Pull-down plateau of a Toda-like fractal-fractional oscillator and its application in grass carp’s roes’ agglomeration

Grass carp’s roes should be agglomerated together for maximizing their survival rate against various predators. Any vibration induced by any environmental perturbation should be attenuated immediately. A Toda-like fractal-fractional oscillator is established, which shows a low-frequency property for most cases; however, the grass carp has evolved a very ability to attenuate the perturbated vibration by sticky adhesion. The pull-down stability of the roes’ vibration is discovered through the results of phase diagrams. The mathematical analysis reveals that there is a pull-down plateau for the attenuating process, the plateau’s height and width are discussed graphically, and the main factors affecting the plateau’s properties are elucidated. The paper offers a totally new window for biomechanics, especially for biomimicking design of chatter vibration systems inspired by the agglomerated roes.