The effect of depth and location of a triple-well potential on vibrational resonance is investigated in a quintic oscillator driven by a low-frequency force and a high-frequency force. The values of low-frequency $\omega$ and amplitude $g$ of the high-frequency force at which vibrational resonance occurs are derived both numerically and theoretically. It is found that: as $\omega$ varies, at most one resonance takes place and the response amplitude at resonance depends on the depth and the location of the potential wells. When $g$ is altered, the depth and location of wells can control the number of resonances, resulting in two, three and four resonances. The system parameters can be adjusted by controlling the depth and position of the wells to achieve optimum vibrational resonance. Furthermore, the changes induced by these two quantities in the tristable system are found to be richer than those induced in bistable systems.

This paper is focussed on investigating the effect of linear time delay on vibrational resonance of a harmonically trapped potential system driven by a biharmonic external force with two wildly different frequencies$\omega$ and $\Omega$ with $\omega \ll \Omega$. Firstly, the approximate analytical expression of the response amplitude $\mathcal{Q}$ at the low frequency $\omega$ is obtained by means of the direct separation of the slow and fast motions, and then we verified the numerical simulation by using the fourth-order Runge–Kutta method and found that it is in good agreement with the theoretical analysis. Next, the influence of the time-delay parameters on the vibrational resonance are discussed. There are some meaningful conclusions. If $\tau$ is a controllable parameter, the response amplitude $\mathcal{Q}$ not only exhibits periodicity but also can be amplified via the cooperation of $F$ and $\tau$ . If the time-delay intensity parameter $r$ is a controllable parameter, the response amplitude $\mathcal{Q}$ is found to be much larger than that in the absence of time delay. Moreover, adjusting $r$ can result in a better response than adjusting $\tau$ . This undoubtedly gives us a superior way to amplify the weak low-frequency signal.

Vibrational resonance is studied in a fractional Mathieu–Duffing oscillator with two types of time delays: fixed and distributed delays. The theoretical expression of the response amplitude is obtained by utilising the methodof direct partition of slow and fast motions. Relative errors between the theoretical prediction and the numerical simulation are introduced to verify the validity of analytical approaches. The relative error of the displacement andthe relative error of the response amplitude are calculated. Small relative errors show that the theoretical analysis is statistically correct. Therefore, the effects of fractional order, linear stiffness coefficient, low-frequency signal, time delay intensity and damping coefficient on the Mathieu–Duffing oscillator with distributed delay are studied successively. In order to better illustrate the impact of distributed time delay on the model, the case of fixed time delay is analysed and compared, and it can be found that the distributed delay has more significant influence than fixed delay on the system. In addition, the influence of distributed delay on the system is more significant than that of the fixed delay.

The effects of two kinds of time-delayed feedbacks and non-Gaussian coloured noise on modulating bifurcations in a birhythmic model have been investigated. The non-Gaussian coloured noise is approximated asan Ornstein–Uhlenbeck process using the path integral approach. Subsequently, by the conjoint of the multiscale method and random average technique, the stationary probability density function (SPDF) of the amplitude is obtained. It is concluded that either in the condition of determinacy or randomness, time delays can effectively control birhythmicity. Especially considered the presence of noise, the varying displacement delay results in multiple bifurcation in the case of single delay, while the velocity delay triggers more complex bifurcation behaviour in the case of double delays. The strength and correlation time of noise present the opposite impacts on bifurcation, so do the two delayed coefficients. Bifurcation under different excitations is analysed theoretically, and the correctness is verified by numerical simulation. Many abundant bifurcations are available by adjusting the time-delayed feedbacks and noise intensity, which may be conductive to achieve the expected phenomenon in the enzymatic–substrate reactions.