Dynamic response and stochastic resonance of a tri-stable system

Stochastic resonance (SR) describes a nonlinear phenomenon in nature, of which the essential ingredients are a nonlinear system, a weak signal, and a source of noise. Using the nonlinear system, the signal-to-noise ratio (SNR) of the output signal of the system will peak at a certain value of noise intensity under a synergistic action of input signal and noise. Besides the traditional Langevin equation, the new SR models such as monostable oscillators, chaotic systems, time-delay systems and bistable Duffing systems, can also produce SR phenomena. In this paper, a normalized symmetrical tri-stable potential function is constructed by using equilibrium parameters p and q, and a tri-stable system model simultaneously driven by weak signal and noise is further proposed. The tri-stable system model can be understood through a cantilever beam structure with three magnets, and deduced from the Brownian motion equation. We study in-depth and summarize the influences of parameters p and q on the potential barrier heights Delta U-1, Delta U-2 and their difference value. By analyzing the steady-state solution of the tri-stable system under invariable input, the concept of system steady-state solution curve (SSS curve) is proposed, and is used to further study the system dynamic response under low-frequency harmonic signal input. In these situations, the system response can be obtained by combining the steady-state solutions of the system following time t under a group of tempolabile inputs. Moreover, with the noise injection, the tri-stable system can realize SR under appropriate parameter condition, which can be demonstrated by the output amplitude curve and also the output SNR curve of the system against noise intensity. The mechanism of noise-induced SR of tri-stable system can be analyzed from the perspective of SSS curve. Finally, we further study the influence of tri-stable SR against system parameters. The value of damping ratio k affects the value of damping force acting on the Brownian particle, thus the tri-stable system needs noise with larger intensity to produce SR under a larger k. The values of equilibrium parameters p and q both affect the shape of the SSS curve, a larger p or a smaller q may result in larger-intensity noise for the system to produce SR.