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0Russian Journal of Nonlinear Dynamics, 2023, vol. 19, no. 3, pp. 389-408. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd230903
MATHEMATICAL PROBLEMS OF NONLINEARITY
MSC 2010: 34C55, 37M20, 37G35, 74H10, 70K30, 70K50
Nonlinear Resonance in a Position-Dependent Mass-Duffing Oscillator System with Monostable Potentials Driven by an Amplitude-Modulated Signal
K. Suddalai Kannan, S. M. Abdul Kader, V. Chinnathambi, M. V. Sethu Meenakshi, S. Rajasekar
This study examines the phenomenon of vibrational resonance (VR) in a classical position-dependent mass (PDM) system characterized by three types of single-well potentials. These potentials are influenced by an amplitude-modulated (AM) signal with Q » w. Our analysis is limited to the following parametric choices: (i) wfi, ¡3, m0, X > 0 (type-1 single-well), (ii) > 0, ¡3 < 0, 2 <m0 < 3, 1 <X< 2 (type-2 single-well), (iii) W > 0, ¡3 < 0, 0 < m0 < 2, 0 < X < 1 (type-3 single-well). The system presents an intriguing scenario in which the PDM function significantly contributes to the occurrence of VR. In addition to the analytical derivation of the equation for slow motions of the system based on the high-frequency signal's parameters using the method of direct separation of motion, numerical evidence is presented for VR and its basic dynamical behaviors are investigated. Based on the findings presented in this paper,
Received July 07, 2022 Accepted September 04, 2023
Karuppasamy Suddalai Kannan
ksramar1992k@gmail.com
Sheik Mohamed Abdul Kader
aksac.physics@gmail.com
Veerapadran Chinnathambi
veerchinnathambi@gmail.com
Department of Physics, Sadakathullah Appa College,
Tirunelveli, Tamilnadu, 627011 India
Manonmaniam Sundaranar University,
Tirunelveli, Tamil Nadu, 627012 India
M. Vaidyanathan Sethu Meenakshi sethumeenakshi91@gmail.com
Department of Mathematics, Fatima College, Madurai, Tamilnadu, 625018 India
Shanmuganathan Rajasekar srj.bdu@gmail.com
School of Physics, Bharathidasan University, Tiruchirapalli, Tamilnadu, 620024 India
the weak low-frequency signal within the single-well PDM system can be either attenuated or amplified by manipulating PDM parameters, such as mass amplitude (m0) and mass spatial nonlinearity A. The outcomes of the analytical investigations are validated and further supported through numerical simulations.
Keywords: position-dependent mass system, amplitude-modulated signal, vibrational resonance, hysteresis, chaos
1. Introduction
Resonances stand out as one of the most fundamental phenomena exhibited by nonlinear systems. They represent specific instances of maximal response within a system due to its capacity to store and transmit energy received from an external driving source. Resonances hold particular significance across physical, engineering, and biological systems. They can yield advantages in various applications, while also potentially leading to instabilities and even catastrophic events in others [1]. When resonance manifests in a nonlinear system, it takes on the label of nonlinear resonance. Recent research has demonstrated that various types of external forces can induce distinct forms of resonance. For instance, a weak input signal can be magnified by the presence of noise, particularly when tuned to an optimal level. This phenomenon of noise-induced resonance is referred to as stochastic resonance [2, 3]. Resonance becomes achievable by substituting the noise term with a high-frequency periodic force, leading to what is termed vibrational resonance [4, 5]. Another resonance occurrence arises when a high-frequency deterministic force aligns with the absent frequency of the input signal, known as ghost-vibrational resonance [6, 7]. There's also the possibility of generating a chaotic signal that emulates the probability distribution of Gaussian white noise. This chaotic signal can induce a resonant effect akin to noise-induced resonance and is labeled chaotic resonance [8, 9]. While introducing noise to a system often increases its unpredictability, in certain instances, the system's behavior synchronizes with the noise level. This intriguing phenomenon is called coherence resonance [10, 11]. Resonance is achievable in specific nonlinear systems exposed to external energy sources and specific parameter settings. This kind of resonance is termed parametric resonance [12, 13]. Moreover, the capability of a nonlinear oscillator to sustain resonance due to variations in its structural and/or excitation parameters is referred to as autoresonance or self-sustained resonance [14, 15]. In this paper, our focus centers on vibrational resonance (VR). To achieve VR, a nonlinear system is subjected to both low-frequency (w) and high-frequency (Q) driving forces. When the amplitude or frequency of the high-frequency force is systematically adjusted from a low value, the response amplitude at the low-frequency Q(w) point in the system maximizes at one or more values of the control parameter.
The phenomenon of vibrational resonance (VR) was initially observed by Landa and Mc-Clintock [16] within a bistable system. Subsequently, Gitterman [17] conducted an analytical investigation to validate VR. Following these pioneering studies, the characteristics of VR have been explored theoretically, numerically, and experimentally across various nonlinear systems. For instance, vibrational resonance has been identified in a double-well Duffing oscillator [16-18], spatially extended systems [19], excitable systems [20], overdamped bistable systems [20-22], and overdamped two-coupled anharmonic oscillators [23, 24]. The influence of noise on vibrational resonance has also been scrutinized in specific systems [20, 25, 26]. Experimental confirmation of vibrational resonance has been demonstrated in a bistable vertical cavity surface emitting laser [26], as well as in an optical system [27]. In the context of a diode laser and logistic
map, it was observed that high-frequency force induces noise-free stochastic resonance within an intermittency region [28].
In recent years, a significant amount of attention has been focused on the phenomenon of vibrational resonance across various scientific groups. For instance, researchers have discovered that vibrational resonance can induce undamped low-frequency signal propagation in both oneway-coupled [29] and globally coupled [30] bistable systems. The concept of vibrational ratchet motion has been explored in specific systems featuring spatially periodic potentials driven by a bi-harmonic force and subjected to Gaussian white noise [31]. In the context of a pendulum system subjected to a high-frequency periodic force and noise, the utilization of vibrational mechanics has revealed that both mobility and diffusion coefficient exhibit an extraordinarily high sensitivity to mass, even under conditions of substantial damping [32]. Investigation into vibrational resonance has extended to encompass time-delayed systems [30, 33-36] and networks [37, 38] as well. Additionally, frequency-resonance-enhanced vibrational resonance has been observed in an overdamped bistable system [39]. The control of vibrational resonance within a calcium ion system has also been documented [40]. Notably, recent findings indicate the occurrence of multiresonance in systems characterized by spatially periodic potentials [41, 42].
The concept of a varying mass pertains to alterations in mass concerning velocity, position, time, or a combination of position and time. When mass changes in relation to position, it is termed position-dependent mass (PDM). In recent years, systems featuring position-dependent mass (PDM) have garnered the attention of numerous researchers and scientists due to their significance across various physics disciplines. Position-dependent mass functions, denoted as m(x), give rise to "forces quadratic in velocity", consequently leading to the emergence of nonlinear differential equations of motion within the Newtonian framework. One of the most renowned instances of such equations, introduced by Mathews and Lakshmanan [43], involves the nonlinear oscillator equation (1 + Xx2) x — Xxx2 + a2x = 0. This equation can be derived from
the Lagrangian L = ^ ( i+L-- ) ~~ o?x2). Variations in mass corresponding to position are
also applicable to the kinetic energy of dynamic systems existing within curved spaces, whether possessing constant curvature [44-46] or nonconstant curvature [47, 48]. Analogous relationships surface in the realm of geometric optics, where the refractive index's position-dependent nature can be interpreted as a varying mass [49]. In the domain of semiconductor theory, it has been established that coherent superpositions of states linked to distinct masses are disallowed [50]. Beyond the aforementioned works, applications encompass rocket motion [51], the raindrop problem [52], the oscillator with variable mass [53], the inversion potential for NH3 within density theory [54], the two-body problem related to binary system evolution [55], the consequences of galactic mass loss [56], neutrino mass oscillations [57], and the interaction of a rigid body with a liquid free surface [58], among others.
In certain experiments, the applicability of heightened position-dependent mass (PDM) has been observed. For instance, in the deuteron-deuteron scattering experiment, the augmentation of electron mass has been successfully elucidated [59]. The increased effective mass finds utility in explaining certain phenomena within quantum field theory through a fluid-based approach [60]. Furthermore, the amplification of the quasiparticle mass in BaFe2(As1_^P^)2 has been documented at the quantum critical point, impacting the critical temperature of the superconducting state [61, 62]. Predictions indicate that the effective mass of excitons within a semiconductor coupled quantum well experiences enhancement under the influence of an electromagnetic field [63]. Additionally, the escalated mass in quasiparticles has implications for entanglement within the Kondo problem [64]. Similarly, reports have indicated enhancements in the energy of
electrons within quasicrystals [65], hydrogen atoms [66], and quantum LC circuits [67]. Building upon these findings, this paper delves into the investigation of vibrational resonance (VR) in a position-dependent mass-Duffing oscillator featuring three variations of monostable potentials. The system is subjected to an amplitude-modulated signal and is examined through both analytical and numerical approaches. Given the escalating interest in position-dependent mass systems and amplitude-modulated signals, we believe that the research presented in this paper holds significance and relevance.
The paper is organized as follows: In Section 2, we begin by introducing the classical position-dependent mass model and presenting its corresponding mathematical equations. Moving on to Section 3, we investigate a system subjected to an AM signal comprised of both low-frequency and high-frequency components, denoted as F(t) = f sin(wt) + g sin(Q + w)t. Specifically, for the scenario where Q » w, the system's solution consists of a slow motion denoted as X(t) and a fast motion denoted as ^(t, Qt), characterized by their respective frequencies, w and Q. We derive the equation of motion governing the slow motion and provide an approximate analytical expression for the response amplitude Q of the low-frequency (w) output oscillation. Building upon the analytical expression for Q established in Section 3, Section 4 delves into an analysis of vibrational resonance (VR) within three distinct forms of single-well potentials. These potential types are categorized as: (i) type-1 single-well, featuring parameters w2, /3, m0, A > 0; (ii) type-2 single-well, characterized by w2 > 0, /3 < 0, 2 < m0 < 3, and 1 < A < 2; and (iii) type-3 single-well, defined by w0 > 0, /3 < 0, 0 <m0 < 2, and 0 < A < 1. Proceeding to Section 5, we explore a system exposed to an AM signal consisting of a single low-frequency component (f sin wt) alongside two high-frequency components (g sin(Q + w)t and g sin(Q — w)t), denoted as F(t) = f sin wt + g sin(Q + w)t + g sin(Q — w)t. Here, we examine the realization of VR through a numerical approach. Finally, in Section 6 we briefly summarize the results of our present study.
2. Model and its mathematical equations
Consider a classical PDM system described by the Lagrangian
L(x, x, t)=T- V(x) = m(x)x2 - V(x), (2.1)
where T = ^m(x)x2 is the kinetic energy of the system, V(x) is the system's potential and m{x) is the position-dependent mass function with x being position at time t. The associated Euler-Lagrangian equation can be written as
d (dL \ dL . .
Using the Lagrangian function (2.1) in the Euler-Lagrangian equation (2.2), the corresponding Newton equation of motion is given by
, . 1 i. . 2 dV(x) , ,
m\x)x + -m (x)x H---— = (p. (2.3)
2 dx
In our present work, 0 assumed to be 0 = —aX + (f + 2g cosQt)sinwt, a is the damping coefficient and the amplitudes and frequencies of the AM signal are f and w for the low-frequency component and g and Q for the high-frequency component, respectively. The prime in Eq. (2.3)
implies differentiation with respect to space variable x and the over-dot indicates differentiation with respect to time. Among the various types of mass variation function, we use the following mass variation function in the present study:
= ^ (2.4)
where m0 is a constant mass equivalent to the mass amplitude and X is the strength of the spatial nonlinearity on mass. In the analysis that follows, we consider a Duffing type oscillator potential, i. e.,
V(x) = ^m(x)u$x2 + ^/ir4, (2.5)
where w0 is the oscillator's natural frequency and / is the stiffness constant which plays the role of the nonlinear parameter. For a constant mass case, i. e., m(x) = 1 (i. e., m0 = 1, X = 0), Eq. (2.3) reduces to a forced, damped Duffing oscillator and is realized in many physical, engineering and biological systems. Originally, the Duffing equation was introduced by the German engineer Georg Duffing in 1918 [68] to describe the hardening spring effect in many mechanical problems. From the above one can easily show that the equation of motion of the PDM-Duffing oscillator can be written as
m(x)x — m2(x)xYXx2 + ax + m2(x)Yw"^x + /x3 = (f + 2g cosQt)sinwt, Q » w, (2.6)
where 7 = With the use of the formula 2 cos Qt sin ujt = sin(Q + uj)t + sin(Q — uj)t, Eq. (2.6) takes the form
m(x)x — m2(x)xYXx2 + ax + m2(x)Yw"^x + /x3 = f sin wt+g sin(Q +w)t+g sin(Q — w)t, Q » w.
(2.7)
When X = 0 and the unit mass m(x) = 1, Eq. (2.7) reduces to the well-known Duffing oscillator equation driven by an AM signal. The physical system, Eq. (2.7), describes a dual frequency-operated gas bubble in which the mass of the bubble depends on the radius of the bubble, which is a spatial coordinate. Equation (2.7) does not apply to the general structure of the method of separation of motions (MSM). After some mathematical manipulations given in Ref. [69], this equation can be fitted into the general framework of MSM, which can be expressed in the form
x — X(x — Xx3 + X2x5) x2 + aY (1 + Xx2) x + w2x + 5x3 + £x5 =
= y (1 + Xx2) (f sin wt + g sin(Q + w)t + g sin(Q — w)t), Q » w, (2.8)
where 5 = /y — Xw2 and £ = /yX + X2w^. Equation (2.8) is also known as the PDM-Duffing oscillator equation. The corresponding potential of the system is
V{x) = + + i^6- (2-9)
2 4 6
The shape of the potential V(x) depends on the parameters w2, 5 and £. It can be a single-well, double-well, single-well with double-hump, double-well with double hump and inverted singlewell potentials. Recently, Roy-Layinde et al. [69] examined and analyzed the VR phenomenon in a double-well PDM-Duffing oscillator system driven by a biharmonic signal.
3. PDM-Duffing oscillator with one high-frequency signal
Consider the system (2.8) with F(t) = f sin ut + g sin(Q + u)t. An approximate solution of Eq. (2.8) for Q » u can be obtained by the method of separation of motions, where the solution is written as a sum of slow motion X(t) and fast motion t(t, t),
x(t) = X(t) + ^(t, t).
(3.1)
2n
Assume that the mean value of tp, defined as (tp) with respect to time r, is (tp) = ^ f ipdr = 0.
0
Because t is rapidly varying, we approximate the equation of motion for t as tp = Yg sin(Q +
2n
—jfa cos{n + f.M- 7/, = —J2L ™fr> I »/. - 1
0
^'av = 2(nX)4 ; ^ = 0; i'tv = sifhw aIld = 2(nX)a ' TheI1 the eClUati°n for the sl°W
motion is
+ u)t, which gives -ip = cos(Q + w)i; tp = - (n^)a sin(Q + u)t, ipav = r2- J ipdr = 0,
X - A (C\X + C2X3 + X2X5) XX2 + aY (C3 + AX2) XX + niX + n2X3 + n3X5 =
= Y (C3 + XX2) f sin ut, (3.2a)
where
15A27V 1 3A72g2 1
8 _(Q + w)8_ 2 (Q +w)4_
C\ =
C2 = 5A2Y2g2
ATv 2
2 „2
+ 1,
1
(Q + u)4 1
- A,
XCiY g
_(Q + u)4_ 1
C3 = 1 +
ni =
% =
% =
and the effective potential is
2 _(Q + u)2
A C27y 1
2 (Q + w)2
A 37V 1
2 _(Q + u)2_
2 15^Y4g4
+ Wq +
8
1
+ 5CY2g2 +£
1
_(Q + u)4_
_(Q + u)8
+ 5,
+
35y2g2
_(Q + u)4_
Feff(.,) = |x2 + |x4 + |x6.
(3.2b) (3.2c) (3.2d) (3.2e) (3.2f) (3.2g)
(3.3)
The shape, the number of local maxima and minima and their location of the potential V(x) (Eq. (2.9)) depend on the parameters u2, S, and £. For the effective potential (Vff) (Eq. (3.3)) these depend also on the parameters g and Q. Consequently, by varying g or Q new equilibrium states can be created or the number of equilibrium states can be reduced.
The equilibrium points about which slow oscillations take place can be calculated from Eq. (3.2). The equilibrium points of Eq. (3.2) are given by
x: = 0,
X2,3 = ±
+ Vnl - 4r?i%
2%
1/2
X4,5 = ±
- V,ril - 4r?i^3 2%
i/2
(3.4)
1
2
Suppose, we choose n3 > 0. Then we have the following cases: Case (i): n1, n2 > 0 or n1 > 0, n2 < 0 with n2 < 4^n3. X* = 0 is the only equilibrium point. Case (ii): n1 < 0, n2 is arbitrary-There are three equilibrium points: X*, X* 3.
Case (iii): n1 > 0, n2 < 0 with n2 > 4n1 n3-There are five equilibrium points given by Eq. (3.4).
We obtain the equation for the deviation of the slow motion X from an equilibrium point X*. Introducing the change of variable Y = X — X* in Eq. (3.2), we get
Y — A (a1 + a2Y + a3Y2 + a4Y3 + a5Y4 + A2Y5) Y2 + aY (a1 + o2Y + AY2) F+
+ P1 + P2Y + p3Y2 + p4Y3 + p5Y4 + n3Y5 = Y (o1 + o2Y + AY2) f sin ut, (3.5a)
where
a1 = C1X * + C2 X *3 + A2X *5, a2 = C1 + 3C2X *2 + 5A2X *4, a3 = 3C2X * + 10A2 X *2, a4 = C2 + 10A2X *2, a5 = 5A2 X *, o 1 = C3 + AX *2, o 2 = 2 AX *, P1 = n1 X * + n2X *3 + %X *5, P2 = n1 + 3n2 X *2 + 5% X *4, P3 = 3%X * + 10%X *2, P4 = V2 + 10^3 X *2, P5 = 5% X *. (3.5b)
For f ^ 1 and in the limit t ^^ we assume that \Y\ ^ 1 and neglect the nonlinear terms in Eq. (3.5). Then the solution of the linear version of Eq. (3.5) in the limit t ^^ is AL cos(ut — d), where
AL = --f-TW (3-6)
(u2 — u2)2 + tx2u2
and the resonant frequency is ujt = n = cryC3 and F = 7C3f. If the slow motion takes place around the equilibrium point A"* = 0, then ujt = sJff[. The response amplitude Q is
Q = ^ = -r-—-(3-7)
f
2 1/2 ' (w2 — W2) + /J,2W2
In the following sections, we analyze the occurrence of VR in the PDM-Duffing oscillator system (2.8) driven by the signals F(t) = f sin ut+g sin(Q+u)t and F(t) = f sin ut+g sin(Q+u)t+ + g sin(Q — u)t with specific emphasis on single-well forms of the potential V(x). We consider the parametric choices (i) u2,/,m0,A > 0 (type-1 single-well) (ii) u0 > 0, 3 < 0, 2 < mo < 3, 1 < A < 2 (type-2 single-well) (iii) u0 > 0, /3 < 0, 0 < m0 < 2, 0 < A < 1 (type-3 single-well). The reason for our interest in single-well cases of this system is that VR is generally studied in bistable and multistable systems where multiple resonances are observed for certain parameter choices. In the present work we analyze the possibility of multiple vibrational resonances in the single-well cases of the system (2.8).
4. Analysis of VR
To compare with the theoretical Q given by Eq. (3.7), we compute Q from numerical solution of Eq. (2.8). The numerical Q(w) is given by
mT
2
®8 = mT j sin(wi) dt> 0
mT
2
Qc = — / x(t) cos(wi) dt,
(4.1a) (4.16)
where T = ^ is the period of the response and m is a positive integer. Then
VQ's+Qc
Q =
f
(4.1c)
4.1. VR with type-1 single-well potential
For w2, m0, X > 0, V(x) is a type-1 single-well potential with a local minimum at x = = 0. The system potential (2.9) is shown in Figs. 1a and 1b for different values of the PDM parameters such as the mass amplitude m0 (= 0.5, 1, 2) with X = 1 and the strength of the spatial nonlinearity, X (= 0.5, 1, 2) with m0 = 1. From Fig. 1, the width of the potential increases as m0 increases, whereas it decreases as X increases. The quantities C1 and C2 given by Eq. (3.2) are now positive. Therefore, the effective potential is always a single-well potential when m0 or X or g or Q is varied. The slow motion occurs around the equilibrium point X\ = 0 and ojr = sJff[.
I.Ottt
0.0
V.I A = 1 1 \ "'M \\\ A mo=0.5—j i: I' ■ m0 = l-—U: m0 = 2~—jf; i; y''' (a)
1.0
IT 0.5 £
0.0
A = 0.5—¡-}l A=i—rj A=2— /(b)
0
x
0
x
Fig. 1. The system type-1 potential (2.9) for (a) m0 = 0.5, 1, 2 and X =1, (b) X = 0.5, 1, 2 and m0 = 1. The values of the other parameters are ¡3 = w2 = 1
First, we analyze the occurrence of VR for X = 0 (i. e., the system with constant mass m(x) = = m0) and we fix a = 0.2, ¡3 = 1, = 1, f = 0.05, w = 1.5 and Q = 15. Figure 2a presents both theoretically and numerically calculated Q as a function of g with X = 0. The continuous curve represents theoretical results obtained from Eq. (3.7). The dotted lines represent numerically calculated Q. The numerically computed Q is in good agreement with the theoretical approximation. For X = 0, resonance is observed at only one place. The theoretical and numerical values of gVR are 87.55 and 85.35. Figure 2b shows the plot of w^ versus g. At g = gVR (at which resonance occurs) we find that, w,2 = w2. At resonance Qmax = = 3.033.
Fig. 2. (a) Variation of the response amplitude Q with g for A = 0. (b) Plot of the theoretical value of <JT versus g. The continuous line represents the analytically computed response amplitude Q from Eq. (3.7), while the dotted lines represent the numerically computed response amplitude Q from Eq. (2.8) using Eq. (4.1)
Fig. 3. Variation of the response amplitude Q (a) with g for three values of mass amplitude m0 with A = = 0.5, u =1.5, Q = 30. The values of m0 for the curves 1-3 are 0.5, 1.0 and 1.5, respectively. (b) with g for four values of the spatial nonlinearity A with m0 = 0.5, u = 1.5, Q = 30. The values of A for the curves 1-4 are 1.0, 1.5, 2.0 and 4.0, respectively. Numerical results for the variation of the response amplitude Q with g for (c) four values of u with Q = 30 and (d) for four values of Q with u = 1.5
Now we analyze the effect of the PDM parameters like the mass amplitude (m0) and the mass spatial nonlinearity (A) on the observed resonances. First, we consider the effect of the mass amplitude m0 on the known resonances for A = 0.5 as presented in Fig. 3a. Figure 3a shows the plot of Q for three values of m0 (= 0.5, 1.0, 1.5) computed from Eq. (3.7) and superimposed with their corresponding numerical values computed from Eq. (2.8) using Eq. (4.1) for comparison. The continuous lines represent the analytically computed response amplitude Q from Eq. (3.7), while the dotted lines represent the numerically computed response amplitude Q from Eq. (3.7) using Eq. (4.1). Other parameters of the system (2.8) are set as a = 0.2, / = 1, u"^ = 1, u = 1.5 and Q = 30. One can obviously see that the theoretical values of Q are in close agreement with the numerical values of Q. However, as the value of m0 increases the position of the peaks shifts further from the appearance. Then we show the effect of mass spatial nonlinearity (A) on the
known resonances for a fixed value of m0. This is presented in Fig. 3b for four values of A (= 1.0, 1.5, 2.0, 4.0) with m0 = 0.5. From Fig. 3b we can clearly observe that the shape of the response curve and the maximum response amplitude Qmax all depend on A. Furthermore, the possibility of initiating resonance through the variation of the low-frequency (w) and high-frequency (Q) components of the signal with the fixed value of PDM parameters (m0, A) is confirmed by the numerical results presented in Figs. 3c and 3d. Due to these frequencies, the influence of g on Q is clearly seen in Fig. 3c and 3d.
We then proceed to investigate the influence of the position-dependent mass (PDM) parameters (m0, A) on the underlying dynamics of the PDM-Duffing oscillator system (2.8) governed by the type-1 potential. This analysis involves plotting a bifurcation diagram to unveil the intricate behavior of the system. Through simulations conducted in the scenario of a system subjected to a single high-frequency signal, we discern a diverse array of dynamics that include bifurcation bubbles, periodic, multiperiodic, reverse period doubling, and chaotic motion. Additionally, phenomena such as hysteresis, multistability, and the coexistence of periodic, multiperiodic, and chaotic attractors are observed. Figure 4 showcases the bifurcation diagram of the system, depicting the variation of g within the domain 0 < g < 200, while considering different combinations of m0 and A. The remaining parameters of the system are held constant at w"^ = 1, / = 1, a = 0.2, w = 1.5, Q = 15.0, and f = 0.05. For the case of m0 = 0.5 with A = 0, Fig. 4a reveals the presence of bifurcation bubbles. As the value of m0 increases to 1.0, the bifurcation diagram (Fig. 4b) illustrates the occurrence of periodic, period-doubling, reverse period-doubling, and chaotic states. Upon introducing the mass spatial nonlinearity (A), as shown in Figs. 4c and 4d, only periodic states are observed within the system's dynamics. In essence, the bifurcation diagram provides a comprehensive visualization of the intricate dynamics influenced by the variations in PDM parameters (m0, A).
-0.02
200
200
Fig. 4. Bifurcation diagram of the type-I single-well PDM-Duffing oscillator driven by F(t) = f sin wt + + g sin(Q + w)t for (a) m0 = 0.5, A = 0.0 and (b) m0 = 1.0, A = 0.0, (c) m0 = 0.5, A = 0.5 and (d) m0 = 1.0, A = 0.5. Other parameter values of the system are fixed as w0 = 1, 3 = 1, a = 0.2, w = 1.5, Q = 30.0 and f = 0.05
4.2. VR in a type-2 single-well potential
In this section we consider the system (2.8) with a type-2 single-well potential of the form shown in Fig. 5 where w2, / < 0, 2 <m0 < 3 and 1 < A < 2. In this type, the sign of both Cl and C2 can be changed by varying g or m0 or A. The effective potential can change into other
Fig. 5. (a) The system potential (2.9) for / = -1, w2 = 1, A =1 and m0 = 1, 2, 2.5, 3, 4. (b) The system potential (2.9) for a = 0.2, / = -1, w2 = 1, m0 = 2 and A = 0.5, 1, 2
Fig. 6. Shape of the effective potential Veff for two values of (a) g = 40 and (b) g = 100. The other parameter values are fixed as a = 0.2, / = -1, w2 = 1, m0 = 2, A = 0.5, 1, 2
forms. Figure 6 depicts Veff for three values of A with g = 40 and g = 100. The values of other parameters of the system are w2 = 1, 3 = -1, a = 0.2, w = 1.5, Q = 30 and m0 = 0.5. By varying the strength of mass nonlinearity A for g = 40 and g = 100, the shape and depth of the effective potential can be altered from single-well to inverted single-well or double-well potential, which is clearly seen in Figs. 6a and 6b.
Fig. 7. Variation of the response amplitude Q with g for three values of (a) mass amplitude m0 with A = = 0, (b) mass amplitude m0 with A =1 and (c) mass spatial nonlinearity parameter A with m0 = 2. Other parameter values of the system are fixed as w(2 = 1, / = -1, a = 0.2, f = 0.05, w = 1.5, Q = 15. The continuous lines represent the analytically computed response amplitude Q from Eq. (3.7), while the dotted lines represent the numerically computed response amplitude Q from the Eq. (2.8) using Eq. (4.1)
We embark on our exploration of the vibrational resonance (VR) phenomenon within the system represented by Eq. (2.8), commencing with the consideration of a constant mass scenario where m(x) = m0 and A = 0. The outcomes of this investigation are portrayed in Fig. 7a for
three distinct values of m0 (= 2, 2.5, 3). The solid lines represent analytically computed Q-values derived from Eq. (3.7), while the dotted lines depict the corresponding numerical values computed using Eq. (2.8) with the aid of Eq. (4.1). Clearly, the theoretical Q-values and numerical Q-values align closely. However, as the value of m0 increases, the positions of the resonance peaks further shift away from the origin. For m0 (= 2.0, 2.5, 3.0), the response amplitude Q reaches its maximum at g = 252.4, 291.5, 498.7, respectively. It's evident that gVR increases as m0 is elevated, a trend that is evident in Fig. 7a. Subsequently, we delve into the influence of the position-dependent mass (PDM) parameters (m0, X) on the observed resonances, as presented in Figs. 7b and 7c. Figure 7b illustrates the dependence of the response amplitude Q on g for three different values of m0 (= 2, 2.5, 3), while X is fixed at 1. Across all values of m0, resonances exhibit significant enhancement as m0 increases, accompanied by a pronounced shift in the value of g at which Q reaches its maximum. Figure 7c provides insight into the relationship between the response amplitude Q and the high-frequency amplitude g, considering varying values of X, while keeping m0 = 2.0, a = 0.2, w = 1.5, and Q = 15.0 constant. This figure illustrates that the shape of the response curve and the maximum response amplitude Qmax are contingent upon the value of X. Additionally, the positions of the resonance peaks shift closer to the origin as X increases.
4.3. VR in a type-3 single-well potential
In this section, we proceed to verify the existence of VR in the presence of a type-3 singlewell (double-hump single-well) potential of the form shown in Fig. 8. We choose mass parameter regimes within which the system potential is a type-3 single-well potential, so that 0 <m0 < 2 and 0 < X < 1 for = 1 and 3 = —1. First, we analyze the existence of VR in the PDM-Duffing oscillator (Eq. (2.8)) with a constant mass, that is, m(x) = m0 and X = 0. Then we extend it to the system in which X = 0 corresponding to the system with variable mass like m(x) = = 1 . Figures 9a and 9b represent the variation in Q depending on g for Q = 30, uj = 0.75, X = 0, 0.5 with w"2 = 1, / = —1 and a = 0.2. From Fig. 9a we see that single peaks appear with significant enhancement for m0 (= 0.5, 1, 1.5) with X = 0. The shape of the resonance curve, the maximum response amplitude Qmax and gVR all depend on m0. The occurrence of VR is demonstrated in Fig. 9b for the system (2.8) with X = 0.5 by varying g for three values of m0 = 0.5, 1.0, 1.5. The observed single resonances are typical of VR induced by the values of mass amplitude m0. The value of the mass nonlinearity strength X determines the occurrence of single resonances with m0 = 0.5 as shown in Fig. 9c. What is important is that, for the variable mass amplitude m0 (X = 0), resonance occurs at lower values of g, in contrast to the scenario of constant mass where resonance emerges at higher g values.
0.3
^0.0
U=i (a)/
\ —'Win = * A ■''A -2 r\j /-/ * < f~\ * / \ 1 '
■1 / j ; h-rn0 = 0.5 '» : ' '
i-J-J—-m0 =i.s 1; i' ■
-2
0
x
H
0.50r
0.25
0.00
-0.25_2
m0=i= 1 -A = 2 -A = 1.B ib)
1 ' ,' A = 0.5—'
Fig. 8. (a) The system potential (2.9) for / = —1, w2 = 1, X =1 and m0 = 0.5, 1, 1.5. (b) The system potential (2.9) for / = —1, w"2 = 1, m0 = 1 and X = 0.5, 1, 1.5, 2
Fig. 9. Variation of the response amplitude Q with g (a) for three values of mass amplitude m0 with A = = 0.0, w = 1.5, Q = 15, (b) for three values of m0 with A = 0.5, w = 1.5, Q = 21 and (c) for three values of A with m0 = 0.5, w = 1.5, Q = 21. Other parameter values of the system are fixed as w0 = 1, ¡3 = — 1, a = 0.2, f = 0.05. The continuous lines represent the analytically computed response amplitude Q from Eq. (3.7), while the dotted lines represent the numerically computed response amplitude Q from the Eq. (2.8) using Eq. (4.1)
5. Single-well system with two high-frequency signals
In the preceding discussions we showed clearly that the VR phenomenon can occur in the single-well PDM-Duffing oscillator with one high-frequency signal. Now we consider the PDM-Duffing oscillator system subjected to the amplitude-modulated (AM) periodic signal, that is, for the system governed by (Eq. (2.8)), then the equation for X and ^ with the additional signal g sin(Q — w)t. In this case,
0 = " sin(Q + w)i - sin(Q - w)i (5.1)
w2 = ni +3n2 + 5%X*4 (5.2)
and Q is given by Eq. (3.7). It becomes evident that the theoretical estimation of Q markedly differs from the numerically computed Q in this case. As a result, the analytical expression for Q derived for the situation with a single high-frequency signal is no longer applicable when dealing with a system driven by two high-frequency signals. Therefore, we resort to numerical integration of the system described by Eq. (2.8) under the influence of two high-frequency signals. This approach yields a multitude of interesting and significant outcomes regarding VR as induced by amplitude-modulated signals, particularly in contrast to the system driven by a single high-frequency signal. To illustrate this, we first consider the type-1 single-well potential (Fig. 1) system with constant mass (A = 0) and can compare Fig. 2a with Fig. 10a (curve 1). When only one high-frequency signal is present, Q(w) decays continuously as g increases beyond gVR (at which resonance occurs). Here only one resonance peak is possible, which is clearly shown in Fig. 2a for w = 0.5 and A = 0. In Fig. 10a corresponding to the system subjected to two high-frequency signals, Q(w) does not decrease continuously beyond the first resonance peak. Q(w) is the maximum at more than one value of g. Further, Q(uj) » ¿b not only at the first resonance, but also over a wide range of values of g. That is, an enhanced response amplitude at the low-frequency w can be achieved by applying two high-frequency signals. Similarly, the response amplitude curves for type-2 and type-3 single-well potentials (Figs. 5 and 8) are presented in Figs. 10b and 10c, respectively, providing further insight into the ramifications of employing amplitude-modulated signals within these contexts.
Q
j
3.0
Q 1.5
A = 0
(b) 3
A = 0
(c)
:.3
0 200 400 600 9
Q 1
0.
it • ij ■
o ,J ■
2 ,1 .
800 1600 "0 200 400 600 800 9 9
Fig. 10. Response amplitude curves for the system (2.8) with constant mass (X = 0) for (a) type-1 single-well potential (Fig. 1). The values of m0 for the curves 1-3 are 0.5, 1.0 and 1.5, respectively, and w = 1.5, Q = 15, 3 =1. (b) type-2 single-well potential (Fig. 5) with the values of m0 for the curves 1-3 are 2.2, 2.5 and 2.8, respectively, and w = 0.75, Q = 15, 3 = —1. (c) type-3 single-well potential (Fig. 8) with the values of m0 for the curves 1-3 are 0.5, 1.0 and 1.5, respectively, and w = 0.75, Q = 15, / = —1. The other parameter values of the system are fixed as w2 = 1, f = 0.05
20 x 0
-20,
m0 0.5. A 0.0 „,
V
— - "«I J* £ , ¡wiMili «
M
25
x 0
mm
0
600 9
1200 250
m0 = 1.0, A = 0.0 (by
i
600 9
1200
Fig. 11. Bifurcation diagram of PDM-Duffing oscillator with a type-1 single-well potential for (a) m0 = = 0.5, X = 0.0 and (b) m0 = 1.0, X = 0.0. Other parameter values of the system are fixed as w2 = 1, 3 =1, d = 0.2, w = 0.5, Q = 15.0 and f = 0.05
We now proceed to explore the diverse bifurcation behaviors exhibited by the system described by Eq. (2.8) when subjected to two high-frequency signals. In Fig. 11, the presented bifurcation diagrams portray the amplitude of oscillations within the Poincare cross-section as a function of the high-frequency signal amplitude (g). The analysis is conducted for two distinct values of m0 (= 0.5, 1.0). Within these figures, one can discern both periodic and chaotic oscillations, each demonstrating distinct visual patterns. Figure 11a illustrates the scenario where m0 = 0.5. For values of g lower than 399.96, the system behavior primarily exhibits periodic motion characterized by period-T behavior. As the value of g is gradually increased beyond 399.96, the system transitions into chaotic motion. However, it's important to note that short intervals of periodicity persist even as g exceeds 399.96. Contrastingly, the bifurcation diagram depicted in Fig. 11b corresponds to the case where m0 = 1.0. Here, the dynamics are reversed compared to Fig. 11a. Larger values of g lead to periodic motion, while smaller values of g induce chaotic behavior. This intriguing behavior indicates that the system can be utilized either as a chaos generator or as a means to suppress the chaotic dynamics present within the system (2.8).
In addition by activating the mass spatial nonlinearity (X), we numerically analyze the effect of three types of single-well potential in the system (2.8) on the observed resonances by varying the high-frequency amplitude g of the AM signal. The results are presented in Fig. 12. For the type-1 single-well potential, the variation of the response amplitude Q with g for three values of m0 (= 0.5, 1.0, 1.5) and X = 0.1 is presented in Fig. 12a. Resonances with a single peak can be seen for all the values of m0 with different Qmax. Notably, as the mass amplitude m0
5.0 Q2.5 0.0,
A=01 3: ..(a)
2 !\ ! \
0
100 200
9
Q
0.3y
0.2
1 .
0.1
0.0„
A = 1.0 2( ; 3 (C) ¡1
1 11 • * 1 1 ■ . 1 I- ■
m„ = 2.0 2, ,1 (d) 0.2i
3 Q 0.1
* ) l
„ 0.0.
80
100 200 9
3 (e)
0 50 100 150 9
0 75 150 9
Fig. 12. Variation of the response amplitude Q with g for few values of m0 and A in the system with two high-frequency signals. (a), (b) type-1 single-well potential, (c), (d) type-2 single-well potential and (e), (f) type-3 single-well potential. The values of m0 for the curves 1-3 are 0.5, 1.0, 1.5 and the values of A for the curves 1-3 are 0.2, 0.3 and 0.5
increases, both the position and width of the resonances grow, as clearly depicted in Fig. 12a. The influence of the mass spatial nonlinearity A on the response amplitude Q is depicted in Fig. 12b. Within this framework, the system exhibits two resonances for A = 0.2, 0.3, while a single resonance is observed for A = 0.5, with m0 = 0.5. Similarly, for the type-2 and type-3 single-well potentials, the relationship between the response amplitude Q and g is explored for various combinations of m0 and A, as shown in Figs. 12c, 12d and Figs. 12e, 12f, respectively. Once again, the system reveals a mix of single and multiple resonances when subjected to the influence of type-2 and type-3 single-well potentials, as depicted in the context of Eq. (2.8).
5.1. Hysteresis and jump phenomenon
In this section, we delve into the analysis of the hysteresis and jump phenomena within the context of the system described by Eq. (2.8), both under the influence of a single high-frequency signal and two high-frequency signals. To illustrate this phenomenon, we focus our attention on the type-1 single-well potential. We initiate the investigation by considering the type-1 singlewell system solely under the influence of a single high-frequency signal (g sin(Q + w)t). The parameter values for the system (2.8) are fixed as wq = 1, / = 1, d = 0.2, f = 0.05, m0 = = 0.5, A = 0, w = 1.5, and Q = 30.0. Figure 13a depicts the variation of Q as g is altered both in the forward and reverse directions. The continuous curve illustrates the outcome of varying the control parameter g in the forward direction, while the dashed curve reflects the response from altering g in the reverse direction. Notably, hysteresis and a jump phenomenon become evident in this depiction. As g is modified in the forward and reverse directions, Q follows divergent trajectories, exemplifying the hysteresis effect and a jump phenomenon. This intriguing phenomenon extends to the scenario of the type-1 single-well system under the influence of two high-frequency signals, as portrayed in Fig. 13b. Similarly to the previous case, hysteresis and a jump phenomenon are discernible in the response amplitude curve as g is varied both in the forward and reverse directions. It's worth noting that this phenomenon is also observable within the contexts of the type-2 and type-3 single-well potentials, thereby illustrating its general
Fig. 13. Response amplitude curves obtained by varying the control parameter g from 0 to 1200 (continuous curve) and from 1200 to 0 (dashed curve) for the type-1 single-well PDM-Duffing oscillator system with (a) one high-frequency signal and (b) two high-frequency signals. Other parameter values of the system are fixed as w2 = 1, 3 = 1, d = 0.2, m0 = 0.5, X = 0.0, w = 1.5, Q = 30.0 and f = 0.05
Fig. 14. Phase portraits of the system (2.8) for four values of g chosen in Fig. 13b. The system is a type-1 single-well PDM-Duffing oscillator driven by the AM signal. The parameter values of the system are as in Fig. 13
applicability across different potential forms. To gain deeper insights into the dynamics of the type-1 single-well system subjected to the AM signal, we turn our attention to the alteration in the phase diagram of the system. This exploration is showcased in Fig. 14, where phase portraits are presented for four distinct values of g within the range g G [754.73, 1200]. For g = 800.96, the trajectory within the x — X plane exhibits a distinctive structure comprising three distinct segments. Notably, one portion of the trajectory encircles both solid circles. Concurrently, two other segments of the trajectory enclose solely the left solid circle and the right solid circle, respectively. This configuration is reminiscent of an orbit that navigates around two equilibrium states. The separation between these two points diminishes (leading to a decrease in Q as well) as the value of g surpasses 800.96. This trend is conspicuously illustrated in Figs. 14a-14c. Upon reaching g = 1005.56, as depicted in Fig. 14d, the two previously separate points converge at the origin. Subsequently, the orbit adopts a pattern akin to that observed within a single-well potential, revolving around the origin. At this particular value of g, Q attains a local minimum. This depiction of evolving phase portraits and the associated changes in the behavior of the system at varying values of g offers valuable insights into the intricate dynamics of the type-1 single-well system under the influence of the AM signal.
6. Conclusion
In VR studies, nonlinear systems are typically driven by a weak periodic signal and further exposed to a high-frequency signal with Q » w. However, less attention has been given to diverse signal types, including amplitude, frequency, and pulse modulated signals. Furthermore, the majority of previous VR investigations have been conducted on nonlinear systems with
constant mass, and only a limited number of studies have been reported for systems with position-dependent mass. In the present work, we have analyzed the occurrence of vibrational resonance in the position-dependent mass PDM-Duffing oscillator system with three forms of single-well potential driven by the amplitude-modulated (AM) signal. The effective potential of the system allowed us to obtain an approximate theoretical expression for the response amplitude Q at the low-frequency w. By analyzing the analytical expression of Q, we determined the values of g at which vibrational resonance occurs for the PDM and AM parameters.
The amplitude-modulated signal can be reformulated as F(t) = f sin wt + g sin(Q + w)t + + g sin(Q — w)t. We compared the VR induced by the two high-frequency signals with that of the single high-frequency signal. The VR phenomenon triggered by two high-frequency signals has many more interesting features than a single high-frequency signal. Specifically, the system exhibits multiple resonance peaks and maintains nondecaying behavior of Q(w) even for large values of the control parameter g. Additionally, the bubble structure and reverse period-doubling bifurcation are observed in the system with a single high-frequency signal, but are not present in the system with two high-frequency signals. The use of the amplitude-modulated signal has shown that the various complex phenomena obtained can be controlled. Future research will analyze VR in the PDM-Duffing oscillator system with a double-well potential driven by an AM signal.
Acknowledgments
The authors are thankful to the referee for the useful suggestions and comments which improved the presentation of the present paper.
Conflict of interest
The authors declare that they have no conflict of interest.
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