Chaos, Solitons and Fractals 41 (2009) 2075–2080

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Chaos, Solitons and Fractals

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

Response of a stochastic Duffing–Van der Pol elastic impact oscillator

Liang Wang *, Wei Xu, Gaojie Li, Dongxi LiDepartment of Applied Mathematics, Northwestern Polytechnical University, Xi’an, 710072, PR China

a r t i c l e i n f o

Article history:Accepted 11 August 2008

Communicated by Prof. Ji-Huan He

0960-0779/$ - see front matter � 2008 Elsevier Ltddoi:10.1016/j.chaos.2008.08.013

* Corresponding author. Tel.: +86 13992816184; fE-mail address: lovenicholaswang@mail.nwpu.ed

a b s t r a c t

The response of a Duffing–Van der Pol elastic impact system with one random parameter isinvestigated. The system is transformed to two high dimension deterministic systemsaccording to Laguerre polynomial approximation, through which the response can bederived from deterministic numerical method. It is found that the behavior of the systemchanges from chaos to periodic through inverse period-doubling bifurcation when thespring stiffness of elastic impact force increases, and the idea that all the bifurcation pointsare advanced by random factor is presented. Numerical results show that the Laguerrepolynomial approximation is an effective approach to solving the problems of elasticimpact systems with random parameters.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

As is well known, non-smooth systems are often encountered in the engineering fields. Therefore, a great deal of attentionhas been paid to such systems [1–11]. Many important phenomena in these systems, including grazing bifurcation, torusbifurcation and Hopf bifurcation, have been studied in the past years [1–3]. Some effective methods were presented to inves-tigate the behavior of these systems. For instance, mean Poincaré map and superposition theorem were adopted to investi-gate a kind of non-smooth stochastic systems in Ref. [4]. A certain transformation, which could reduce an impact system toone without impact, was applied to a series of nonlinear non-smooth systems in Refs. [5,6]. However, not all non-smoothsystems can be studied by these methods because of the application scopes.

In this paper, a kind of non-smooth systems, named elastic impact system [12], is considered. The model of this system isshown in Fig. 1. The elastic impact occurs when the displacement x(t) is equal to the gap D. As we know, random factor is inev-itable in actual condition, so a system with a random parameter is proposed in the following discussion. The equation is given by

€xþ a _xþ bxþ cx3 þ dx2 _xþ f ðx; _xÞ ¼ F cosðxtÞ; ð1Þ

where

f ðx; _xÞ ¼k1ðx� DÞ þ c1 _x; x P D

0; x < D;

�ð2Þ

is the elastic impact force with spring stiffness k1 and damping constant c1. It exists only when the impact happens. Theequation implies that the system is a Duffing–Van der Pol oscillator. Here a,b,c are constants, Fcos(xt) is a given harmonicexcitation, d is a random parameter which can be written as d ¼ �dþ ru, where �d is a constant and u is a random variable thatsubjects to an exponential distribution with the intensity r.

It is well known that the orthogonal polynomial approximation is an available method for the systems with random param-eter under both smooth condition [13,14] and non-smooth condition [15]. Here we apply the Laguerre polynomial approxima-

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ax: +86 29 88495453.u.cn (L. Wang).

Fig. 1. Schematic of the elastic impact system.

2076 L. Wang et al. / Chaos, Solitons and Fractals 41 (2009) 2075–2080

tion [13] to change the stochastic system into two equivalent high dimension deterministic systems, and then we adopt numer-ical method to obtain the response. The main purpose of this paper is to find out the effects of elastic impact on the motions ofthe system. Furthermore, the differences between the stochastic system and the deterministic system are investigated.

The paper is organized as follows: Section 2 gives the concepts and some features of exponential distribution probabilitydensity function (PDF) and the Laguerre polynomial; in Section 3 we change the stochastic system to two high dimensiondeterministic systems; Section 4 analyzes the response of the systems by numerical simulation; and the conclusions are gi-ven in Section 5.

2. Exponential distribution PDF and Laguerre polynomial

Various random factors exist in practical engineering. If the sample of the random variable is positive and the PDF ismonotone decreasing to zero, the random variable could be supposed to subject to exponential distribution [16]. The PDFis shown in Fig. 2 and the expression is

pðuÞ ¼e�u; u P 00; u < 0:

�ð3Þ

The Laguerre polynomial is the best orthogonal polynomial basis [16] for this kind of PDF and can be given by

LnðuÞ ¼ eu dn

dun une�uð Þ; n ¼ 0;1;2; � � � ð4Þ

The recurrent formula is

Lnþ1ðuÞ ¼ ð1þ 2n� uÞLnðuÞ � n2Ln�1ðuÞ; n ¼ 1;2;3; � � � ð5Þ

The orthogonality of this polynomial can be shown as follows:

Z þ1

0LiðuÞLjðuÞpðuÞdu ¼

0; i–j

ðj!Þ2; i ¼ j:

�ð6Þ

Fig. 2. The exponential distribution PDF curve for random variable u.

L. Wang et al. / Chaos, Solitons and Fractals 41 (2009) 2075–2080 2077

A weighted orthogonal relationship is implied in Eq. (6), and the weight function is just the exponential distribution PDF.The expectation of the product Li(u)Lj(u) is put in the left-hand side of Eq. (6).

Therefore, any measurable function f(u) can be expanded into the following form according to the feature of the Laguerrepolynomial

f ðuÞ ¼Xþ1i¼0

ciLiðuÞ; ð7Þ

where ci ¼Rþ1

0 pðuÞf ðuÞLiðuÞdu. Eq. (7) is called an orthogonal decomposition of the random function f(u), which is the bestapproximation in the mean-square sense.

3. Laguerre polynomial approximation for the stochastic Duffing–Van der Pol system with elastic impact

Now we change the system presented by Eq. (1) into two stochastic systems as follows:

€xþ a _xþ bxþ cx3 þ ð�dþ ruÞx2 _x ¼ F cosðxtÞ; x < D; ð8Þ

€xþ a _xþ bxþ cx3 þ ð�dþ ruÞx2 _xþ k1ðx� DÞ þ c1 _x ¼ F cosðxtÞ; x P D: ð9Þ

First, we solve Eq. (8). According to the Laguerre polynomial approximation, the response of this system can be written as

xðt;uÞ ¼XN

i¼0

xiðtÞLiðuÞ: ð10Þ

The series mentioned above is the exact solution to Eq. (8) whenN ? +1. If N is a finite value, the series is only anapproximation. For the feasibility of calculation and precision requirement, we take N = 4. Then substituting Eq. (10) intoEq. (8), we have

d2

dt2 þ addtþ b

" #X4

i¼0

xiðtÞLiðuÞ þ cX4

i¼0

xiðtÞLiðuÞ !3

þ 13

�dddt

X4

i¼0

xiðtÞLiðuÞ !3

þ 13ru

ddt

X4

i¼0

xiðtÞLiðuÞ !3

¼ F cosðxtÞ:

ð11Þ

According to the property of the Laguerre polynomial, the nonlinear term in the left-hand side of Eq. (11) can be trans-formed into

X4

i¼0

xiðtÞLiðuÞ !3

¼X12

i¼0

XiðtÞLiðuÞ; ð12Þ

where the expression Xi(t) is the coefficient of Li(u)(i = 0,1,� � �,12). By the recurrent formula, the term with random parameteru in Eq. (11) can be rewritten as follows:

uddt

X4

i¼0

xiðtÞLiðuÞ !3

¼ uX12

i¼0

_XiðtÞLiðuÞ !

¼X12

i¼0

_XiðtÞ½uLiðuÞ� ¼X12

i¼0

_XiðtÞ½ð2iþ 1ÞLiðuÞ � Liþ1ðuÞ þ i2Li�1ðuÞ�; ð13Þ

and _XiðtÞ represents the derivative of Xi(t)(i = 0,1,� � �,12) with respect to t. Then multiplying both sides of Eq. (11) byLi(u)(i = 0,1,2,3,4) in sequence and taking expectation, owing to Eqs. (12) and (13), we will get a high dimension determin-istic system equivalent to Eq. (8), which is

d2

dt2 þ addtþ b

!x0 þ cX0 þ

13

�d _X0 þ13rð _X0 � _X1Þ ¼ F cosðxtÞ;

d2

dt2 þ addtþ b

!x1 þ cX1 þ

13

�d _X1 þ13rð3 _X1 � 4 _X2 � _X0Þ ¼ 0;

d2

dt2 þ addtþ b

!x2 þ cX2 þ

13

�d _X2 þ13rð5 _X2 � 9 _X3 � _X1Þ ¼ 0;

d2

dt2 þ addtþ b

!x3 þ cX3 þ

13

�d _X3 þ13rð7 _X3 � 16 _X4 � _X2Þ ¼ 0;

d2

dt2 þ addtþ b

!x4 þ cX4 þ

13

�d _X4 þ13rð9 _X4 � 25 _X5 � _X3Þ ¼ 0:

ð14Þ

The ensemble mean response (EMR) of Eq. (8) can be denoted by

2078 L. Wang et al. / Chaos, Solitons and Fractals 41 (2009) 2075–2080

E½xðt; uÞ� ¼ EX4

i¼0

xiðtÞLiðuÞ" #

¼ x0: ð15Þ

Thus, the constraint in Eq. (8) can be found as

E½xðt; uÞ� < D() x0 < D: ð16Þ

Then we deal with Eq. (9) in a similar way, finally we will obtain another equivalent deterministic system, which is

d2

dt2 þ addtþ bþ k1 þ c1

ddt

!x0 þ cX0 þ

13

�d _X0 þ13rð _X0 � _X1Þ � k1D ¼ F cosðxtÞ;

d2

dt2 þ addtþ bþ k1 þ c1

ddt

!x1 þ cX1 þ

13

�d _X1 þ13rð3 _X1 � 4 _X2 � _X0Þ ¼ 0;

d2

dt2 þ addtþ bþ k1 þ c1

ddt

!x2 þ cX2 þ

13

�d _X2 þ13rð5 _X2 � 9 _X3 � _X1Þ ¼ 0;

d2

dt2 þ addtþ bþ k1 þ c1

ddt

!x3 þ cX3 þ

13

�d _X3 þ13rð7 _X3 � 16 _X4 � _X2Þ ¼ 0;

d2

dt2 þ addtþ bþ k1 þ c1

ddt

!x4 þ cX4 þ

13

�d _X4 þ13rð9 _X4 � 25 _X5 � _X3Þ ¼ 0:

ð17Þ

EMR of Eq. (9) can be expressed by Eq. (15) and the constraint is

E½xðt; uÞ�P D() x0 P D: ð18Þ

According to the above work, the stochastic Duffing–Van der Pol elastic impact system, i.e., Eq. (1), has been transformedto two equivalent high dimension deterministic systems as shown in Eqs. (14) and (17). Then numerical method canbe applied to obtain the responses of these equations, through which we can get EMR of Eq. (1).

Fig. 3. Bifurcation diagram: (a) DR; (b) EMR with r = 0.01; (c) EMR0.

L. Wang et al. / Chaos, Solitons and Fractals 41 (2009) 2075–2080 2079

4. Response analysis

When r = 0, the stochastic system (1) degenerates into a deterministic system which can be written as

€xþ a _xþ bxþ cx3 þ �dx2 _xþ f ðx; _xÞ ¼ F cosðxtÞ; ð19Þ

where the function f ðx; _xÞ is shown in Eq. (2). It is well known that the response of Eq. (19), which can be named the deter-ministic response (DR), can be obtained by numerical method directly. By setting the parameter r = 0 in Eqs. (14) and (17),we can get the deterministic EMR, denoted by EMR0. The validity of the polynomial approximation for this non-smooth sys-tem can be verified by comparing EMR0 with DR. The differences between EMR (obtained by Eqs. (14) and (17) whenr = 0.01) and DR show the effect of the random factor on the system.

The system with parameters a = �0.1, b = �0.5, c = 0.5, d ¼ 0:1; F ¼ 3:5, x = 0.415 and D = 0 has been chosen for analysis.In the following research, we fix c1 = 0.04, with the increase of k1 from 0.15 to 0.85, we investigate the changes of eachresponse. The bifurcation diagrams are shown in Fig. 3. We can see that the responses of both the random and deterministicsystems change from chaos to periodic with the increase of k1, but there are still some differences among the figures. It isshown that Fig. 3(c) is the same as 3(a), in other words, EMR0 is consistent with DR, which indicates the validity of the poly-nomial approximation in non-smooth system; while Fig. 3(b) is quite different from Fig. 3(a) because of the effect of randomfactor. The detailed analysis is as follows: with the increase of parameter k1 from 0.15 to 0.3343, Fig. 3(a) and (c) exhibit thechaotic motions respectively, while the corresponding interval in Fig. 3(b) is k1 2 [0.15,0.2767). We set k1 = 0.286 and makethe Poincaré section to display the difference directly, see Fig. 4. Obviously, DR and EMR0 are both chaotic, as is implied inFig. 3(a) and (c). But EMR shows the periodic motion with period 8T(T = 2p/x) in Fig. 4(b). As the spring stiffness k1 increasesprogressively, the inverse period-doubling bifurcation is found in each response, but the bifurcation points are not identical,as is shown in Table 1 clearly.

According to Table 1 and Fig. 3, we can see that the values of corresponding bifurcation points in EMR are smaller thanthose in DR and EMR0. This result implies that the random factor can advance the occurrence of inverse period-doublingbifurcation with the increase of k1. Another obvious difference among the three responses is that EMR exhibits a skip atk1 = 0.4385, which doesn’t happen in Fig. 3(a) and (c).

Fig. 4. Poincaré section, k1 = 0.286: (a) DR; (b) EMR with r = 0.01; (c) EMR0.

Table 1The intervals of k1 for the motions of different responses

Motions Period

Chaos 8T 4T 2T T

ResponsesDR 0.15 6 k1 < 0.3343 0.3343 6 k1 < 0.3454 0.3454 6 k1 < 0.3997 0.3997 6 k1 < 0.8307 0.8307 6 k1 6 0.85EMR 0.15 6 k1 < 0.2767 0.2767 6 k1 < 0.2911 0.2911 6 k1 < 0.3543 0.3543 6 k1 < 0.7742 0.7742 6 k1 6 0.85EMR0 0.15 6 k1 < 0.3343 0.3343 6 k1 < 0.3454 0.3454 6 k1 < 0.3997 0.3997 6 k1 < 0.8307 0.8307 6 k1 6 0.85

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5. Conclusion

The response of a stochastic Duffing–Van der Pol elastic impact system is considered by changing the system to twoequivalent high dimension deterministic systems. The validity of this method is verified by comparing EMR0 with DR. Thedifferences between EMR and DR exhibit the effects of random parameter on the system. The results of numerical simulationshow that the elastic impact can change the motions of both the random and deterministic systems from chaos to periodicwith the increase of k1, and inverse period-doubling bifurcation is born in this process. Furthermore, the idea that the ran-dom factor can advance the occurrence of this phenomenon is proposed.

Acknowledgement

This research is supported by the National Natural Science Foundation of China (Grant Nos. 10472091 and 10332030),NSF of Guangdong Province (Grant No. 04011604).

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