The International Journal of Engineering and Science (IJES)

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The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.

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The International Journal of Engineering and Science (IJES)

  1. 1. The International Journal of EngineeringAnd Science (IJES)||Volume|| 1 ||Issue|| 2 ||Pages|| 81-92 ||2012||ISSN: 2319 – 1813 ISBN: 2319 – 1805 Effect of Depth and Location of Minima of a Double-Well Potential on Vibrational Resonance in a Quintic Oscillator 1, L. Ravisankar, 2,S. Guruparan, 3, S. Jeyakumari, 4,V. Chinnathambi 1,3,4, Department of Physics, Sri K.G.S. A rts College, Srivaikuntam 628 619, Tamilnadu, India 2, Depart ment of Chemistry, Sri K.G.S. Arts Co llege, Srivaikuntam 628 619, Tamilnadu, India--------------------------------------------------------Abstract-----------------------------------------------------------------We analyze the effect of depth and location of minima o f a double-well potential on vibrational resonance in alinearly damped quintic oscillator driven by both low-frequency force f sin t and high-frequency force f sin t with    . The response consists of a slow motion with frequency ω and a fast motion withfrequency  . We obtain an analytical expression for the response amplitude Q at the low-frequency ω. Fromthe analytical exp ression Q , we determine the values of ω and g (denoted as  VR and g VR ) at whichvibrational resonance occurs. The depth and the location of the min ima of the potential well have distinct effecton vibrational resonance. We show that the number of resonances can be altered by varying the depth and thelocation of the minima of the potential wells. A lso we show that the dependence of  VR and g VR by varyingthe above two quantities. The theoretical predictions are found to be in good agreement with the nu mericalresult.Keywords - Vibrational resonance; Quintic oscillator; Double-well potential; Biperiodic force.-------------------------------------------------------------------------------------------------------------------------------------- Date of Submission: 28, November, 2012 Date of Publication: 15, December 2012--------------------------------------------------------------------------------------------------------------------------------------1. Introduction In the last three decades the influence of noise on the dynamics of nonlinear and the chaotic systemswas extensively investigated. A particular interesting example of the effects of the noise within the frame wo rkof signal processing by nonlinear systems is stochastic resonance (SR) ie., the amplification of a weak inputsignal by the concerted actions of noise and the nonlinearity of the system. Recently, a great deal of interest hasbeen shown in research on nonlinear systems that are subjected to both low- and h igh-frequency periodic signalsand the associated resonance is termed as vibrational resonance (VR) [1, 2]. It is important to mention that two-frequency signals are widely applied in many fields such as brain dynamics [3, 4], laser physics [5], acoustics[6], teleco mm-unications [7], physics of the ionosphere [8] etc. The study of occurrence of VR due to abiharmonical external force with two different frequencies  and  with    has received much interest.For examp le, Landa and McClintock [1] have shown the occurrence of resonant behavior with respect to a low-frequency force caused by the high-frequency force in a bistable system. Analytical treat ment for thisresonance phenomenon is proposed by Gitterman [2]. In a double-well Duffing oscillator, Blekh man and Landa[9] found single and double resonances when the amplitude or frequency of the high -frequency modulation isvaried. So far this phenomenon has been studied in a monostable system [10], a mu ltistable system [11],coupled oscillators [12, 13], spatially periodic potential system [14, 15], t ime-delayed systems [16, 17], noiseinduced structure [18], the Fit zHugh-Nagu mo equation [19], asymmetric Duffing oscillator [20], bio logicalnonlinear maps [21] and so on. In this paper we investigate the effect of depth of the potential wells and the distance between thelocation of a min imu m and a local maximu m of the symmet ric double-well potential in a linearly dampedquintic oscillator on vib rational resonance.www.theijes.com The IJES Page 81
  2. 2. Effect Of Depth And Location Of Minima Of A Double-Well Potential…The equation of motion of the linearly damped quintic oscillator driven by two periodic forces is   d x  A0 x  Bx3  Cx5  f cos t  g cos t , (1) x  2where    and the potential of the system in the absence of damping and external force is 1 1 1 V (x )  A0 x 2  Bx 4  Cx6 2 (2) 2 4 6 0  1,   1 and   1 (a) A  B  C  1 2 Fig.1: Shape of the double-well potential V ( x ) for and (b) A  1 , B  1 , C  1 . In the sub plots, the values of 1 (a) and 2 (b) for 2 2 4 2 62 continuous line, dashed line and painted circles are 0.5, 1 and 1.5 respectively.For 0  1, 2 , , A, B, C  0 the potential V (x) is of a symmetric double-well form. When  5 1 0  1,   1,   1 and A  B  C  1 there are equilibriu m points at (0,0) and 2  ,0 . At (0,0),  2     5 1   1 , so this point is a saddle. At   , 0  ,   0.935i , so these points are centres. Recently,  2   Jeyakumari et al [10, 11] analysed the occurrence of VR in the quintic oscillator with sing le-well, double-welland triple well forms of potential. When A=B=C  1 potential has a local maximu m at x0  0 and two *   2  40 2minima at x   * . The depths of the left-and right-wells denoted by DL and DR 2   2  40 2respectively are same and equal to 1[60 p  3p  2p ] / 12 where p  2 2 3 . By varying the 2 *parameter 1 the depths of the two wells can be varied keeping the values of x unaltered. We call the damped 1 1 1system with A=B=C  1 as DS1. We call the damped system with A  ,B  ,C  as DS2 in which 2 2 4 2 6 2   2  40 2case x0  0 where as x   2 and DL  DR  [60 p  3p  2p ] / 12 is independent of * * 2 2 3 2  2 . Thus, by varying  2 the depth of the wells of V (x) can be kept constant while the distance between thelocal maximu m and the minima can be changed. Figures (1a) and (1b) illustrate the effect of 1 and  2 .www.theijes.com The IJES Page 82
  3. 3. Effect Of Depth And Location Of Minima Of A Double-Well Potential… In a very recent work, Rajasekar et al [22] analysed the role of depth of the wells and the distance of aminimu m and local maximu m of the symmetric double-well potential in both underdamped and overdampedDuffing oscillators on vibrational resonance. They obtained the theoretical expression for the responseamp litude and the occurrence of resonances is shown by varying the control parameters , g ,  and  . In thepresent work, we consider the system (1) with 1 and  2 arbitrary, obtain an analytical expression for thevalues of g at wh ich resonance occurs and analyze the effect of depth of the wells and the distance between thelocation of a minimu m and local maximu m of the potential V (x) on resonance. The outline of the paper is as follows, for    the solution of the system (1) consists of a slowmotion X (t ) and a fast motion (t , t ) with frequencies  and  respectively. We obtain the equation ofmotion for the slow mot ion and an approximate analytical exp ression for theresponse amplitude Q of thelow-frequency () output oscillation in section II. Fro m the theoretical expression of Q we obtain thetheoretical expressions for the values of g and  at which resonance occurs and analyse the effect of depth ofthe potential wells in section III and the distance between the location of a minimu m and a local maximu m ofthe symmetric double-well potential in section IV. We show that the number of resonances can be changed byvarying the above two quantities such as 1 and  2 . Finally section V contains conclusion.2. Theoretical Description Of Vibrational Resonance An approximate solution of Eq. (1) for    can be obtained by the method of separation wheresolution is written as a sum of slo w motion X (t ) and fast motion (t , t ) : x(t )  X (t )  (t , t ) (3) We assume that  is a periodic function with period 2  or 2 -period ic function of fast time  t and its mean value with respect to the time τ is given by 2  1  d   0. (4) 2 0Substituting the solution Eq. (3) in Eq. (1) and using Eq. (4), we obtain the following equations of motion for X and  :   X  d X  ( A 0  3B 2  5C 4 ) X 2  (B  10C 2 ) X 3  CX 5  B3  C5  f cos t , (5)   d   A0   3B X 2 (   )  3B X ( 2   2 )   2  B(3  3 )  5C X 4 (   )  10C X 3 ( 2   2 )  10C X 2 (3  3 )  5CX ( 4   4 )  C (5  5 )  10CX 2 3  g cos t . (6)Because  is a fast motion we assume that   , ,  ,  ,  ,  and neglect all the terms in the 2 3 4 5  left-hand-side of Eq. (6) except the term  . This appro ximation called inertial approximation leads to the equation   g cos t the solution of which is given by  g  cos t . (7) 2www.theijes.com The IJES Page 83
  4. 4. Effect Of Depth And Location Of Minima Of A Double-Well Potential…For the  given by Eq. (7) we find g2 3g 4 2  , 3  0,  4  , 5  0. (8) 2 4 84Then Eq. (5) for the slow mot ion becomes   X  d X  C1X  C2 X 3  C X 5  f cos t , (9a )where 3Bg 2 15Cg 4 5Cg 2 C1  A0  2  .(9b ) , C2  B  24 88 4The effective potential corresponding to the slow motion of the system described by the Eq. (9) is 1 1 1 Veff (X )  C1 X 2  C2 X 4  CX 6 . (10) 2 4 6The equilibriu m points about which slow oscillat ions take place can be calculated from Eq. (9). The equilibriu mpoints of Eq. (9) are g iven by 1/ 2    C 2  C 2  4CC1   2 X *  0, * X 2, 3     , 1  2C    1/ 2    -C2  C 2  4CC1  2 * X 4, 5    2C  (11)    The shape, the number of local maxima and minima and their location of the potential V (x) (Eq. (2)) depend onthe parameters 0 ,  and  . For the effective potential (Veff ) these depend also on the parameters g and  . 2Consequently, by varying g or  new equilibriu m states can be created or the number of equilibriu m statescan be reduced.We obtain the equation for the deviation of the slow motion X fro m an equilibriu m point X * . Introducing thechange of variable Y  X  X * in Eq. (9a) we get Y  dY  1  2 2  3 3  4 4  C Y 5  Y Y Y Y  f cos t , (12a )where 1  C 1  3C 2 X *2  5C X *4 , (12b ) 2  3C 2 X *  10C X *3 , (12c ) 3  C 2  10C X *2 , 4  5C X *. (12d )For f  1 and in the limit t   we assume that Y  1 and neglect the nonlinear terms in Eq. (12). Then,the solution of linear version of Eq. (12a) in the limit t   is AL cos(t  ) wherewww.theijes.com The IJES Page 84
  5. 5. Effect Of Depth And Location Of Minima Of A Double-Well Potential… f 2  1 AL ,   tan 1 ( ) (13) [(r  2 )2  d 2 2 ]1 2 2 dand the resonant frequency is r  1 . When the slow motion takes place around the equilib riu m pointX *  0 , then r  C1 .The response amplitude Q is AL 1 Q  . (14) f [(r  2 )2  d 22 ]1 2 23. Effect Of Depth Of The Potential Wells On Vibrational Resonance In this section we analyze the effect of depth of the potential wells on vibrational resonance in thesystem DS1. Fro m the theoretical expression of Q we can determine the values of a control parameter at whichthe vibrational resonance occurs. We can rewrite Eq. (14) as Q  1 S where S  (r  2 )2  d 22 2 (15)and  r is the natural frequency of the linear version of equation of slow motion (Eq. (9)) in the absence of theexternal fo rce f cos t . It is called resonant frequency (of the low-frequency oscillat ion). Moreover,  r isindependent of f ,  and d and depends on the parameters 0 , , , g ,  and 1 . When the control 2parameter g or  or 1 is varied, the occurrence of vibrational resonance is determined by the value of  r .Specifically, as the control parameter g or  or  or 1 varies, the value of  r also varies and a resonanceoccurs if the value of  r is such that the function S is a min imu m. Thus a local minimu m o f S represents aresonance. By finding the minima of S , the value of g VR or VR or VR at which resonance occurs can bedetermined. For examp le d2 d2  VR  r  2 , r  2 . (16) 2 2For fixed values of the parameters, as  varies fro m zero, the response amplitude Q becomes maximu m at  VR given by Eq. (16). Resonance does not occur for the parametric choices for which r2  d 2 2 .When  is varied fro m zero, r remains constant because it is independent of  . In figure (2), VR versusg is plotted for three values of d with 1  0.75 and 2.0. The values of the other parameters are 0  1,     1 and   10 . VR is single valued. Above a certain critical value of d and for certain 2range of fixed values of g resonance cannot occur when  is varied. For examp le, for d  0.5, 1.0, 1.5 and 1  0.75 the resonance will not occur if g [64.03, 70.49],[56.87, 79.81] and [42.54, 91.28] respectively.For 1  2.0 and d  0.5,1.0,1.5 the resonance will not occur if g [65.11, 67.26], [62.96,72.28] andwww.theijes.com The IJES Page 85
  6. 6. Effect Of Depth And Location Of Minima Of A Double-Well Potential… Fig.2: Plot of VR versus g for three different values of d with (a) 1  0.75 and (b) 2  2.0 .The value of the other parameters are 0  1,     1 and   10 . 2[57.95, 78.02] respectively. Analytical exp ression for the width of such nonresonance regime is difficult toobtain because 2  1 is a complicated function of g . In fig. (2), we notice that the nonresonance interval of rg increases with increase of d but it decreases with increase of 1 . We fix the parameters as0 2  1,     1, f  0.05, g  55 and   10. Figures (3a) and (3b ) show Q versus  for d  0.5, 1.0, 1.5with 1  0.75 and 2.0 . Continuous curves represent theoretical result obtained from Eq . (14). Painted circlesrepresent numerically calculated Q . We have calculated numerically the sine and cosine components QS andQC respectively, fro m the equations nT  2 QS  x(t ) sin t dt , (17a ) nT 0 nT  2 QC  x(t ) cos t dt , (17b ) nT 0where T  2  and n is taken as 500 .Then QS  QC 2 2 Q (17c ) fnumerically co mputed Q is in good agreement with the theoretical approximat ion. In fig. (3a), for 1  0.75and g  55 resonsnce occurs for d  0.5 and 1 while for d  1.5 the value of Q decreases continuouslywhen  is varied. For 1  0.75 and g  55 resonance occurs for d  0.5 and the response amplitude Q isfound to be maximu m at   1.02 and 0.75 . For 1  2.0 and g  55 resonance occurs forwww.theijes.com The IJES Page 86
  7. 7. Effect Of Depth And Location Of Minima Of A Double-Well Potential… Fig.3: Response amplitude Q versus  for three values of d with (a) 1  0.75 and (b) 2  2.0 .Continuous curves represent the theoretically calculated Q fro m Eq. (14) with r  C1 , wh ile painted circles represent numerically computed Q fro m Eq. (17). The values of the other parameters are 0  1,     1, f  0.05 , g = 55 and 2   10. d  0.5, 1.0, 1.5 .The response amplitude Q is maximu m at   1.62, 1.5 and 1.35 . Both VR and Q at the resonance decreases with increase in d . The above resonance phenomenon is termed as Vibrational Resonance as it is due to the presence of the high-frequency external periodic force. No w we co mpare the change in the slow motion X (t ) described by the Eq. (9) and the actual motion x(t ) of the system (1). The effect ive potential can change into other forms by varying either g or  . Figure 4 depicts V eff for three values of g with 1  0.75 (fig.4a) and 2.0 (fig.4b). The other parameters are 0  1,     1 and   10 . V eff is a double-well potential for g  55 while it becomes a single-wel l 2 potential fo r g  70 and g  90 . For 1  0.75, d  0.5 and g  55 the value of VR  0.765 and for 1  2.0, d  0.5 and g  55 , VR is 1.25 . The system (1) has two co-existing orbits and the associated (a) (b) Fig.4: Shape of effective potential V eff for 0  1,     1 and   10 with (a) 2 1  0.75 and (b) 1  2 for three values of g . slow motion takes place around the two equilibriu m points X 2,3 . This is shown in fig. (5a) for   0.5, 1.0 and * 1.25 . The corresponding actual motions 2 the system (Eq.(1)) are shown in figs. (5(b)-5(d)). For a wide range V for of 1,     1 eff 0and   10 with (a) 1  0.75 and (b) 1  2 for three valuesof g . Fig.6: Phase portraits of (a) slow motion and (b-d) actual motion of the system (1) for few values of  with g  90 and d  0.5. www.theijes.com The IJES Page 87
  8. 8. Effect Of Depth And Location Of Minima Of A Double-Well Potential… Fig.5: Phase portraits of (a) slow mot ion and (b-d) actual mot ion of the system (1) for few values of  with g  55 and d = 0.5. d  0.5. Fig.6: Phase portraits of (a) slow motion and (b-d) actual motion of the system (1) fo r few values of  with g = 90 and d = 0.5.of values of  including the values of VR , x(t ) is not a cross-well motion, that is not crossing both theequilibria ( x * , y * ( x * ))(0.7521,0). When g  90 , V eff is a single-well potential and VR  0.9925 for 1  0.75 and for 1  2.0 , VR is 1.72 . The slow oscillation takes place around X 1  0 [fig.(6a)] and * x(t ) encloses with the minimu m ( x  0.7521) and the local maxima ( x  0) of the potential for all valuesof * [figs. (6b)-6(d)]. Next , we determine g VR which are the roots of S g  dS  4(r  2 )r rg  0 with 2 dgS gg  0 where VR  d r . The variation in r with g for four values of 1 is shown in fig.(7). For g  gVR dgeach fixed value of 1 as g increases from zero, the resonant frequency r decreases upto g = g0  65.69 .The value of g (= g0 ) at which V eff undergoes bifurcation fro m a double-well to a single-well is independentof 1 . For g  g 0 , V eff becomes a single-well potential and r increases with increase in g . Resonance will .take place whenever r   Further, for g > g0 , V eff is a single-well potential and r  C1 . In this case an 2 .analytical exp ression for g VR can be easily obtained fro m r  C1   and is given by 1 2  B   B 2  10C ( A 0  2 ) / 3  2   g VR   2   ,  5C / 2    2  A 0 2 (18)www.theijes.com The IJES Page 88
  9. 9. Effect Of Depth And Location Of Minima Of A Double-Well Potential…For g < g0 the resonance frequency is 1 . The value of g = g0 at which V eff undergoes bifurcation fro ma double-well to a single-well is independent of 1 . The analytical determination of the roots of Sg  0 andg VR is difficult because C1 and C 2 and X 2,3 are function of g and 2  1 is a co mplicated function of g . * r Fig.7: Plot of g versus r for few values of 1  0.25, 0.75, 2 and 3 . The values of the other parameters are 0  1,   1,   1, f  0.05 and   10 . 2 Fig.8: (a) Plot of theoretical g VR versus 1 for the system (1) and (b) theoretical and numerical response amplitude Q versus g for a few values of 1  0.25, 0.75 and 2 with d  0.5 . Continuous curves are theoretical result and painted circles are nu merical values of Q .Therefore we determine the roots of Sg  0 and g VR numerica lly. We analyze the cases (r   )  0 and 2 2 rg  0 . In fig. 8(a), g VR computed numerically for a range of 1 is plotted. For 1  1c there are tworesonances - one at value of g < g0 ( 65.69) and another at a value of g > g0 . In fig. (7) for 1  0.25  1c  0.4059 the r curve intersects the   1 dashed lineat only one value of g > g 0 and so weget only one resonance for 1   c . Figure (8b) shows Q versus g for 1  0.25, 0.75 and 2 .Continuouscurve represents theoretical results obtained from Eq. (18). Painted circles represents numerically calculated Qwww.theijes.com The IJES Page 89
  10. 10. Effect Of Depth And Location Of Minima Of A Double-Well Potential…fro m Eq. (17). For large values of 1 , the two g VR values are close to g 0  65.69 . As 1 decreases the valuesof g VR move away from g 0 . For 1  0.25 we notice only one resonance. In the double resonance cases thetwo resonances are almost at equidistance from g 0 and the values of Q at these resonances are the same.However, the response curve is not symmetrical about g 0 .4. Effect Of Location Of Minima Of The Potential On Vibrational Resonance For DS2 as  2 increases the location of the two min ima of V (x) move away fro m the orig in inopposite direction, ie., the distance between a min imu m and local maximu m x0  0 of the potential increases *with increase in  2 . When the slow oscillation occurs about x*  0 then 2  1 and analytical determination rof g VR is difficult because  is a comp licated function of g . In this case numerically we can determine the 1value of g VR . Figure (9) shows the plot of g VR versus  2 Fo r  2   2c there are two resonances while for 2   2c only resonance. The above prediction is confirmed by numerical simu lation. Figure (9b) shows thevariation of r with g for three values of  2 . The bifurcation point g 0 increases linearly with  2 . That is, at2  0.75, 1.2 and 1.6 , the values of g 0 are 48.54, 78.66 and 105.44 respectively. Samp le response curves for *three fixed values of  2 are shown in fig. (9c). The difference in the effect of the distance of x  fro m orig inover the depth of the potential wells can be seen by comparing the figs. (9a) and (8a). In DS1 t wo resonancesoccur above certain critical depth 1c of the wells. In contrast to this in DS2 two resonances occur only for * 2   2c . In fig. (9a) we infer that as  2 increases fro m a small value (ie., as x  moves away fro m 1origin) g VR increases and reaches a maximu m value 96.5 at  2  1.15 . Then with further increase in  2 , Fig.9: (a) Plot of theoretical g VR versus  2 for the system (1). (b) Theoretical and numerical r versus g for a few values of 2  0.75, 1.2 and 1.6 with 0  1,   1,   1, f  0.05 and 2   10 . The horizontal dashed line represents r    1 (c) Theoretical and numerical response amp litude Q versus g for a few values of 2  0.75, 1.2 and 1.6 with d  0.5 . Continuous curves are theoretical result and painted circles are nu merical values of Q .it decreases and  0 as  2   2c . Similarly g VR 2 increases and reach a maximu m value 98.5 at 2  1.35 .Then with further increase in  2 , it decreases and  0 at  2  1.7015 . We note that g VR and 1g 2 of DS1 continuously decreases with increase of 1 . But in DS2,  2 increases from a s mall value, VRg 1VR and g 2 increase and reaches a maximu m value. Then with further increase in  2 , it decreases and  0 . VRwww.theijes.com The IJES Page 90
  11. 11. Effect Of Depth And Location Of Minima Of A Double-Well Potential…In double resonance case the separation between the two resonances increas es with increase in  2 . Theconverse effect is noticed in DS1. Next we consider the dependence of VR on g . VR is given by Eq. (16).Figure (10) shows plot of ωVR versus g for three different values of d with  2  0.75 and 1.2 . Ford  0.5, 1, 1.5 and  2  0.75 the nonresonance intervals occur at g [48.98, 51.14], g [46.84, 54.72]and g [43.25,59.02] and for 2  1.2, the nonresonance intervals occur at g [75.92,84.12], g [66.82,95.95] and g [49.50, 111.43] . That is the nonresonance intervals of g increases with increasein 1 . But in DS1, nonresonance intervals of g decreases with increase in 1 . Figures (2) and (10) can becompared. Fig.10: Plot of VR versus g for three d ifferent of d with (a) 2  0.75 and (b) 2  1.2 . The values of other parameters are 0  1,   1,   1 and   10 . 25. Conclusion We have analysed the effect of depth of the wells and the distance of a min imu m and the localmaximu m of the symmet ric double-well potential in a linearly damped quintic oscillator on vibrationalresonance. The effective potential of the system allowed us to obtain an approximate theoretical expression forthe response amplitude Q at the low-frequency  . Fro m the analytical expression of Q , we determined thevalues of  and g at wh ich v ibrational resonance occurs. In the system (1) there is always one resonance ata value of g , g > g 0 , wh ile another resonance occurs below g 0 for a range of values of  . The two quantities1 and  2 have distinct effects. The dependence of g VR and VR on these quantities are exp licit lydetermined. The number of resonance and the value of g VR and VR can be controlled by varying theparameters 1 and  2 . Qmax is independent of 1 and  2 in the system (1). g VR of DS1 is independent of1 while in DS2 it depends on  2 .References[1] P.S. Landa And P.V.E. Mcclintock, “ Vibrat ional Resonance”, J. Phys. A: Math. Gen., 33, L433, (2000).[2] M. Gittermann, “ Bistable Oscillator Driven By Two Periodic Fields”, J. Phys. A: Math. Gen., 34, L355, (2001).[3] G. M. Shepherd, “The Synaptic Organiz-Ation Of The Brain, Oxford Univ. Press”, New -Yo rk, 1990.[4] A. Knoblanch And G. Palm, “Bio System”, 79, 83, (2005).[5] D. Su, M. Ch iu And C. Chen, J. Soc., Precis. Eng., 18, 161, (1996).[6] A. Maksimov, “Ultrasonics”, 35, 79, (1997).[7] V. M ironov And V. So kolov, “Radiotekh Elektron”, 41, 1501 (1996) . (In Russian)[8] V. Gherm, N. Zernov, B. Lundborg And A.Vastborg, “J.Atmos. So lar Terrestrial Phys.”, 59, 1831, (1999).[9] I.I. Blekh man And P.S. Landa, “ Conjugate Resonances And Bifurcations In Nonlinear Systems Under Biharmonical Excitations”, Int. J. Nonlinear Mech., 39, 421, (2004).10] S. Jeyaku mari, V. Chinnathamb i, S. Rajasekar And M.A.F.Sanjuan, “ Single And Multiple Vibrational Resonance In A Quintic Oscillator With Monostable Potentials”, Phys. Rev.E, 80, 046698, (2009).www.theijes.com The IJES Page 91
  12. 12. Effect Of Depth And Location Of Minima Of A Double-Well Potential…[11] S. Jeyaku mari, V. Ch innathambi, S. Rajasekar, M.A.F. San juan, “ Analysis Of Vibrational Resonance In A Quintic Oscillator”, Chaos, 19, 043128, (2009).[12] V.M. Gandhimathi, S. Rajasekar And J. Kurths, “ Vib rational Resonance And Stochastic Resonance In Two Coupled Overdamped Anharmonic Oscillators”, Phys. Lett. A, 360, 279, (2006).[13] B. Deng, J. Wang And X. Wei, “ Effect Of Chemical Synapse On Vibrational Resonance In Coupled Neurons”, Chaos, 19 013117, (2009) .[14] S. Rajasekar, K. Abirami And M.A.F. Sanjuan, “ Novel Vib rational Resonance In Multistable Systems”, Chaos, 21, 033106, (2011).[15] J.H. Yang And X.B. Liu, “ Sequential Vibrat ional Resonance In Multistable Systems”, Arxiv: 1106.3431V1 [Nlin.CD], (2011).[16] J.H.Yang And X.B. Liu, “ Delay Induces Quasi Periodic Vibrational Resonance”, J. Phys. A: Math. Theor., 43, 122001, (2010).[17] C. Jeevarathinam, S. Rajasekar, M.A.F. Sanjuan, “ Theory And Numerics Of Vibrational Resonance In Duffing Oscillator With Time-Delay Feedback”, Phys. Rev. E, 83, 066205, (2011) .[18] A.A. Zaikin, L. Lopez, J.P. Baltanas, J. Kurths And M.A.F. Sanjuan, “ Vibrational Resonance In A Noise Induced Structure”, Phys. Rev. E, 66, 011106, (2002).[19] E. Ullner, A. Zaikin, J. Garcia-Ojalvo, R. Bascones And J. Kurths, “ Vibrat ional Resonance And Vibrat ional Propagation In Excitable Systems”, Phys. Lett. A, 312, 348, (2003).[20] S. Jeyaku mari, V. Ch innathambi, S. Rajasekar, M.A.F. Sanjuan, “ Vib rational Resonance In An Asymmetric Duffing Oscillator”, Int. J. Bifu r. And Chaos, 21, 275, (2011).[21] S. Rajasekar, J. Used, A. Wagemakers And M.A.F. Sanjuan, “ Vibrat ional Resonance In Bio logical Maps”, Commun. Nonlinear Sci.:Nu mer. Simu lat. 17, 435, (2012).[22] S. Rajasekar, S. Jeyaku mari, V. Ch innathambi And M.A.F. Sanjuan, “ Ro le Of Depth And Location Of M inima Of A Double-Well Potential On Vibrat ional Resonance”, J. Phys. A : Math. Theore., 43 , 465101, (2010).www.theijes.com The IJES Page 92

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