Pramana – J. Phys. (2019) 93:43 © Indian Academy of Sciences https://doi.org/10.1007/s12043-019-1802-7

Multiple vibrational and antiresonance in a coupled anharmonic oscillator under monochromatic excitation

M PAUL ASIR, A JEEVAREKHA and P PHILOMINATHAN ∗

Theoretical Research Laboratory, Department of , AVVM Sri Pushpam College, Poondi, Thanjavur 613 503, India ∗Corresponding author. E-mail: [email protected]

MS received 26 October 2018; revised 22 January 2019; accepted 26 February 2019; published online 21 June 2019

Abstract. We examine the existence of multiple vibrational resonance (VR) and antiresonance in two coupled overdamped anharmonic oscillators where each one is individually driven by a monochromatic sinusoidal signal with widely separated (  ω). In contemporary VR, superposed periodic waves are adopted to infer resonance, but herein we employ non-superposed periodic waves to acquire the elevated response. We study two coupling schemes namely, unidirectional and bidirectional, to substantiate the occurrence of multiple VR and antiresonance. Such occurrences have been shown and the results were ascertained with supportive numerical and experimental outcomes. We also illustrate the effect of coupling strength on the observed phenomenon.

Keywords. Coupled anharmonic oscillators; multiple vibrational resonance; antiresonance; mean residence time.

PACS Nos 05.45.−a; 05.90.+m

1. Introduction extensive applications in communication and engineer- ing. The development of atmospheric disturbance has Resonance, the maximum displacement in the magni- also been described in terms of VR [15]. VR concern- tude spectrum of a system at particular frequencies, ing the nonlinear response of the system was reported found a spotlight in the arena of physics, engineering and recently [16], where symmetry-breaking effect due to communication technologies. Among the several known high- component was manifested as the non- resonance phenomena, stochastic resonance (SR) [1,2] linear response. attracted a great deal of attention among the research Further, non-autonomous coupled dynamical sys- community as it exploits the productive role of noise tems are pervasive in nature, and unravelling the in the amplification of a weak input signal. Recently, physical phenomena associated with them becomes a the ‘deterministic’ approach to describe and predict the non-trivial problem. Several intriguing observations, SR phenomenon, based on replacing noise by high- including synchronisation, death, chimera frequency excitations, has been studied [3]. As noise states and so on could be witnessed in such sys- is inexorable and ubiquitous, SR has drawn interdisci- tems [17–19]. Among them, the resonant dynamics plinary attention from climate modelling [4] and signal of coupled nonlinear systems has been a subject of processing [5] to neurophysiology [6–8]. After a brief extensive research in recent times due to its wide interval, Landa and McClintock [9] reported a homolo- potential applications. To cite a few, stronger cou- gous phenomenon, namely vibrational resonance (VR) pling strength in globally coupled Hodgkin–Huxley in a bistable system, where the amplification of a weak neurons can induce coherence resonance and synchro- periodic component can be stimulated in the presence of nisation [20], theoretical and experimental verification an optimal high-frequency signal. Following the obser- of fano in coupled plasmonic system [21], vation of VR in the bistable system, the study has been chemical synaptic coupling in three Fitzhugh–Nagumo extended to monostable systems [10], excitable systems neurons shows enhanced response under the influence of [11], time-delayed systems [12,13], maps [14] and so on, high-frequency driving [22], one-way coupled bistable as structures employing widely varying frequencies find system attributed to high- and low-frequency displays 43 Page 2 of 12 Pramana – J. Phys. (2019) 93:43 improved signal transmission [23]. In particular, VR and SR have been ascertained in nonlinearly cou- pled overdamped anharmonic oscillators driven by Oscillator 1 Oscillator 2 amplitude-modulated periodic force and noise [24, driven driven ω Ω 25]. In addition, multiple VR with extreme appli- by fcos( t) by gcos( t) cation facets in signal transmission and communi- cation technologies have been reported in discrete systems [26], one-way coupled n-Duffing oscillator Figure 1. Case 1: (Blue line) One-way coupling scheme of the oscillators. Case 2: (Red line) Mutually interacting oscilla- [27], quintic oscillator with monostable potential [28] ω and in a neuron model induced by dynamics of Na+ tors. The oscillators are driven by harmonic signals ( f cos t and g cos t) of f , g and frequencies ω, . and K+ ions [29]. On the other hand, antiresonance leads to umpteen number of real practical applica- tions including designs of chemotherapeutic protocols and experimental support. Finally, conclusions are given [30], robust dynamic model updating [31], desyn- in §4. An effort has been made to derive an empirical chronising undesired [32], etc. have been analytical expression for response amplitude Q and crit- reported. Specifically, Zhang et al [33] indicate the ical high-frequency amplitude gmax for both coupling antiresonance behaviour as mode localisation in the schemes and are presented in appendices. frequency domain of weakly coupled micromechanical resonator sensors. As an added credit, antiresonance has been used for minimising undesired vibrations of cer- 2. Unidirectionally coupled oscillators tain parts of the system in aerospace and mechanical engineering. Unidirectionally coupled structures do play a key role A sweeping generalisation made out of the aforemen- in potential applications ranging from the construction tioned studies is that either VR or SR is observed in of electronic sensors [35] to modelling of repressilators nonlinear systems when excited by superposed bihar- [36]. In the present case, we consider a system of uni- monic signals or a periodic signal combined with noise. directionally coupled anharmonic oscillators in which But, a question can be raised whether VR can be induced the first oscillator is modulated by a high-frequency sig- in a coupled system, when both of them are driven by nal (g cos(t)) and the other with the low-frequency distinct and unique frequencies (figure 1). The present periodic forcing component ( f cos(ωt)). The dynam- study aims at addressing the constraints such as (i) ical equations of unidirectionally coupled oscillators restriction on the modulation of additional periodic sig- are nals in coupled oscillating systems and (ii) denial of x˙ = x − x3 + g cos(t), (1) accessing the individual nodes or subsystems in a hybrid ˙ = − 3 + γ + (ω ), or complex network. Motivated by these aspects, we had y y y x f cos t (2) set the following objectives: where γ is the coupling coefficient that plays a crucial • To investigate the VR and antiresonance phenomena role in response amplitude. To substantiate the pres- in a unidirectionally coupled overdamped oscillator ence of VR and antiresonance in the chosen model, ( ) driven at distinct frequencies with one greater than we have calculated the response of y t from the sine the other (  ω). and cosine components of Fourier amplitude, which are • To probe an interesting system with a bidirectional given by  diffusive coupling to concur single and multiple VR. 2 nT B (ω) = y(t) sin(ωt)dt, (3) s nT Furthermore, a system of linear coupled anharmonic 0 oscillators is capable of imitating the dynamics of 2 nT B (ω) = y(t) cos(ωt)dt, (4) many naturally existing systems, including the dynam- c nT ics of interacting species in a fluctuating environ- 0 ment to find the probability of the survival of species where n is an integer and T = 2π/ω. The response [34]. amplitude is calculated using the relation  The lay-out of the paper is as follows: in §2, a detailed B2 + B2 description of the observation of VR and antiresonance Q = s c , (5) in one-way coupled system is presented with numer- f ical and experimental results. Section 3 discusses the whereas the shift with respect to the input signal −1 detection of multiple VR and antiresonance in a bidi- ( f cos ωt)isgivenbyθ(ω) = tan (Bs(ω)/Bc(ω)).To rectionally coupled system with the necessary numerical analyse the response amplitude in the parametric space Pramana – J. Phys. (2019) 93:43 Page 3 of 12 43 of f and g, we fix the parameters as ω = 500 and (a) (d) = 2000. These frequencies were chosen to provide γ γ convenient experimental concurrence, and the results follow in the next section. By varying f ∈[0, 1] and g ∈[0, 20], one can observe multiple resonance peaks as inferred from figure 2a. It is observed that the coupling strength γ has an effective role in influencing the mag- nitude of Q but none with the number of resonant peaks. γ γ Therefore, is fixed as 1.5. Larger response (Qmax) can (b) γ (e) be obtained at f = g = 0.02 that is measured to be 1.8 and the interval of peaks agrees with the period of low- frequency signal. Figure 2b is plotted by varying ω in the range [200, 1000] and ∈[1400, 2600] for fixed values of f = g = 0.02. The plot displays two promi- ω Ω nent peaks with Qmax measured as 6.98 at ω = 440 and 940 within the chosen spectrum of . The response amplitude is calculated for different ω in the range of (c) (f) [100, 500] as a function of γ ∈ [0.01, 1.2] to study the effect of coupling strength and is shown in figure 2c. The plot depicts multiple VR at regular intervals of ω, but the variation in the magnitude of Q changes abruptly ω γ due to nonlinear . γ The antiresonance frequencies of the system can be spotted by fixing ω = 550, = 2000, f = 0.18 and Figure 2. (a) The response amplitude Q in parametric space scanning g ∈[0, 20]. Figures 2d and 2e show the res- f ∈[0, 1] and g ∈[0, 20] when ω = 500, = 2000 γ = . ω ∈[ , ] onance peak at ω = 500 and antiresonance locus at and 1 5; (b) Q in parametric space 200 1000 , ∈[1400, 1600] when f = 0.02, g = 0.02 and γ = 1.5; ω = 550 for two different coupling strengths γ = 0.1 ω ∈[ , ] γ [ . , . ] . (c) Q as a function of 100 500 and 0 01 1 5 when and 1 5, respectively. The mechanism of antiresonance = 2000, f = g = 0.02; (d) Q for g ∈[0, 20] with can be ascribed as the vigorous energy transfer from f = 0.18, = 2000 for γ = 0.1, resonance peak (black the oscillator driven by high-frequency signal and at solid line) at ω = 500 and antiresonance locus (red solid particular frequencies, the amplitude of the oscillator line) at ω = 550; (e) Q and antiresonance curve for γ = 1.5 driven by low-frequency signal is limited to a mini- and (f) elevation of Q as a function of γ when g = 13.0, = . ω = = mum. The behaviour of linear increase in magnitude f 0 18, 500 and 2000. of the response amplitude as a function of γ can be affirmed from figure 2f, due to the absence of to and dx Rf C R = g cos(t) 1 fro vibrational energy transfer, as limited by unidirec- 1 4 dt R tional coupling. The rate of change in Q is measured 2 by fitting a straight line to the data and quantified as Rf 1 Rf 1 − x3α2 + x , (6) 0.015. R1 R3

dy Rf 2 C2 R9 = f cos(ωt) 2.1 Experimental results dt R6

Rf 2 Rf 2 Rf 2 To ascertain the numerical results, an experimental hard- − y3α2 + y + x , (7) R R R ware implementation of the chosen system has been 5 8 7 realised. The circuit comprises four dual BIFET TL082 where α is the multiplier constant. To design the cir- JN operational amplifiers, four AD633 JN multipliers cuit with available off-shelf components, the values are (with 10 times scaling factor), two capacitors, eleven fixed as R1 = R5 = 1k, R f 1 = R f 2 = R2 = linear resistors and two ATTEN function generators R6 = R3 = R8 = R4 = R9 = 100 k, R7 = 1M to generate low- and high-frequency harmonic signals and C1 = C2 = 10 μF. The coupling coefficient (γ ) (figure 3). Functionally, two pairs of operational ampli- is determined by the ratio Rf 2/R7 and is fixed at 0.1 fiers ((OA1,OA3) and (OA2,OA4)) act as the adder and to compare the results with the numerical outcome. the integrator, respectively. The governing dynamical Following the procedure illustrated in refs [37,38], the equations due to Kirchhoff’s laws are as follows: capture of response amplitude has been achieved. The 43 Page 4 of 12 Pramana – J. Phys. (2019) 93:43

Vcc Vcc GND AD633GND GND AD633 -Vcc -Vcc GND R1 GND

R2 Rf1 C1 gcosΩt R3 − R4 − OA1 x + A OA2 +

Vcc Vcc GND AD633 GND GND AD633 -Vcc -Vcc GND R5 GND

R6 Rf2 C2 fcosωt R7 R8 − R9 − OA3 y + B OA4 +

Figure 3. Analogue simulation of one-way coupled overdamped oscillators.

(a) (b) bidirectional coupling scheme enables us to understand 0.858 ω/2π=80 HZ 0.1388 ω/2π=87.5 HZ much about many natural processes, including neural

0.855 transmissions [39], interspecies competition, population Q 0.1384 dynamics of the species [34] and so on. Despite vari- 0.852 ous coupling schemes, bidirectional diffusive coupling 0.138 is pervasive, and that can facilitate synchronisation and 0.849 0 5 10 15 20 0 5 10 15 20 phase coherence, and enhance the detection of weak g g periodic signals. The occurrence of VR in the chosen model can be assisted by the amplitude of both high- Figure 4. Numerical (black solid line) and experimental frequency and low-frequency signals at an adjacent node (red circles) values of Q for g ∈[0, 20] V with fixed that make it distinct from the classical VR. = / π = values of f 180 mVp, 2 318 Hz: (a)VR Let us consider the following mutually coupled driven peak at ω/2π = 80 Hz and (b) antiresonance locus at ω/2π = 87.5Hz. system: x˙ = x − x3 + γ(x − y) + f cos(ωt), (8) ω/ π / π values of 2 , 2 and f are fixed as 80, 318 Hz y˙ = y − y3 + γ(y − x) + g cos(t). (9) and 180 mVp, respectively. The response amplitude of the voltage across capacitor C2 is measured by vary- The potential of the system in the absence of external ing g from 0 to 20 Vp, VR is observed and figure 4a driving is given by compares the numerically and experimentally obtained x2 x4 y2 y4 VR peaks. By fixing the rest of the parameters and U(x, y) =− + − + altering ω/2π = 87.5 Hz, we observed antiresonance 2 4 2 4 phenomenon. The locus is assessed with numerical pre- x2 y2 +γ xy − − . (10) diction and found to be in good agreement as shown in 2 2 figure 4b. Depending on the sign of the coefficients of x and y and coupling strength γ , the potential can have wells, 3. Mutually coupled oscillators humps or a combination of both. Here we are interested in potential with four and two wells explicitly relying Next, we consider a system of mutually coupled over- on γ .Forγ = 0.1, eq. (10) unveils symmetric four-well damped oscillators, each bearing an exclusive mono- potential with a barrier height of 0.2 in turn equivalent chromatic excitation ( f cos(ωt) and g cos(t)). The to fc as evinced from figure 5a, whereas for γ = 1.0 Pramana – J. Phys. (2019) 93:43 Page 5 of 12 43

(a) γ=0.1 a(i) 1 (a) 4 4 3 3 2 x 1 2 0 U(x,y) y 0 0 1 -2 -1 1 -1 γ 0 -1 0 0 =0.1 y 1 -2 -2 x 2 -2 0 (b) 1 γ=1.0 (b) b(i) 30 30 2 x 0 20 20 -1 10 y 0 γ=1.0 U(x,y) 10 0 -2 0 2 4 6 8 10 12 2 -2 0 -2 0 0 2 -2 g y x -2 0 2 x Figure 6. Poincaré SOS of x(t) by setting g as a control parameter in the range g ∈[0, 12] when f = 1.0, ω = 0.1 Figure 5. (a) Three-dimensional view of symmetric four- and = 3.0: (a)forγ = 0.1 and (b)forγ = 1.0. well potential for γ = 0.1 and (b) transformed two-well potential for γ = 1.0. Parts a(i) and b(i) show the respective contour plots (colour box denotes the height of the potential wells). increased for higher coupling strength (γ = 1.0) as a consequence of an increase in the depth of the potential wells as seen by comparing figures 5a and 5b. Moreover, the potential shrinks to a double-well potential due to period-doubling bifurcation or chaos is exempted in the the disappearance of barriers between U++ and U−+ chosen parametric space, because of the stability of the and its symmetric potential (U−− and U+−) (figure 5b). fixed points in this region. Even though the amplitude of One can notice that the barrier height is increased to the orbits differs as a function of coupling strength, the fc = 1.0, when γ is increased. The model has nine time period of remains constant, equivalent equilibrium points, out of which (x1, y1) = (0, 0) and to T = 2π/ω. (x2,3, y2,3) = (±1, ±1) for all γ>0. The stability For numerical evaluation of the response amplitude of (x1, y1) remains unaltered. However, the stability of of x(t) in eq. (8), same set of equations that assesses the (x2,3, y2,3) varies from the stable node (γ<1.01) to Fourier amplitude of the signal is considered (eqs (3)– the saddle (from γ = 1.01 onwards). The values and (5)). To start with, the non-monotonic variation of Q is the stability of the rest of the equilibrium points rely on assessed by simultaneous scanning of f ∈[0.01, 0.5] the value of γ . For a set of initial conditions (x0, y0 = and g ∈[0, 12.0] with fixed values of ω = 500, ±1.0, ±1.0), the system settles down to the equilibrium = 2000 and γ = 1.0, as shown in figure 7a. This points x2,3, y2,3, respectively. figure shows four resonant peaks and among them the Figures 6a and 6b represent the Poincaré surface of peak at g = 4.0 is found to be with maximum response, section (SOS) of the variable x(t) sampled at every Qmax = 15.4. Moreover, upon close inspection one 2π/ω with respect to the control parameter g ∈[0, 12] could visualise the presence of multiple antiresonance corresponding to two different coupling strengths γ = loci in the fabric. To track the response of x(t) in the 0.1and1.0, respectively. The other parameters of the parametric space of frequencies, ω ∈[100, 300] and model are kept fixed as f = 1.0, ω = 0.1and = 3.0. ∈[800, 1100], Q is computed by fixing f = 0.05, Figure 6a denotes the confinement of the orbit in the g = 0.1, γ = 1.0 and shown in figure 7b. The potential well U++ for the preferred initial conditions up plot displays consistent alternative peaks and loci at to the critical threshold vibrational amplitude gc = 7.92 overtones found at ω = 180, 280, etc. with average for γ = 0.1. Beyond gc, the oscillation of the orbit Qmax = 0.62. When compared with unidirectional cou- gets bounded in U−+, whereas at the point of a criti- pling (figure 2b), it is found that the interval between cal threshold the orbits of x(t) coexist and experience harmonic frequencies is reduced and thus one could get cross well motion that illustrates the well renowned sig- more number of resonant peaks due to the mutual trans- nature of VR. When γ = 1.0, gc is shifted to a higher fer of energy. Figure 7c is plotted to analyse the influence value (gc = 9.2) due to the increase in potential bar- of coupling strength on Q in the range of γ ∈[0.01, 1.5] rier height as inferred from figure 6b. As the potential for a spectrum of ω ∈[100, 500] when f = 0.05, is a two-well potential for γ = 1.0, the dynamics of g = 0.1and = 2000. One could infer multiple VR the system get bounded within U−+ and U+− wells. peaks associated with nonlinear variations in Q from The amplitude of the periodic orbit in a single well is the plot due to the intrinsic nonlinearity present in the 43 Page 6 of 12 Pramana – J. Phys. (2019) 93:43

(a) (d) γ=1.0 15 15 0.035 Q 10 10 2.34 5 5 Q

0.2 γ=0.1 12 2.32 0.4 8 f 4 0.0345 0 g 0 4 8 12

(b) (e) 3.5 0.038 0.6 0.6 Q 0.036 Q 0.4 0.3 3 0.034 0.2 100 γ=1.0 0.032 1100 2.5 200 1000 900 0 4 8 12 ω 300 Ω 800 g

(c) (f)

3 1.35 3 2 Q 2 1 Q 1.3 1

1.2 1.25 150 0.8 300 0.4 γ 0.4 0.8 1.2 1.6 2 ω 450 γ

Figure 7. (a) The response amplitude Q by scanning f ∈[0.01, 0.5] and g ∈[0, 12] when ω = 500, = 2000 and γ = 1.0; (b) Q in parametric space ω ∈[100, 300], ∈[800, 1100] when f = 0.05, g = 0.1 and γ = 1.0; (c) Q as a function of ω ∈[100, 500] and γ [0.01, 1.5] when = 2000, f = 0.05 and g = 0.1; (d) Q for g ∈[0, 12] with f = 0.05, = 2000 for γ = 0.1, resonance peak at ω = 500 and antiresonance locus at ω = 150; (e) same parameters were chosen as if to compute (d) except with γ = 1.0, triple resonant peak is observed for ω = 500 and a single antiresonant locus for ω = 150 and (f)plotofQ as a function of γ ∈[0.1, 2] for fixed values of g = 8.1, f = 0.1, ω = 500 and = 2000.

model. Higher magnitude of Qmax is observed at one of and unlike unidirectional coupling, the plot resembles a the harmonic frequencies (ω = 380) that can be affirmed cubic polynomial function that is symmetric about the from the figure. inflection point. This behaviour of the system could be In an attempt to determine the antiresonant frequen- manifested from the cubic nonlinearity present in the cies, we have chosen the values of parameters as = model. 2000, f = 0.05 and γ = 0.1. When ω = 150, the model To determine the random distribution of time spans exhibits antiresonance locus as seen from figure 7d. (τMR) spent by the orbit in respective potential wells, For the same set of parameters, but when ω = 500, we have chosen the parameters of the resonance curves the model exhibits resonance peak with Qmax = 2.35. shown in figure 7d. The initial conditions were taken Interestingly, at higher coupling strength γ = 1.0, triple such as the orbit lies in the well U−+ at g = gc,to resonant peaks were identified as shown in figure 7e. We account for the variation in the behaviour of τMR.When found that Qmax is linearly decreasing with increasing g = gc, τMR in U++ is very short and in U−+ it is close g. Nevertheless, a single antiresonant locus is detected to 2π/ω (figure 8a). As g → gmax,thevalueofτMR in irrespective of coupling strength as witnessed from the both the wells are found to be equal to the time period figure. We study the dependence of Qmax on γ by fix- of the low-frequency driving, as evident from figure 8b. ing g = 8.1, f = 0.1, ω = 500 and = 2000, The logarithmic plot of τMR vs. the factor 1/(g − gc) Pramana – J. Phys. (2019) 93:43 Page 7 of 12 43

(a) a(i) 1600 0 1200 −0.5 x 800 −1 frequency 400 −1.5 0 50 100 150 200 250 300 −2 −1.5 −1 −0.5 0 0.5 (b) b(i)

1 200

x 0 100 −1 frequency

0 50 100 150 200 250 300 −1.5 −1 −0.5 0 0.5 1 1.5 (c) c(i) 600 1 400 x 0 200 −1 frequency

0 50 100 150 200 250 300 −1.5 −1 −0.5 0 0.5 1 1.5 time x

Figure 8. Time traces of the trajectory x for a fixed set of parameters fitting the resonance curve of figure 7d: (a)at g = gc = 7.0, (b)atg = gmax = 8.2 and (c)atg = 12. The corresponding subplots depict the distribution of points in respective potential wells.

(a)0.7 (b) 0.48

0.65 0.45 ) 0.6 MR τ

0.55

Log ( 0.42

0.5

0.45 0.39 0 20 40 60 80 100 0 20 40 60 80 100

1/(g-gc) 1/(g-gc)

Figure 9. Logarithmic mean residence time of the orbit: (a)inthewellU++ and (b)inthewellU−+ vs. 1/(g − gc).The arrow mark in the plot indicates τMR at g = gmax. confirms the gradual decrement in mean residence times of the orbit in the active potential wells at g = 12 are beyond gmax in U++, whereas in U−+ it decreases as shown in figure 8c. gc → gmax and then increases to the maximum value at g = gmax (figure 9). When g is increased above gmax, 3.1 Experimental results the value of τMR decreases in both the wells. To reassert, we found that as g increases from gc, τMR in the well Equivalent analogue simulation of eq. (8) is achieved by U−+ remains close to T, whereas in U++ it increases to employing six dual BIFET TL082 JN operational ampli- T until g reaches gmax.Atg = gmax, the time span of the fiers, four AD633 JN multipliers (10 times scaling), two orbit in both the wells is equal to T and then decreases to capacitors, 20 linear resistors and two ATTEN function T/2 beyond gmax. The time trace and spatial distribution generators as shown in figure 10. 43 Page 8 of 12 Pramana – J. Phys. (2019) 93:43

Vcc Vcc GND AD633GND GND AD633 -Vcc -Vcc GND Rf R1 GND 2 R R R 2 f1 6 C1 R fcosωt 3 − OA3 R4 − R7 R5 + OA1 − x + OA2 R + 8

Vcc Vcc GND AD633 GND GND AD633 -Vcc -Vcc GND Rf R9 GND 4

R10 Rf 3 R14 C2 R11 − gcosΩt OA6 R12 − R15 R13 + OA4 − y + OA5 + R16

Figure 10. Analogue simulation of mutually coupled overdamped oscillator.

(a) (c) γ=0.1 0.0351

2.34

0.0348 Q (v)

2.32 γ=0.1 0.0345 0 4 8 12 0 4 8 12

(b) 3.6 (d) 0.039

3.2 0.036

Q (v) 2.8 γ=1.0 0.033 γ=1.0 2.4

0 4 8 12 0 4 8 12 g (volts) g (volts)

Figure 11. Numerical (solid line) and experimental (circles) values of Q for various values of g ∈[0, 12] Vfor f = 50 mVp and /2π = 318 Hz. (a) Single resonance peak for γ = 0.1 and ω/2π = 80 Hz, (b) triple resonant peaks for γ = 1.0 and ω/2π = 80 Hz and (c, d) antiresonance locus for ω/2π = 24 Hz with γ = 0.1 and 1.0, respectively.

On applying Kirchhoff’s laws to the circuit, we obtain dy Rf 3 3 2 Rf 3 Rf 3 C2 R13 = g cos(ωt) − y α + y dt R10 R9 R12 dx Rf 1 3 2 Rf 1 Rf 1 C1 R5 = f cos(ωt) − x α + x dt R2 R1 R4 R R Rf 4 Rf 3 + f 2 ( − ) f 1 , + (y − x) . (12) x y (11) R R R6 R3 14 11 Pramana – J. Phys. (2019) 93:43 Page 9 of 12 43

The values of the resistors and the capacitors are fixed termed as VR. This sort of behaviour is the manifes- as: R1 = R9 = R5 = R13 = 10 k, Rf 2 = Rf 4 = tation of non-monotonic dependence of the model’s R6 = R14 = R8 = R16 = 1k, Rf 1 = Rf 3 = R2 = effective stiffness on the amplitude of high-frequency R4 = R10 = R12 = 100 k, C1 = C2 = 0.01 μF, forcing. Interestingly, we found multiple VRs and anti- and the coupling strength (γ ) between the oscillators resonance loci for the given choice of parameters in is determined by the ratio Rf 1/R3 and Rf 3/R11.Two both coupling schemes due to the multistability of the unit gain differential amplifiers (OA3 and OA6) were chosen model. The response of the low-frequency com- imposed for mutual diffusive coupling. The magnitude ponent is found to be influenced by the high-frequency spectrum of the voltage drop across capacitor C1 is eval- amplitude in a resonant way at harmonic frequencies. uated using the AGILENT (DSO-X-2006) oscilloscope We had effectively probed the influence of coupling spectrum analyser. Initially, the response Q of the signal strength on the resonant interaction of coupled oscil- is traced as a function of g for γ = 0.1 by keeping the lators, and it is found that stronger coupling will lead to values of R3 = R11 = 1M. The control parameters higher response amplitude. Even though unidirectional are fixed as ω/2π = 80 Hz, /2π = 318 Hz and f = coupling may induce larger responses, the mutual dif- 50 mVp. By altering the value of g from 0.5 to 12 Vp,we fusive coupling increases the probability of occurrence obtained resonance peak with gmax at 8.0 Vp. Figure 11a of resonance in the model. In other words, bidirectional compares the response of the low-frequency signal diffusive coupling enhanced the interaction between the with numerically simulated results and the outcomes coupled oscillators ensuring more spans of entrainment agree with each other. With the same set of parame- domains. Moreover, it has been detected that the intra- tersandbyreducingω/2π = 24 Hz, we obtain the well motion between the potential wells is assimilated antiresonant locus that can be affirmed from figure 11c. as the mechanism of VR in mutual coupling. Upon cal- Further, we are interested in larger coupling strength, culating the mean residence time spent by an orbit in say γ = 1.0, which can be achieved by fixing the val- active potential wells, we found that at g = gmax,the ues of R3 = R11 = 100 k. By following the same value of τMR is equal to integer multiples of the time procedure, we obtained three consecutive VR peaks period of the low-frequency driving (T) in mutually cou- with enhanced response when compared to γ = 0.1. pled oscillators. Antiresonance locus as a counterpart of Figure 11b portrays the perfect match between numer- resonance peaks can be used to identify the symmetry of ically simulated and experimentally obtained multiple the coupled oscillators and the spot of perturbations in VR peaks. However, a single antiresonant locus can be the system. Critically, antiresonance is induced in the found experimentally at ω/2π = 24 Hz and figure 11d model as a consequence of the excessive vibrational verifies the correlation of the values between experiment energy transfer to an adjacent oscillator which reduces and numerics. Moreover, from experimental heuris- the amplitude to a minimum value at particular frequen- tics, we found that the accuracy of the experimental cies. The experimental substantiation of the remarked outcome enhances when the measure of Q is pro- phenomenon has also been accomplished by means of nounced, that is when Q is measured in a scale of a analogue simulations and the results are in good agree- few volts. ment with numerical computations. Since interacting dynamical systems are ubiquitous in nature, we believe that the results of the present investigation can find significant applications in communication technologies, 4. Conclusion neuronal dynamics and mechanical engineering.

Generally, in VR, biharmonic signals are deliberately modulated to evince the resonance phenomenon. On the Acknowledgement contrary, here, efforts were made to prove that no super- position of signals is required to achieve non-monotonic A Jeevarekha thanks the Department of Science and variation in the system’s response. Rather, interacting Technology for the support provided in the form of DST oscillators can induce resonance in one of the subsys- INSPIRE Fellowship. tems in accordance with the high-frequency amplitude variation. To establish the observation, we had chosen two coupled anharmonic oscillators, each of which is Appendix A. Unidirectionally coupled oscillators— driven by distinct frequencies employing unidirectional Theoretical approach and bidirectional coupling. We had detected the non- monotonic variation in the system’s response driven We have made an attempt to provide an empirical solu- by a low-frequency signal in both cases, classically tion to the considered models. While solving eq. (1), the 43 Page 10 of 12 Pramana – J. Phys. (2019) 93:43 solution for x˙ is determined to be Theory g cos(t − θ ) 12 x =  1 , (A1) Nmerics 2 (1 + ) 9 where θ = tan−1(). Upon substituting the value of x, 1 Q 6 the standard method of variable decomposition y(t) = ( ,ω )+ψ( , ) ( ,ω ) Y t t t t is applied to eq. (2), where Y t t 3 and ψ(t,t) correspond to slow and fast moving com- ponents of the response, respectively [40]. Employing 0 the decomposition method and averaging the fast mov- 0 1 2 3 ing components, one gets g Y˙ = Y (1 − 3ψ¯ 2) − Y 3 + f cos(ωt), (A2) Figure 12. Comparison of numerical and analytical values = . ω = . = . ψ˙ = g cos(t) (A3) of Q for fixed values of f 0 05, 0 1 and 0 5 1 √ when the coupling strength γ = 1.0. 2 ¯ 2 with g1 = γ g/( 1 + ). Considering the value of ψ from eqs (A3), (A2) becomes and phase shift now become Y˙ − Yc + Y 3 = f (ωt), 1 cos (A4) 1 Q =  , where c = 1 − (3g2/22) and it is quite evident (A9) 1 1 ω2 + 2 from the above equation that the shape of the poten- c1 ω tial depends on the value of g, γ and .Whenc1 > 0, −1 θ2 = tan . (A10) a double-well potential is obtained with its√ maxima and c1 ∗ = ∗ =± minima located at Y1 0andY2,3 c1, respec- < The response amplitude is calculated using eq. (A9) tively. On the other hand, when c1 0, the shape of the when ω = 0.1, = 0.5, f = 0.05 and the outcomes potential becomes a single well with its minima located ∗ = are compared in figure 12. at Y1 0. Also, from eq. (A4), it is evident that the effective parameters of the system get affected due to resonance. Appendix B. Mutually coupled oscillators— The critical value of g at which the resonance is Theoretical approach observed can be deduced from the following relation:  Further, as the process of obtaining analytical solution 2(2 + ) 2 1 for mutually coupled oscillators is tedious, a highly gmax = . (A5) 3γ 2 approximated method of solution is proposed here. It As the oscillators are one-way coupled, f and ω do starts with the addition of eqs (8) and (9) as < 3 3 not affect gmax. As previously mentioned, if g√ x˙ +˙y = (x + y)−(x + y )+ f cos ωt +g cos t. (B1) ( 2(2 + ))/ γ 2 ± 2 1 3 , there exist two minima in c1 = ( ,ω ) + ψ( , ) = ( ,ω ) + and the deviation from the minima Y ∗ can be calculated Replacing x X t t t t and y Y t t 2,3 ψ(t,t) and on resolving slow and fast components, we by substituting Z = Y −Y2,3 in eq. (A4). Upon substitu- tion of z and using the linearisation, one gets a modified obtain equation as X˙ + Y˙ = (X + Y )[1 − 3ψ¯ 2]−(X 3 + Y 3)+ f cos ωt, z˙ = s1z + f cos ωt, (A6) (B2) ˙ g cos t where s1 = 2c1. From the above equation, the response ψ = . (B3) amplitude and the phase shift are analytically calculated 2 to be After substituting the value of ψ using eq. (B3) and by 1 considering X + Y = A, the linearised form of equa- =  , ˙ Q (A7) tion will be obtained as A = c1 A + f cos ωt. Then the ω2 + 2 4c1 solution A is determined to be − ω (ω − φ) θ = tan 1 . (A8) = f cos t , 2 s A (B4) 1 2 + ω2  c1 Else when g > (22(2 + 1))/3γ 2, there exists ∗ =[( 2/ 2) − ] φ = −1(ω/ ) only one minimum at Y1 and the response amplitude where c1 3g 4 1 and tan c1 . Pramana – J. Phys. (2019) 93:43 Page 11 of 12 43

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