effects on the bifurcation and chaos in forced duffing oscillator due to nonlinear damping

Commun Nonlinear Sci Numer Simulat 17 (2012) 2254–2269

Contents lists available at SciVerse ScienceDirect

Commun Nonlinear Sci Numer Simulat

journal homepage: www.elsevier .com/locate /cnsns

Short communication

Effects on the bifurcation and chaos in forced Duffing oscillatordue to nonlinear damping

Anjali Sharma, Vinod Patidar ⇑, G. Purohit, K.K. SudDepartment of Physics, School of Engineering, Sir Padampat Singhania University, Udaipur 313 601, Rajasthan, India

a r t i c l e i n f o

Article history:Received 31 August 2011Received in revised form 29 October 2011Accepted 29 October 2011Available online 12 November 2011

Keywords:Melnikov analysisForced Duffing oscillatorChaosNonlinear damping

1007-5704/$ - see front matter � 2011 Elsevier B.Vdoi:10.1016/j.cnsns.2011.10.032

⇑ Corresponding author. Fax: +91 2957 226094.E-mail addresses: [email protected], v

a b s t r a c t

In this communication, the two-well Duffing oscillator with non-linear damping term pro-portional to the power of velocity is considered. We mainly focus our attention on how thedamping exponent affects the global dynamical behaviour of the oscillator. In particular,we obtain analytically the threshold condition for the occurrence of homoclinic bifurcationusing Melnikov technique and compare the results with the computational results. We alsoidentify the major route to chaos and the regions of the 2D parameter space (consists ofexternal forcing amplitude and damping coefficient) corresponding to the various typesof asymptotic dynamics under linear (viscous or friction like) and nonlinear (drag like)damping. We also attempt to analyze how the basins of attraction patterns change withthe introduction of nonlinear damping. We also present our analysis for the physicallyless-interesting cases where damping is proportional to the 3rd and 4th power of velocityfor the sake of generalizing our findings and establishing firm conclusion.

� 2011 Elsevier B.V. All rights reserved.

Study of chaotic behaviour in nonlinear systems has been an active area of research during the second half of the lastcentury. The chaotic behaviour is mainly attributed due to the nonlinear effects in the physical systems. Most of the realphysical systems exhibit nonlinear effects to some extent. The identification of these nonlinear effects in some systemsmay or may not be straight forward as these effects may enter into the physical systems in many different ways. For amechanical system the nonlinear effects may be due to the nonlinear elastic/spring elements, nonlinear damping, systemswith fluids, nonlinear boundary conditions, etc. whereas in an electromagnetic system the nonlinear resistive, inductive,capacitive elements, hysteresis properties of ferromagnetic materials, nonlinear active elements like vacuum, diode, transis-tor, etc. may be responsible for nonlinearities in the system.

During last three-four decades a large number of studies (analytical, numerical and experimental) have been carried outon various nonlinear chaotic systems to understand the complex behaviour of these systems. Forced Duffing oscillator is oneof the most explored nonlinear dynamical systems, which serves as a prototype model for various physical and engineeringproblems such as particle in a forced double well, particle in a plasma, a defect in solids, dynamics of a buckled elastic beam,etc. [1]. Although number of studies have been done on Duffing oscillator for many decades, but even today it is an inter-esting and preferred system to explore hidden mysteries of chaos in nonlinear physical systems. The Duffing oscillatorcan be interpreted as a damped oscillator with a complicated potential. The damping or dissipation here is very importantas it decides the border of stability and instability. Most of the studies on Duffing system have been done by considering theviscous damping i.e., damping is linearly proportional to the velocity. However the consideration of nonlinear damping/dis-sipation is necessary in several engineering applications such as rolling in ship dynamics [2], vibration isolators [3], dragforces in flow induced vibrations [1], etc. In many of the real physical systems dissipative forces (e.g. drag like friction, which

. All rights reserved.

[email protected] (V. Patidar).

http://dx.doi.org/10.1016/j.cnsns.2011.10.032

mailto:[email protected]

mailto:[email protected]

http://dx.doi.org/10.1016/j.cnsns.2011.10.032

http://www.sciencedirect.com/science/journal/10075704

http://www.elsevier.com/locate/cnsns

A. Sharma et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 2254–2269 2255

is proportional to the square of the velocity) introduce the nonlinear damping. Several models of nonlinear damping areavailable in literature [3,4]. One of the simplest empirical mathematical models of the nonlinear dissipative force isf(v) / �vjvjp�1, where v represents the velocity (with sign) and p is an integer known as damping exponent. A few of the re-cent important studies involving nonlinear damping in Duffing/Duffing-like oscillators are discussed here. The role of non-linear damping in soft Duffing oscillator with a simultaneous presence of viscous damping has been discussed in [5], whichconcludes that the threshold value of forcing amplitude to generate period-doubling decrease with the increase in dampingexponent p. The study of universal escape oscillator with nonlinear damping has been considered in [6], in which the effectsof nonlinear damping on the thresholds of period doubling bifurcations, fractal basin boundaries and the destruction of thebasin of attraction have been analyzed and concluded that increasing the power of nonlinear damping has similar effects asof decreasing the damping coefficient for a linearly damped case. The analytical estimates of nonlinear damping effect forDuffing oscillator and simple pendulum using Melnikov technique have been investigated in [7]. Baltanas et al. [8] have stud-ied the effect of including a nonlinear damping term proportional to the power of velocity in the dynamics of double-wellDuffing oscillator, in particular they have studied the energy dissipation over a cycle and concluded that the energy dissipa-tion depends on the power of damping term. Litak et al. [9] and Borowieck et al. [10] have considered the generalized Duffingoscillator with a nonlinear damping term proportional to vjvjp�1 with fractional damping exponent (p) covering the gap be-tween viscous, dry friction and turbulent damping. They also explained that for p ? 0, the system mimics the dry frictionphenomena. Siewe et al. [11] have studied the effect of nonlinear dissipation on the basin boundaries of a driven two-wellRayleigh–Duffing oscillator in which a cubic damping and linear damping terms are simultaneously present. Litak et al. [12]have examined the Melnikov criterion for the global homoclinic bifurcation and a possible transition to chaos in a single de-gree of freedom nonlinear oscillator with a symmetric double-well potential and simultaneously subject to parametric peri-odic forcing and self excitation via negative damping term. Later Siewe et al. [13] have also examined the chaotic behaviourof extended Rayleigh oscillator in a three-well potential under additive parametric and external periodic forcing using Mel-nikov technique and showed that a transition from regular to chaotic motion is often associated with increase in the energyof oscillator. Siewe et al. [14] have used the Melnikov technique to detect the necessary condition for chaotic motion in theRayleigh–Duffing oscillator and showed that basin boundaries are fractal as the damping increases above the threshold ofMelnikov chaos.

Fig. 1. (a) The Duffing two-well potential, (b) phase trajectories of undamped and unforced Duffing oscillator.

2256 A. Sharma et al. / Commun Nonlinear Sci Numer Simulat 17 (2012) 2254–2269

In this communication, the two-well Duffing oscillator with non-linear damping term proportional to the power of veloc-ity is considered. In particular, we find analytically the threshold condition for the occurrence of homoclinic bifurcation usingMelnikov technique and compare the results with the computational results. We also focus our attention to identify the

Fig. 2. (a) Critical value of forcing amplitude Fc (as a function of damping coefficient a) obtained using the Melnikov analysis for x20 ¼ 1; b ¼ 1 and x = 1

and p = 1, 2 and 3. (b) Critical value of forcing amplitude Fc (as a function of external frequency x) obtained using the Melnikov analysis for x20 ¼ 1; b ¼ 1

and x = 1 and p = 1, 2 and 3.

Fig. 3. Comparison of critical values of forcing amplitude Fc for the homoclinic tangency obtained theoretically using the Melnikov analysis and numericallyusing the Largest Lyapunov exponent computation (as a function of damping coefficient a) for x2

0 ¼ 1; b ¼ 1 and x = 1 and p = 1, 2 and 3.


regions of the 2D parameter space (consists of external forcing amplitude and damping coefficient) correspond to the varioustypes of asymptotic dynamics under linear (viscous or friction like) and nonlinear (drag like) damping. We also attempt toanalyze how the basins corresponding to various attractors change with the introduction of nonlinear damping. We alsopresent our analysis for the physically less-interesting cases where damping is proportional to the 3rd and 4th power ofvelocity for the sake of generalizing our findings.

To study the effects of nonlinear damping on the dynamics of two-well forced Duffing oscillator, we consider the follow-ing generalized form of model equation

Fig. 4.a = 0.5:

€xþ a _xj _xjp�1 �x20xþ bx3 ¼ F cos xt; ð1Þ

Dynamical behaviour of Duffing oscillator under linear/viscous damping (p = 1) as a function of forcing amplitude (F) for x20 ¼ 1; b ¼ 1; x ¼ 1 and

(a) bifurcation diagram showing route to chaos, (b) all three Lyapunov exponents and (c) the Kaplan–Yorke dimension.


where a is the damping coefficient, p is the damping exponent, x0 is the natural frequency, b is the stiffness constant whichplays the role of nonlinear parameter and F and x respectively are the external amplitude and frequency. The unperturbedsystem (1) (i.e., a, F = 0) can be written as

Fig. 5.a = 0.5:

_x ¼ y;

_y ¼ x20x� bx3;

ð2Þ

which corresponds to the undamped, unforced motion of a unit mass particle in the potential VðxÞ ¼ �x20x2

2 þbx4

4 . The unper-

turbed system (2) has three equilibrium points: one saddle (0,0) and two � x0ffiffibp ;0

� �are centres for all x2

0; b > 0 (double

well potential). In Fig. 1(b), we have depicted the phase space trajectories, for the system (2) with x20 ¼ 1and b = 1 (a sym-

Dynamical behaviour of Duffing oscillator under drag like damping (p = 2) as a function of forcing amplitude (F) for x20 ¼ 1; b ¼ 1; x ¼ 1 and

(a) bifurcation diagram showing route to chaos, (b) all three Lyapunov exponents and (c) the Kaplan–Yorke dimension.


metric double-well potential which has minima at x = ±1 and a local maximum at x = 0 as shown in Fig. 1(a)). The solution ofthe unperturbed system (2) obtained by integrating the system (2) is given as

Fig. 6.1; x ¼

x0ðtÞ ¼

ffiffiffi2b

sx0 sec hx0t; ð3Þ

_x0ðtÞ ¼ y0ðtÞ ¼

ffiffiffi2b

sx2

0 sec hx0t tanh x0t: ð4Þ

Now we use Melnikov’s method [15,16] to detect the homoclinic bifurcation in the forced Duffing oscillator system withnonlinear damping (Eq.(1)). The Melnikov function gives the distance between the stable and unstable manifolds of the

Dynamical behaviour of Duffing oscillator under nonlinear (third-power) damping (p = 3) as a function of forcing amplitude (F) for x20 ¼ 1; b ¼

1 and a = 0.5: (a) bifurcation diagram showing route to chaos, (b) all three Lyapunov exponents and (c) the Kaplan–Yorke dimension.


saddle point (which in the present case is the origin) in forced systems, hence the simple zeros of the Melnikov function givethe critical value of the forcing amplitude (for a given set of other system parameters a, p, b, x and x0) for which the distancebetween stable and unstable manifolds becomes zero and a homoclinic orbit originates. The Melnikov function for the forcedDuffing oscillator system with nonlinear damping (Eq.(1)) is given by

Fig. 7.1; x ¼

Mðt0Þ ¼Z 1

�1_x0ðt � t0Þ �a _x0ðt � t0Þj _x0ðt � t0Þjp�1 þ F cos xt

h idt; ð5Þ

Mðt0Þ ¼ �aZ 1

�1½ _x0ðt � t0Þ�2j _x0ðt � t0Þjp�1dt þ F

Z 1

�1_x0ðt � t0Þ cos xt dt: ð6Þ

Dynamical behaviour of Duffing oscillator under nonlinear (fourth-power) damping (p = 4) as a function of forcing amplitude (F) for x20 ¼ 1; b ¼

1 and a = 0.5: (a) bifurcation diagram showing route to chaos, (b) all three Lyapunov exponents and (c) the Kaplan–Yorke dimension.


Here it should to be noted that for p = 2 the system under consideration becomes non-smooth [9] and according to Kunzeand Küpper [17] additional terms could appear in the Melnikov integral but in this particular case of non-smooth dampingterm these additional term are quenched to zero. Using Eq. (4) and then evaluating the integrals, we obtain

Fig. 8.0:691 a

Mðt0Þ ¼ �2ab

x30

ffiffiffi2b

sx2

0

��p�1

Bpþ 2

2;pþ 1

2

� �þ F

ffiffiffi2b

sx2

0 sin xt0

Pbx0�1=2cn¼0 ð�1Þn 2px

x20

cosh ð2nþ1Þpx2x0

� �1þ cosh xp

; ð7Þ

Here B(�,�) and b�c respectively, are the Euler beta and floor functions. The simple zeros of the above function (7) (the con-dition of homoclinic tangency) give the critical value of the forcing amplitude (Fc) and when we cross this critical value whileincreasing the forcing amplitude, the first homoclinic orbit appears. At this value of critical forcing amplitude the homoclinicbifurcation occurs for the limit cycle which may trigger a chaotic behaviour in the system.

For a given set of parameters F, a, p, b, x and x0, M(t0) will vanish if a real solution can be found for t0. Assuming thatF; a; b; x2

0 > 0, the above condition will be met if jsinxt0j 6 1. Hence

F ¼ Fc ¼2ab x3

0

ffiffi2b

qx2

0

�� p�2B pþ2

2 ; pþ12

� ð1þ cosh xpÞPbx0�1=2c

n¼0 ð�1Þn 2pxx2

0cosh ð2nþ1Þpx

2x0

� � : ð8Þ

In Fig. 2, we have shown the variation of critical values of forcing amplitude (Fc) obtained using Eq.(8) for a symmetric dou-ble-well potential described by x2

0 ¼ 1and b = 1 (as shown in Fig. 1(a)). Particularly in Frame (a) the variation of Fc with

damping coefficient (a) for a specific choice of external forcing frequency x = 1 (i.e., Fc ¼ ð1=pÞffiffiffi2p� �p

B ðpþ 2Þ=2;ððpþ 1Þ=2Þ coshðp=2ÞaÞ and different values of damping exponent p = 1, 2 and 3, however in Frame (b) the variation of Fc

with external forcing frequency (x) for a specific choice of damping coefficient a = 0.5 (i.e., Fc ¼ ð1=pxÞffiffiffi2p� �p�2

B ðpþ 2Þ=2; ðpþ 1Þ=2ð Þ coshðpx=2ÞÞ and different values of damping exponent p = 1, 2 and 3 have been depicted. It is clearthat Fc linearly increases with the damping coefficient (a), however Fc depends on external forcing frequency (x) in a

(a) Poincare map for the Duffing phase space attractor having the highest value of Kaplan–Yorke dimension for x20 ¼ 1; b ¼ 1; x ¼ 1; a ¼ 0:5; F ¼

nd p = 1, (b) the correlation dimension of the attractor shown in frame (a).


nonlinear way: first it decreases with the increase in x and after attaining a minimum at x = 0.76374 (independent of p aswell as a), it starts increasing. It is also very clear that if we increase the damping exponent (p) by keeping all the otherparameters fix, the value of Fc lowers. i.e., due to the introduction of nonlinear damping, the critical value of forcing ampli-tude (Fc), for which homoclinic tangency occurs, shifts towards zero. Hence we may anticipate the behaviour of a particle insymmetric double-well potential with drag like damping (p = 2) may become chaotic at lower value of forcing amplitude ascompare to the same system with viscous or friction like damping (p = 1). It is also found true for the higher order of non-linear damping (p = 3). The similar conclusion has been reported by Ravindra and Mallik [5] for the soft Duffing system witha simultaneous presence of linear (viscous) and nonlinear (p = 3) damping. To verify the theoretical predictions of Melnikov’smethod, we have also obtained the critical value of forcing amplitude by computing the largest Lyapunov exponent (LLE) byincreasing the forcing amplitude and recording the value of forcing amplitude where LLE shows a first transition from neg-ative to positive (10�4). For the calculation of Lyapunov exponents, we have integrated the system and averaged the resultsup to t = 2p � 102 with a time step Dt = 2p � 10�3 for each set of parameters. The comparisons of the critical values of forcingamplitude (Fc) obtained using theoretical analysis (Melnikov function) and computational approach (using Lyapunov expo-nent) for various cases under consideration, have been shown in Fig. 3. It is very clear that Melnikov analysis provides anoverall good estimate of the lower bound on the regions of chaos in F � a parameter space. Specifically for the linear (viscousor friction like i.e., p = 1) damping it provides remarkably correct estimate of the critical forcing amplitude however for thenonlinear damping cases (p = 2 and 3, it slightly diverges from the computational results. We observe sudden peaks/dips inthe computational curves in Fig. 3. These are due to the limited number of time steps used for the calculation of largestLyapunov exponent (we have integrated the system and averaged the results up to t = 2p � 102 with a time stepDt = 2p � 10�3 for each value of forcing amplitude to save the computational time). Also we have used the same numberof steps for all the values of forcing amplitude, however in general the convergence is not achieved for all the values of forc-ing amplitude with the same number of steps. At the critical value of forcing amplitude where the system changes the sign ofits largest Lyapunov exponent, the average value of largest Lyapunov exponent is very small and hence very sensitive to thenumber of time steps if convergence is not achieved. Hence the resultant curves are not smooth.

Now we analyze various features (such as route to chaos, topological shape and fractalness of the phase space attractor,Lyapunov exponents, Kaplan–Yorke dimension, etc.) of the dynamical behaviour of particle in double-well without and with

Fig. 9. (a) Poincare map for the Duffing phase space attractor having the highest value of Kaplan–Yorke dimension for x20 ¼ 1; b ¼ 1; x ¼ 1; a ¼ 0:5; F ¼

0:297 and p = 2, (b) the correlation dimension of the attractor shown in frame (a).


nonlinear damping. For this we have done computation for the bifurcation plots, Lyapunov exponents and Kaplan–Yorkedimensions and the results are shown in Figs. 4–7 respectively for p = 1, 2, 3 and 4 for fixed values of damping coefficienta = 0.5, external forcing frequency x = 1 and by varying the external forcing amplitude (F). From the bifurcation plots (Frame(a) of each figure), we may clearly observe that for the case of linear (viscous or friction like) damping i.e., p = 1 the perioddoubling route to chaos leading to single band (one well) chaos followed by double-band (double well) chaos is observed aswe increase the forcing amplitude. However for the cases of nonlinear damping p = 2, 3 and 4, we do not observe the firsttransition to chaos through period-doubling route, however the dominating route to chaos remains period-doubling forhigher values of F. One more important feature we observe form all the bifurcation diagrams that chaotic behaviour becomesmore fragile (i.e., number of periodic windows increases) as we increase the damping exponent (p). We also observe that thetotal region corresponding to chaos (i.e., the range of F values for which chaos exists) also increases with the increase in thedamping exponent. The above features have been found true for all other values of damping constant a also. In Frames (b) ofFigs. 4–7, we have shown the results of all three Lyapunov exponents for p = 1, 2, 3 and 4. Here for the calculation of Lyapu-nov exponents, we have integrated the system and averaged the results up to t = 1 � 104 with a time step Dt = 10�2 (which isdifferent than the one used for Fig. 3, hence we may find some variations in the results. In fact the results in these figures aremore accurate, as the averaging has been done for very long time). We see a clear one-to-one correspondence between thebifurcation plots and the largest Lyapunov exponent curve. In Frame (c) of each figure, we have shown the result of Kaplan–Yorke dimension of the phase space attractor of the particle in two-well Duffing potential. The Kaplan–Yorke dimension [18]

has been calculated using DKY ¼ mþPm

i¼1ki

jkmþ1 j, where m 6 n is the largest index such that

Pmi¼1ki P 0 is valid. The dimension of

the chaotic attractor is always between 2 and 3 i.e., the chaotic phase space attractor is topologically complicated more thana limit cycle and less than a three-dimensional object. To compare the geometrical complexity of the phase space attractors,we have identified the cases corresponding to the highest values of the Kaplan–Yorke dimensions for each case p = 1, 2, 3 and4 and these are respectively 2.252, 2.314, 2.325 and 2.360 at forcing amplitudes F = 0.691, 0.297, 0.355 and 0.280 (all fora = 0.5). To observe the complexity of these phase space attractors, we have also obtained the Poincare maps (as the com-plete phase space attractors are not very easy to compare, however the Poincare maps, which reduce the dimension of phasespace attractor by 1, are easy to compare) for all these cases corresponding to the highest values of Kaplan–Yorke dimen-

Fig. 10. (a) Poincare map for the Duffing phase space attractor having the highest value of Kaplan–Yorke dimension for x20 ¼ 1; b ¼ 1; x ¼ 1; a ¼ 0:5; F ¼


Fig. 11. (a) Poincare map for the Duffing phase space attractor having the highest value of Kaplan–Yorke dimension for x20 ¼ 1; b ¼ 1; x ¼ 1 a ¼ 0:5; F ¼



sions. The results have been shown in Figs. 8–11 respectively for p = 1, 2, 3 and 4. We have also computed the correlation-dimension (DC) [19] of these attractors in the Poincare surface of section using the following set of formula.

1 For

DC ¼ limr!0

log10CðrÞlog10r

; ð9Þ

where the correlation function C(r) is given by

CðrÞ ¼ limN!1

1N2

XN

i¼1

XN

j¼1

Hðr � jXi � XjjÞ; ð10Þ

here r is the radius of spheres with their centres as Xi, N is the total number of points and H(�) is the Heaviside step function.For the computation of the correlation dimension, we have used N = 105 points for each attractors. The values of the corre-lation dimension obtained are 1.1268, 1.15049, 1.15937 and 1.19959 respectively for p = 1, 2, 3 and 4. The results of Kaplan–Yorke and correlation dimensions clearly suggest that the geometrical complexity/fractalness of the phase space attractorincreases with the increase in the damping exponent.

We have also identified the regions of the parameter space consisting of external forcing amplitude and damping coef-ficient (F � a) corresponding to chaotic and periodic behaviour for the forced Duffing two-well oscillators with linear andvarious nonlinear damping cases using the extensive Lypunov calculation for 0 6 F 6 1.0 and 0.15 6 a 6 1.0 with step sizesDF = 0.01 and Da = 0.01. For the calculation of largest Lyapunov exponent (LLE), we have integrated the system and averagedthe results up to t = 2p � 102 with a time step Dt = 2p � 10�3 for each set of parameter. In Fig. 12 we have shown the results,Frames (a) to (d) respectively show the results for p = 1, 2, 3 and 4. Here the periodic regions have been represented by thelight-blue/cyan1 (LLE 6 �0.2) and blue (�0.2 < LLE 6 0) shades however the chaotic regions have been depicted through theyellow (0 < LLE 6 0.125) and red (0.125 < LLE 6 0.25) shades. We may clearly observe that the chaos becomes more fragile

interpretation of color in Fig. 12, the reader is referred to the web version of this article.

Fig. 12. Parameter space consisting of external forcing amplitude and damping coefficient (F � a) corresponding to chaotic and periodic behaviour in theforced Duffing two-well oscillator for x2

0 ¼ 1; b ¼ 1; x ¼ 1 under various kinds of damping: (a) linear/viscous damping (p = 1), (b) drag like damping(p = 2), (c) nonlinear (third power) damping (p = 3) and (d) nonlinear (fourth-power) damping (p = 4).


i.e., number of periodic windows increases as we increase the damping exponent (p). We also observe that with the increasein the nonlinearity in the damping, chaos is more global in the parameter space. We have also calculated the percentage ofthe regions of parameter space corresponding to chaotic and periodic motions. The results are summarized in Table 1. Weshould note here that this comparison is only for the parameter space defined by 0 6 F 6 1.0 and 0.15 6 a 6 1.0. The chaoticregions also exist outside the parameter space considered here. It can be verified by comparing the bifurcations diagramsshown in Frames (a) of Figs. 4–7, which are showing the chaotic and periodic behaviours for a = 0.5 and varying the forcingamplitude).

To observe the effect of nonlinear damping on the basin of attraction patterns, we have computed the basin of attractioncorresponding to various combination of parameters for all four cases i.e., p = 1, 2, 3 and 4. We have observed that the basinboundaries become fractal as we increase the value of forcing amplitude beyond the critical forcing amplitude (FC) obtainedusing the Melnikov analysis. In Figs. 13–15, we have tried to depict the comparisons of various fractal basin of attraction

Table 1Comparison of regions of parameter space corresponding to periodic and chaotic motions.

Colour p = 1 p = 2 p = 3 p = 4

Chaos Red 0.125 < LLE 6 0.25 9.51% 25.10% 11.18% 40.38% 2.40% 38.02% 16.36% 64.21%Yellow 0 < LLE 6 0.125 15.59% 29.20% 35.61% 47.85%

Periodic Dark blue �0.2 < LLE 6 0 42.39% 74.90% 47.64% 59.62% 54.06% 61.98% 36.01% 35.79%Light blue LLE 6 �0.2 32.51% 11.98% 7.93% 1.78%

Fig. 13. Comparison of the basin boundaries of a period-3 attractor in the forced Duffing oscillator under linear/viscous damping(x2

0 ¼ 1; b ¼ 1; x ¼ 1; a ¼ 0:5, F = 0.55 and p = 1) and basin boundaries of a period-4 attractor in the forced Duffing oscillator under drag-like damping(x2

0 ¼ 1, b = 1, x = 1, a = 0.5, F = 0.55 and p = 2).


patterns for the periodic attractors observed after the homoclinic bifurcation for p = 1, 2, 3 and 4. In Fig. 13, we have shownone such comparison between the basin of attraction patterns for the period-3 and period-4 attractors (shown in Frames (a)and (d)) observed in the forced Duffing oscillators with viscous (p = 1) and drag (p = 2) like damping respectively for the set ofparameters a = 0.5 and F = 0.55. As both the periodic attractors are not of same kind (it is not possible to have same kind ofperiodic motion at same values of parameters for p = 1 and p = 2) hence the comparison is not absolute, but it gives a fair ideaabout the role of nonlinear damping in changing the basin patterns. For drawing the basin of attractions we have considered501 � 501 different initial conditions defined by �2.5 6 x 6 2.5 and �2:5 6 _x 6 2:5 with steps Dx = 0.01 and D _x ¼ 0:01. Eachpoint of the initial condition (x; _xÞ defined in the above manner has been coloured according to its fate after the 73 drive cy-cles (t = 73 � 2p/x) i.e., if a trajectory ends up after 73 cycles in the extreme right well with x = 1.5 or in the extreme left wellwith x = �1.5 respectively then it is coloured bright red or bright green. Other points between x = �1.5 and 1.5 have been

Fig. 14. Comparison of the basin boundaries of a period-5 attractor in the forced Duffing oscillator under drag-like damping(x2

0 ¼ 1; b ¼ 1; x ¼ 1; a ¼ 0:5; F ¼ 0:275 and p = 2) and basin boundaries of a period-3 attractor in the forced Duffing oscillator under nonlinear third-power damping (x2

0 ¼ 1; b ¼ 1; x ¼ 1 ¼ 0:5; F ¼ 0:275 and p = 3).


coloured as per the vertical gradient bars shown in the Frames (b) and (e). We may clearly observe from the visual inspectionof these basin patterns that due to the nonlinear damping the basin of attraction patterns become more complex. To quantifythe fractalness of these basin patters, we have also computed the correlation dimension for these patterns by convertingthem into binary images (each point in these patterns has been considered as a pixel and the right-well and left-well pointsrespectively are denoted by 1 and 0). Then we calculate the correlation dimension of the object represented by all nonzeropixels using the Eqs. (9) and (10). The results have been depicted in the Frames (c) and (f) of Fig. 13 respectively for the basinpatterns shown in Frames (b) and (e) of Fig. 13 and the correlation dimensions respectively are 1.8739 and 1.9183. In Fig. 14,we have shown a similar comparison between the basin of attraction patterns for the period-5 and period-3 attractors(shown in Frames (a) and (d)) observed in the forced Duffing oscillators with p = 2 and p = 3 (both nonlinear damping cases)respectively for the set of parameters a = 0.5 and F = 0.275. In this comparison also both the periodic attractors are not of

Fig. 15. Comparison of the basin boundaries of a period-8 attractor in the forced Duffing oscillator under drag-like damping(x2

0 ¼ 1; b ¼ 1; x ¼ 1; a ¼ 0:5; F ¼ 0:517 and p = 3) and basin boundaries of a period-5 attractor in the forced Duffing oscillator under nonlinear third-power damping (x2

0 ¼ 1; b ¼ 1; x ¼ 1; a ¼ 0:5; F ¼ 0:517 and p = 4).


same kind hence the comparison is not absolute. Here the calculated results of the correlation dimensions for p = 2 and p = 3respectively are 1.90939 and 1.86019. In Fig. 14, we have shown one more such comparison between the basin of attractionpatterns for the period-8 and period-5 attractors (shown in Frames (a) and (d)) observed in the forced Duffing oscillatorswith p = 3 and p = 4 (both nonlinear damping cases) respectively for the set of parameters a = 0.5 and F = 0.517 and the cal-culated results of the correlation dimensions for p = 3 and p = 4 respectively are 1.91709 and 1.90446. Hence the above anal-ysis show that increase in the order of nonlinear damping does not always increases the fractalness of the basin boundaries.

In general the occurrence of chaos, fractal attractors and fractalness in the basin boundaries, is mainly due to presence ofnonlinearity in the system. In the Duffing oscillator with linear damping, source of nonlinearity is the double-well potentialwhich leads to a nonlinear force on the particle. However in the Duffing oscillator with nonlinear damping, the sources ofnonlinearity are two: one due to the double-well potential and another is the presence of nonlinear damping force. Hence


the nonlinearly damped Duffing oscillator is rich in nonlinearity and that may be the possible cause for the qualitativechanges in the overall dynamical behaviour, as observed above, of the system. In physical words, the nature of energy dis-sipation changes the overall dynamical behaviour of Duffing two-well system.

In conclusion our analysis to observe the effect of nonlinear damping on the bifurcation and chaos in forced Duffing oscil-lator system reveal that the nonlinear damping: (i) lowers the critical value of the forcing amplitude corresponding to firsttransition to chaos, (ii) increases the chaos in parameter space, (iii) increases the fractalness of the phase space attractor, (iv)increases the fragileness of chaotic regions in the parameter space and also (v) affects the route to chaos in the system. How-ever we have found that the introduction of nonlinear damping in the system does not necessarily increases the fractalnessof the basin boundaries.

Acknowledgements

Anjali Sharma and Vinod Patidar acknowledge to the Science and Engineering Research Council (SERC), Department ofScience and Technology (DST), Government of India respectively for the Junior Research Fellowship (JRF) and Fast TrackYoung Scientist Research Grant (SR/FTP/PS-17/2009).

References

[1] Moon FC. Chaotic and fractal dynamics. New York: Wiley; 1992.[2] Thompson JMT, Rainey RCT, Soliman MS. Ship stability criteria based on chaotic transients from incursive fractals. Philos Trans R Soc London A

1990;332:149–67.[3] Mallik AK. Principles of vibrations control. New Delhi: Affiliated Publishers; 1990.[4] Pippard AB. The physics of vibration. Cambridge: Cambridge University Press; 1989.[5] Ravindra B, Mallik AK. Role of nonlinear dissipation in soft Duffing oscillators. Phys Rev E 1994;49:4950–4.[6] Sanjuan MAF. The effect of nonlinear damping on the universal escape oscillator. Int J Bifurcat Chaos 1999;9:735–44.[7] Trueba JL, Rams J, Sanjuan MAF. Analytical estimates of the effect of nonlinear damping in some nonlinear oscillators. Int J Bifurcat Chaos

2000;10:2257–67.[8] Baltanas JP, Trueba JL, Sanjuan MAF. Energy dissipation in nonlinearly damped Duffing oscillator. Physica D 2001;159:22–34.[9] Litak G, Borowiec M, Syta A. Vibration of generalized double well oscillators. Zeitschrift fuer Angewandte Mathematik und Mechanik

2007;87:590–602.[10] Borowiec M, Litak G, Syta A. Vibration of the Duffing oscillator: effect of fractional damping. Shock Vib 2007;14:29–36.[11] Siewe MS, Cao H, Sanjuan MAF. Effect of nonlinear dissipation on the boundaries of basin of attraction in two-well Rayleigh–Duffing oscillators. Chaos

Soliton Fract 2009;39:1092–9.[12] Litak G, Borowiec M, Syta A, Szabelski K. Transition to chaos in the self-excited system with a cubic double well potential and parametric forcing. Chaos

Soliton Fract 2009;40:2414–29.[13] Siewe MS, Cao H, Sanjuan MAF. On the occurrence of chaos in a parametrically driven extended Rayleigh oscillator with three-well potential. Chaos

Soliton Fract 2009;41:772–82.[14] Siewe MS, Tchawoua C, Woafa P. Melnikov chaos in a periodically driven Rayleigh–Duffing oscillator. Mech Res Commun 2010;17:363–8.[15] Melnikov VK. On the stability of the center for time periodic perturbations. Trans Moscow Math Soc 1963;12:1–56.[16] Guckenheimer J, Holmes P. Nonlinear oscillations, dynamical systems and bifurcations of 0vector fields. New York: Springer-Verlag; 1983.[17] Kunze M, Kupper T. Nonsmooth dynamical systems: an overview. In: Fiedler B, editor. Ergodic theory, analysis, and efficient simulation of dynamical

systems, Springer, Berlin-New York, 2001. p. 431–452.[18] Kaplan JL, Yorke JA. Chaotic behavior of multidimensional difference equations. In: Peitgen H-O, Walther H-O, editors. Functional differential equations

and approximation of fixed points, Lecture Notes in Mathematics, vol. 730; 1979. p. 204–227.[19] Grassberger P, Procaccia I. Measuring the strangeness of the strange attractors. Physica D 1983;9:189–208.

effects on the bifurcation and chaos in forced duffing oscillator due to nonlinear damping

Documents