PHYSICAL REVIEW E 83, 061122 (2011) Frequency-resonance-enhanced vibrational resonance in bistable systems Chenggui Yao, 1,2 Yan Liu, 1,2 and Meng Zhan 1,* 1 Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China 2 Graduate School of the Chinese Academy of Sciences, Beijing 100049, China (Received 30 March 2011; revised manuscript received 14 May 2011; published 16 June 2011) The dynamics in an overdamped bistable system subject to the action of two periodic forces (assuming their frequencies are ω and , and amplitudes are A and B, respectively) is studied. For the usual vibrational resonance, the nonmonotonic dependence of signal output of the low frequency ω on the change of B for a fixed , the condition  ≫ ω is always assumed in all previous studies. Here, removing this restriction, we find that a resonant behavior can extensively occur with respect to the changes of both the frequency  and amplitude B. Especially, the resonance becomes stronger when  is chosen such that it is exactly in frequency resonance with ω. This combinative behavior, called frequency-resonance-enhanced vibrational resonance, is of great interest and may shed an improved light on our understanding of the dynamics of nonlinear systems subject to a biharmonic force. DOI: 10.1103/PhysRevE.83.061122 PACS number(s): 05.40.Ca, 87.16.dj I. INTRODUCTION Stochastic resonance, the phenomenon of the response of nonlinear systems to a weak periodic signal enhanced by an appropriate amount of noise, has drawn much attention in nonlinear sciences for more than 30 years [1–3]. The constructive role of noise has been extensively studied in a variety of nonlinear systems, e.g., bistable systems [4,5], monostable systems [6], excitable systems [7], nondynamical threshold models [8], and ensembles of interacting nonlinear elements [9]. Considerable works on theoretic analysis [10] and experimental observations [11] have been conducted, and now the focus has been moved to its applications in diverse fields. Recently, a high-frequency signal, as another type of excitation, has been proposed in the context of stochastic resonance and found that it can play a similar role as noise [12–15]. The system’s response to a weak low-frequency signal can also become maximal by an appropriate choice of vibration amplitude for the high-frequency signal. This phenomenon is referred to as vibrational resonance. Since biharmonic signals are pervasive in many science and appli- cation fields, such as acoustics [16], neuroscience [17], laser physics [18], engineering [19], and even the Global Positioning System [20], the study of vibrational resonance is of great significance, and indeed it has been widely investigated in various systems, including excitable [21–24], bistable [25–28], and spatially extended systems, etc. [29–31]. Chizhevsky et al. provided the first experimental evidence of vibrational resonance in a bistable vertical cavity laser system [32]. A recent theoretical study demonstrated that with the change of driving amplitude, a high-frequency signal may induce a system transition from bistable to monostable and result in a substantial change of the system response across the critical point [33]. Although vibrational resonance is called resonance, it has little in common with the classical concept of (frequency) resonance [34]. Vibrational resonance means the resonant am- plification of the signal output with respect to the amplitude of * zhanmeng@wipm.ac.cn the external force, which is usually assumed as high frequency, whereas frequency resonance means its resonant amplification with respect to the frequency of the external force, especially when it becomes exactly equal to or is multiples of the natural frequency of the system. Therefore, basically, vibrational resonance is an amplitude effect and frequency resonance is a frequency effect. In frequency resonance, small periodic driving forces can produce a large amplitude output, whereas in vibrational resonance, only sufficiently large periodic driving forces can produce a large amplitude effect. So far, to the best of our knowledge, in all existing works on vibrational resonance, the condition for a biharmonic force with two very different frequencies was always preassumed, and thus the possible connection and interplay between vibrational resonance and frequency resonance remains unclear. In this paper, we attempt to study vibrational resonance within the whole parameter plane constructed by the pa- rameters of amplitude and frequency, in the absence of the condition of two frequencies being very different. We find that vibrational resonance can appear under the condition of only two low frequencies, and more interestingly, the frequency-resonance effect may even be superimposed on the vibrational resonance curve. These findings show the combinative effects of the system nonlinearity and two external competitive signals. II. MODEL We still consider the classical model written by ˙ x = x − x 3 + A cos(ωt ) + B cos(t + φ). (1) The model describes the overdamped motion in the bistable potential U (x ) = x 4 4 − x 2 2 subject to the modulation of two different periodic signals with frequencies ω (ω = 2π/T ) and  ( = 2π/T ′ ), respectively. Their corresponding driving intensities (amplitudes) are A and B . Different with previous studies on vibrational resonance with the restriction of  ≫ ω, here we are interested in the system response with  capable of being freely chosen. Namely, the value of  can be tuned from zero to a very large number. Without losing generality, the first periodic signal is always set to be weak (subthreshold) 061122-1 1539-3755/2011/83(6)/061122(6) ©2011 American Physical Society