis focuses on developing numerical methods to map band-type resonance, which serves as a benchmark for assessing whether a system achieves an energy-optimal state. We describe the mapping of band-type resonance as an optimization problem and propose two primary numerical methods: particle swarm optimization and numerical continuation. We evaluate the accuracy of the numerical solutions via the solution work loops and power waveforms and compare them with analytical approximations to the space of band-type resonant states. Our findings reveal that while the standalone particle swarm method can provide a relatively complete set of estimated solutions, the solution space lacks continuity. The numerical continuation method sacrifices some completeness in finding solution sets to ensure better continuity in the corresponding domain of the output solution set. After comparing these methods' performance in identifying potential solutions for simple cases, we improve them and propose a compound numerical method for solving more complex problems, such as higher harmonic and nonlinear oscillators. Notably, this compound algorithm performs well not only on simple linear cases with known analytical solutions but also on complex problems lacking analytical solutions, offering a valuable numerical tool for estimating the mapping zone of band-type resonance when analytical methods are not feasible. Comparing the results of mapping zones of band-type resonance with wingbeat frequency modulation behaviour observed in actual insect species suggests that such behaviour may be consistent with sustained resonant energy savings by exploiting band-type resonance. This report is written in English.