Magnetic potential tuning in a typical piezoelectric energy harvesting system: bistable versus tristable capability under initial-state sensitivity

• A magnetic coordination mechanism enables switching between bistable and tristable configurations with precise well-depth control, facilitating performance optimization. • At equivalent potential barriers, the bistable system is superior to the tristable one under low excitation due to lower inter-well thresholds, unless near the trivial equilibrium, emphasizing initial-state dependence. • The bistable harvester dominates at low and extreme excitation amplitudes, whereas the tristable configuration excels only within a moderate high-amplitude range. • Globally attractive chaotic responses in the bistable system under extreme amplitudes enhance energy harvesting efficiency, demonstrating that chaos can be beneficial rather than detrimental in multi-stable energy harvesters. This study presents a theoretical framework for analyzing a magnetically coupled multi-stable piezoelectric energy harvesting (PEH) system, comparing bistable and tristable configurations under initial perturbations. Through static bifurcation analysis, extended averaging, and numerical simulations, we reveal how nonlinear dynamics such as saddle-node bifurcations, chaos, and fractal basins of attraction (BAs) collectively influence efficiency. A magnetic coordination mechanism enables controlled switching between bistable and tristable potentials by adjusting the magnet spacing and vertical separation, challenging the conventional superiority of tristable designs. Specifically, the bistable system outperforms at low excitation amplitudes because of lower interwell thresholds, whereas the tristable setup excels only near trivial equilibria or within a moderate high-amplitude range. In an extreme-amplitude environment, the global attractive chaotic response that appears in the bistable system enhances the efficacy of such a system, demonstrating that chaos can benefit energy collection. This work establishes an excitation-amplitude-dependent selection principle for PEH designs and highlights the critical yet overlooked role of initial conditions. The observed fragile BAs and hidden/rare attractors provide new perspectives for enhancing system reliability. A comparative study reveals that a bistable piezoelectric energy harvester (PEH) outperforms its tristable counterpart at low and high excitation amplitudes. This work establishes an amplitude dependent selection principle, emphasizing the critical role of initial conditions and potential well tuning.