Vibrational resonances in driven oscillators with position-dependent mass

The vibrational resonance (VR) phenomenon has received a great deal of research attention over the two decades since its introduction. The wide range of theoretical and experimental results obtained has, however, been confined to VR in systems with constant mass. We now extend the VR formalism to encompass systems with position-dependent mass (PDM). We consider a generalized classical counterpart of the quantum mechanical nonlinear oscillator with PDM. By developing a theoretical framework for determining the response amplitude of PDM systems, we examine and analyse their VR phenomenona, obtain conditions for the occurrence of resonances, show that the role played by PDM can be both inductive and contributory, and suggest that PDM effects could usefully be explored to maximize the efficiency of devices being operated in VR modes. Our analysis suggests new directions for the investigation of VR in a general class of PDM systems. This article is part of the theme issue ‘Vibrational and stochastic resonance in driven nonlinear systems (part 1)’.

UEV, 0000-0002-3944-726X; JAL, 0000-0003-0633-4995 The vibrational resonance (VR) phenomenon has received a great deal of research attention over the two decades since its introduction. The wide range of theoretical and experimental results obtained has, however, been confined to VR in systems with constant mass. We now extend the VR formalism to encompass systems with position-dependent mass (PDM). We consider a generalized classical counterpart of the quantum mechanical nonlinear oscillator with PDM. By developing a theoretical framework for determining the response amplitude of PDM systems, we examine and analyse their VR phenomenona, obtain conditions for the occurrence of resonances, show that the role played by PDM can be both inductive and contributory, and suggest that PDM effects could usefully be explored to maximize the efficiency of devices being operated in VR modes. Our analysis suggests new directions for the investigation of VR in a general class of PDM systems.
This article is part of the theme issue 'Vibrational and stochastic resonance in driven nonlinear systems (part 1)'. analyses of VR in the PDM systems are presented in §3. Section 4 discusses our numerical simulations and §5 summarizes the results and draws conclusions.

Position-dependent mass oscillators
We consider a classical oscillator whose dynamics may be described by the Lagrangian function [20] L(x,ẋ; t) where T = (1/2)m(x)ẋ 2 is the kinetic energy of the system, V(x) is the system's potential and m(x) is an explicitly PDM function with x being its position at time t. In the analysis that follows, we assume a Duffing-type oscillator potential, i.e.
where β is the potential parameter, i.e. the system's coefficient of nonlinearity and ω 0 is the oscillator's natural frequency. The associated Euler-Lagrange equation can be written as where φ accounts for all the external contributions to the motion from dissipative and driving forces, assumed here to be φ = −αẋ + f cos ωt + g cos Ωt. α is the damping coefficient and the amplitudes and frequencies of the external driving signals are f and ω for the weak component and g and Ω for the fast component, respectively. Using the Lagrangian function (2.1) in the Euler-Lagrange equation (2.3), the corresponding Newton's equation of motion may be written as The prime in equation (2.4) implies differentiation with respect to space variable x and the overdot indicates differentiation with respect to time.
As mentioned in the Introduction, §1, the nature of the problem or potential function considered determines the type of mass variation functions to be employed [14]. For instance, m(x) can be a quadratic or exponential function of position x [20,102,103]. The former has been classified on the basis of its singularity property: as either regular mass-functions without singularity or as singular mass-functions with single or dual singularities [20]. Moreover, a classification of finite-gap PDM systems with diverse physical applications, such as the families of trigonometric, hyperbolic and elliptic mass functions, was presented in [15]. In this paper, we adopt the simplest regular mass-function without singularities originally proposed by Mathews & Lakshmanan [14] in relation to relativistic fields of elementary particles. The mass-function (2.5) appears frequently in the modelling of diverse nonlinear mechanical systems (see [19,20] and references therein). Here, m 0 is a constant mass, equivalent to the mass amplitude and λ is the strength of the spatial nonlinearity in mass. m(x) is bounded and defined over the entire real line D(m 1 ) = with its maximum, m 0 , at x = 0 and varnishing as |x| → ∞.
One can easily show that the equation of motion of the PDM-Duffing oscillator can be written as where γ = 1/m 0 . Remarkably, the PDM-Duffing oscillator (equation (2.6)) is consistent with the system described by equation (23) in [19] for a unit mass amplitude (m 0 = 1) and g = 0.
When the strength of nonlinearity in mass is negligible, that is λ = 0, equation (2.6) reduces to the well-studied bi-harmonically driven Duffing oscillator (eqn (1) in [23] system is a generalized version of the model systems considered hitherto in the study of VR. A typical example of a physical system described by equation (2.6) is a dual-frequency-driven gas bubble in which the mass of the bubble is dependent on the bubble's radius-which is a spatial coordinate [104]. The dual-frequency driving force, which can be realized by means of acoustic waves with two frequency components, is applied to control the bubble's properties, including the promotion of acoustic cavitation. We refer the reader to a very recent study of driven bubbles highlighting the state of the art in applications of dual-frequency irradiation [105]. Moreover, the optical properties of semiconductor devices, such as Al x Ga 1−x m, many of which are also characterized by position-dependent effective masses [8,9], can be modulated and controlled effectively by employing external fields consisting of an applied electromagnetic field and a highfrequency intense laser field (ILF). The quantum mechanical counterpart of VR [75,76] would of course be more appropriate for the analysis of the combined impacts of the position-dependent effective mass (PDEM) and applied fields on the properties of semiconductors.
In what follows, we will express equation (2.6) in a form that makes our analytical procedure convenient for the application of the well-established method of separation of motions (MSM). This is the basis of the theoretical analysis. For a nonlinear system whose mass depends explicitly on position or velocity, or both, intuitively one would encounter a position-dependent function (k 1 ± k 2 x p ) n , where k 1 and k 2 are constants, and p and n are positive and negative integers, respectively. This function cannot fit into the general framework of MSM. By dividing equation (2.6) by m(x), we express it as and obtain (k 1 ± k 2 x p ) n = (1 + λx) −1 which can be approximated using the Binomial expansion. Considering only the first three terms of the binomial expansion of (1 + λx 2 ) −1 , we write equation (2.7) asẍ (2.8) Furthermore, by setting δ = βγ − λω 2 0 , and ξ = βγ λ + λ 2 ω 2 0 , in equation (2.8), the PDM-Duffing oscillator can be expressed in the form (2.9) The corresponding potential V(x) of the system is Henceforth, we shall refer to equation (2.9) as the PDM-Duffing oscillator. The system potential is shown in figures 1 and 2 for different values of the PDM parameters: the mass amplitude m 0 (= 1, 1.5, 2, 4) and the strength of spatial nonlinearity λ(= 0, 1, 1.5, 2), respectively, is computed from equation (2.10). The dynamical properties of the system can be altered by adjustment of its potential which, in turn, is largely determined by the PDM parameters (m 0 , λ). We choose mass parameter regimes within which the system potential is double-well, so that 0 < m 0 < 1.5 and 0 < λ < 1 for α = 0.2, β = 1, ω 2 0 = −1.
Next, we linearize equation (3.12) around the equilibrium points (y * ,ẏ * ) in order to obtain an approximate analytic response amplitude Q ana which can also be compared to the response amplitude Q num obtained from the Fourier coefficients of the solution of the full equation of the system (2.9). The system's oscillation can be described in terms of the deviation of slow motion y from the equilibrium points y * by using the deviation variable Y = y − y * in equation (3.12). This yields the motion around equilibrium points in the form where = C 1 y * + C 2 y * 3 + λ 2 y * 5 , 1 = C 1 + 3C 2 y * 2 + 5λ 2 y * 4 , 2 = 3C 2 y * + 10λ 2 y * 2 , 3 = C 2 + 10λ 2 y * 2 , 4 = 5λ 2 y * , Ξ 1 = C 3 + λy * 2 , Ξ 2 = 2λy * , Θ = η 1 y * + η 2 y * 3 + η 3 y * 5 , Θ 1 = η 1 + 3η 2 y * 2 + 5η 3 y * 4 , By ignoring the nonlinear parts of equation (3.15) and using the approximation f 1 such that |Y| 1 in the long-term limit t → ∞, the linearized equation of motion then becomes where the resonant frequency is ω r = √ η 1 , μ = αγ C 3 and F = γ C 3 f when the oscillation is considered around the equilibrium point y * = 0. For y * = 0, ω 2 r = Θ 1 , μ = αγ Ξ 1 and F = γ (Ξ 1 + Ξ 2 Y + λY 2 )f . It is clear that the steady-state solution of equation (3.12) takes the form (3.18) and that the response amplitude can be computed as If we set S = (ω 2 r − ω 2 ) 2 + μ 2 ω 2 in equation (3.19), the qualitative features of Q ana would be determined by S. A local minimum in S implies resonance, i.e. the appearance of a maximum in Q ana . When a variation of any of the system parameters leads the system to resonance, the value of the parameter at which resonance occurs (e.g. λ = λ vr ) can be obtained from the root of the equation S λ = dS/dλ = 0 and S λλ|λ=λ vr > 0.
When the sign of either η 1 or η 2 is changed by varying any of g, Ω, m 0 or λ, the effective potential changes structure from single-well to double-well as shown in figure 3. The value of g vr and λ vr when the effective potential is single-well can be obtained by setting S g = 0 and S λ = 0 and satisfies the condition In addition, the resonance condition for λ vr gives When the effective potential is a double-well, it is challenging to establish analytical conditions in terms of g vr and λ vr . However, they can be computed numerically by analysing the cases ω 2 r − ω 2 = 0 and ω rg = 0 or ω rλ = 0 since S g = 0 ( or S λ = 0) at resonance.

Numerical results and discussions
To validate the analytic results, the theoretical response amplitude Q given by equation (3.19) was compared with the numerical Q computed from the Fourier spectrum of the solution of the main PDM-Duffing equation (equation (2.9)) expressed as coupled first-order autonomous ordinary differential equations (ODEs) of the form dx dt = y, The solution of equation (4.1), corresponding to the output signal of the system, is obtained by numerical integration using the fourth order Runge-Kutta (FORK) scheme with step size t = 0.01T over a simulation time interval T s = nT, where T = 2π ω is the period of the oscillation, ω is the low frequency (LF) of the input signal and n(= 1, 2, 3, . . . ) is the number of complete oscillations. We used (x(0),ẋ(0.1)) initial conditions with a relaxation time of 20T. Except where otherwise specified, the values of fixed system parameters were: α = 0.2, β = 1, ω 2 0 = −1, Ω = 9.842, ω = 0.5 and f = 0.05. The PDM parameters were set as (m 0 , λ) ∈ ((0, 1.5), (0, 1)). These parameter choices ensure that the system remains in the overdamped regime for which only periodic or quasiperiodic motion is admissible and where the system remains a bistable oscillator. The other system parameters were chosen within regimes that optimize the emergence of VR for n = 200.
The response amplitude Q at frequency ω was then obtained from the Fourier sine and cosine coefficients of the output signal with components Q s and Q c given by Conventionally, the amplitude of the output signal is given by while the phase shift is The response amplitude is thus given by The analytically computed response amplitudes from equation (   increases. Thus, for a Duffing oscillator with a unitary particle mass, the mass amplitude plays a complementary role to the HF signal parameters (g, Ω) in the observed resonances.
Furthermore, the possibility of initiating resonance through variation of the PDM mass amplitude, with the cooperation of the HF input signal, is confirmed by the results presented in figure 5. Figure 5a shows the dependence of the response amplitude Q on m 0 for four values of the HF amplitude g(= 20, 40, 60, 80, 100) for a particle with constant mass (λ = 0). Resonances with single peaks at m 0 (Q max ) directly dependent on the HF amplitude g can be seen for each value of g. Although resonances can thus be achieved by varying g, there is no significant optimization, and the impact of g on Q is a shift in the peak position in the direction of increasing m 0 . In addition, by switching on the mass spatial nonlinearity and examining the dependence of Q on m 0 for increasing mass nonlinearity (λ = 0, λ = 0.1, λ = 0.2, λ = 0.5) at g = 20, single resonance peaks indicative of VR for the dependence of Q on g (or Ω, shown in figure 4b) are observed for each value of λ. It is evident that the mass amplitude m 0 can be used to initiate VR or/and can complement the HF input signal parameters in determining the conditions for resonance. This is similar to the effect of constant mass on the VR phenomenon observed in the dynamics of an inhomogeneously damped one-dimensional single particle moving in a symmetrical periodic potential [37,101]. For the observed resonances in figure 5, the PDM mass amplitude m 0 and the HF signal amplitude g are directly related: increasing the value of g corresponds to increasing the value of mass amplitude m 0 .
To gain further insight into the contributions of the PDM to VR, we also considered the effect of the PDM nonlinear strength λ on the observed resonances. First, we showed that the resonances for constant unitary mass can also be realized with a suitable combination of PDM parameters when the mass spatial nonlinearity is activated. This is presented in figure 6a- This implies that, besides the independent impact of the mass amplitude m 0 , the combination of PDM parameters plays a role in determining the conditions for VR. The variation of the response amplitude Q with HF amplitude g for four values of spatial nonlinearity strength λ (λ = 0, λ = 0.05, λ = 0.2, λ = 0.5) is presented in figure 7 for m 0 = 1.1. The shape of the resonance curve, maximum response amplitude Q max , and g(Q max ) all depend on λ. The maximum response amplitude Q max at which VR occurs decreases as the strength of the spatial nonlinearity increases, as presented in the inset (b) of figure 7. Here, we have zoomed the top portions of the numerically computed response curves, i.e. figure 7a.
Finally, figure 8 demonstrates that cooperation between the HF input signal and the PDM parameters can induce VR through the mass spatial nonlinearity strength λ for m 0 = 1. This is presented for four values of the HF input signal amplitude g(g = 20, g = 25, g = 30, g = 35) in figure 8a-d, respectively. The observed single resonances are typical of VR induced by the HF input parameters. In this figure, resonance occurs for a pair of low values of λ and high values of g, illustrating the cooperation effect between the HF driving force and the PDM in the VR process. In general, the strength of the spatial nonlinearity optimizes the effect of HF amplitude when the system is driven into resonance and vice versa.
In figure 9, we present a three-dimensional plot illustrating the numerically computed response amplitude Q as functions of both the strength of the mass nonlinearity λ and the HF signal amplitude g for ω (= 0.   figure 3, the effective potential changes from a double-well structure (for g = 0) to a single-well structure as g takes on larger values, such that g ≥ 80. The transition to a double-well structure is readily enhanced by the cooperation between the PDM parameter λ and the high-frequency amplitude g. Figure 10 presents a broader picture of the effect of the PDM parameters (λ, m 0 ) on the system's response amplitude Q. Here, Q was also computed numerically. It is plotted in 3D as

Summary and conclusion
We have provided a detailed but succinct review of the VR phenomenon [23], which was proposed two decades ago. We cite numerous works exploring and elucidating the mechanism of VR in several different systems, as well as the contributory or inductive roles of diverse system parameters in the occurrence of VR. Practical experimental realizations and applications of VR have also been explored and discussed. We emphasize that many of the above investigations deal with additive driving forces, whereas rather less attention is paid to parametric driving and amplitude-modulated forcing [34,[65][66][67][68][69]. In connection with signal detection, transmission and amplification, parametric driving and amplitude modulated forcing are excellent tools for achieving higher laser modulation bandwidths which are desirable qualities for applications in multigigabit optical fibre transmitters [107] and could be suitable for designing measurement techniques where highfrequency response is required [108]. Thus, exploring high-frequency parametric vibrations could find practical applications in communications systems as well as in the detection and assessment of structural damage in systems with breathing cracks-suggesting a new direction for VR investigations.
Complementing all of the previous VR investigations, where systems had constant mass, we have demonstrated VR in a Duffing oscillator whose mass is position dependent. In particular, we considered the PDM-Duffing, with the mass defined as a regular function comprising mass amplitude m 0 and strength of spatial nonlinearity λ. Based on the generalized Duffing oscillator equation with PDM, we presented and validated the VR phenomenon in a bistable potential by considering the reduced case in which m 0 = 1, λ = 0. We then extended the problem by examining the effects of the mass parameters on the response curves (Q versus g) and explored the resonances induced by the PDM parameters in the presence of the HF input signal. We conclude that, in the generalized PDM-Duffing oscillator, the roles played by PDM are both inductive and contributory. They can with advantage be explored to maximize the efficiency of devices that operate in VR modes. We believe that our new formalism describing VR in PDM systems, and its applications as enumerated above, paves the way to a new body of research on VR.