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Research on parametric resonance in a stochastic van der Pol oscillator under multiple time delayed feedback control
ARTICLE IN PRESS International Journal of Non-Linear Mechanics 45 (2010) 621–627
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Research on parametric resonance in a stochastic van der Pol oscillator under multiple time delayed feedback control X.L. Yang a,, Z.K. Sun b a b
College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, PR China Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, PR China
a r t i c l e in fo
abstract
Article history: Received 7 June 2008 Received in revised form 16 February 2010 Accepted 14 March 2010
Analytical derivations and numerical calculations are employed to gain insight into the parametric resonance of a stochastically driven van der Pol oscillator with delayed feedback. This model is the prototype of a self-excited system operating with a combination of narrow-band noise excitation and two time delayed feedback control. A slow dynamical system describing the amplitude and phase of resonance, as well as the lowest-order approximate solution of this oscillator is firstly obtained by the technique of multiple scales. Then the explicit asymptotic formula for the largest Lyapunov exponent is derived. The influences of system parameters, such as magnitude of random excitation, tuning frequency, gains of feedback and time delays, on the almost-sure stability of the steady-state trivial solution are discussed under the direction of the signal of largest Lyanupov exponent. The non-trivial steady-state solution of mean square response of this system is studied by moment method. The results reveal the phenomenon of multiple solutions and time delays induced stabilization or unstabilization, moreover, an appropriate modulation between the two time delays in feedback control may be acted as a simple and efficient switch to adjust control performance from the viewpoint of vibration control. Finally, theoretical analysis turns to a validation through numerical calculations, and good agreements can be found between the numerical results and the analytical ones. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Parametric resonance Multiple time delayed feedback Random noise
1. Introduction As well known actual systems are inevitably fluctuated by various uncertain factors usually modeled by some kind of stochastic noise. On the other hand, time delay is an intrinsic feature of many mechanical, neural and natural systems [1–5] commonly in consequence of finite propagation speeds of signals, finite processing times in synapses, and so on. Thus, investigation on the interaction influences of random noise excitation and time delayed feedback on non-linear systems is a practical and interesting topic. Up to now, notable attentions among such problems focused on the study of system dynamics in time delay controlled dynamical systems without stochastic forcing, which has fascinated a great deal of researchers in various fields. For instance, Maccari [6,7] examined the resonance response of a parametrically or externally forced van der Pol oscillator with delayed state feedback. Using an asymptotic perturbation method, he pointed out that proper choices of feedback gains and time delay are possible to exclude quasi-periodic motion and reduce the response amplitude. By means of the center manifold technique
Corresponding author.
E-mail address: [email protected] (X.L. Yang). 0020-7462/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2010.03.005
and averaging theorem, Xu and Chung [8] described the effect of linear and non-linear time delayed position feedbacks on a Duffing–van der Pol oscillator. The system considered in [8] demonstrated rich dynamical behaviors under time delayed feedbacks, including phase shifting solution, chaos, period doubling and quasi-periodic motion. Through stability analysis and normal forms, Ji and Hansen [9] considered the effect of time delay on the stability of trivial equilibrium. The result indicated that the trivial equilibrium of the controlled system may lose its stability via a subcritical or supercritical Hopf bifurcation and regain its stability via a reverse subcritical or supercritical Hopf bifurcation as the time delay increases. Using the averaging method, Morrison and Rand [10] reported the dynamics of the time delayed Mathieu system in the neighborhood of 2:1 resonance. Mori and Kokame [11] obtained the delay-independent stability criterion in a system with single time delay, then Hu and Wang [12] proposed a practical delay-independent stability criterion for linear damped single degree of freedom system with two time delays. Comparing with the above abundant results provided by deterministic non-linear systems with time delayed feedback, to the author’s most knowledge, little literature [13,14] is available for time delayed non-linear systems subject to stochastic forcing. In Refs. [13,14], stability, bifurcation and jump of the steady-state response in a stochastic additionally or parametrically forced
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Duffing oscillator under a time delay feedback is studied by means of qualitative analysis. In order to reveal potential dynamics in stochastic non-linear systems with time delayed feedback, especially in the case of multiple time delayed feedback, the present work attempts to explore the non-linear dynamics of a self-excited oscillator in the neighborhood of parametric resonance under a combination of two time delayed feedback control and narrow-band noise excitation. As the first study, a stochastic parametrically excited van der Pol system under two time delayed linear feedback control is introduced. The oscillator of van der Pol originally arose as model in electric circuit theory. Now, many applications consisting of vehicle dynamics, oil-film journal bearings, flutter of plates and shells are all related with this system. The motion of the van der Pol oscillator is dominated by the following equation 2 0 u
u€ þ o
2
_ eað1u Þu_ þ eBuðtt1 Þ þ eC uðt t2 Þ ¼ euxðtÞ
The technique of multiple scales has been well-established by Hayfeh [16] and the references cited therein. Moreover, it has been extended to deal with the dynamics in non-linear systems with time delay (s) [13,14,17]. By the extended technique, the approximate asymptotic solution of system (1) is firstly derived in this section. A two scales expansion of the solution is sought in the form of ð2Þ
in which Ti ¼ ei t,i ¼ 0,1 are different time scales. The slow scale T0 characterizes the modulation in the amplitude and phase caused by the non-linearity and damping. The fast scale T1 is associated with the relatively fast changes in the response. Denoting Dn ¼ @=@Tn , the first and second derivatives with respect to t are given by d2 ¼ D20 þ 2eD0 D1 þOðe2 Þ dt 2
ð5Þ
The quantities a and j are arbitrary to this order of approximation and they will be determined by imposing solvability conditions at the next order of approximation. Substituting Eq. (5) into the second equation of Eq. (4) yields D20 u1 þ o20 u1 ¼ ð2a0 o0 aao0 þ 14a3 ao0 Ba sinðo0 t1 Þ þ Cao0 cosðo0 t2 Þ12Fa sin ZÞ sinðo0 T0 þ jðT1 ÞÞ þ ð2ao0 j0 Ba cosðo0 t1 ÞCao0 sinðo0 t2 Þ þ 12Fa cos ZÞ cosðo0 T0 þ jðT1 ÞÞ þ 14aa3 o0 sinð3ðo0 T0 þ jðT1 ÞÞÞ þ 12Fa cosð3ðo0 T0 þ jðT1 ÞÞÞ cos Z12Fa sinð3ðo0 T0 þ jðT1 ÞÞÞsin Z
ð6Þ
where prime denotes differentiation with respect to T1 and pffiffiffi Z ¼ sT1 þ g WðT1 Þ2jðT1 Þ, g ¼ g= e. In order to eliminate the secular producing terms in Eq. (6), it is required a and Z vary in the slow time scale 8 a a 3 B C F > 0 > > < a ¼ 2 a8 a þ 2o a sinðo0 t1 Þ2 a cosðo0 t2 Þ þ 4o a sin Z 0 0 > > aZ0 ¼ sa B a cosðo0 t1 ÞCa sinðo0 t2 Þ þ F a cos Z þag W 0 ðT1 Þ > : o0 2o0 ð7Þ Once solving a and Z, the lowest-order uniform expansion of the solution can be expressed 1 g ð8Þ ðOtZðetÞ þ pffiffiffi WðetÞÞ þ OðeÞ uðt, eÞ ¼ aðetÞcos 2 2 e
3. Steady-state response and stability analysis
2. Analysis of the lowest-order approximate solution
d ¼ D0 þ eD1 þOðe2 Þ, dt
u0 ðT0 ,T1 Þ ¼ aðT1 Þcosðo0 T0 þ jðT1 ÞÞ
ð1Þ
where u is the displacement of this oscillator, the dot denotes differentiation with respect to time t, 0 o e 5 1 is a small parameter, o0 is natural frequency, a is damped coefficient, B and C are feedback gains, t1 and t2 are the time delays in the paths of displacement feedback and velocity feedback, respectively. The random excitation xðtÞ [15], having the form xðtÞ ¼ F cosðOt þ gWðtÞÞ, is an ergodic narrow-band stochastic process with zero mean, in which W(t) stands for a standard Wiener process with the intensity of g, F and O are amplitude and frequency of the noise, respectively. The next section will discuss the lowest-order approximate solution by virtue of the technique of multiple scales. Presented in Section 3 are the steady-state response and its stability analysis. The comparisons between theoretical results and numerical ones are sketched in Section 4.
uðt, eÞ ¼ u0 ðT0 ,T1 Þ þ eu1 ðT0 ,T1 Þ þ Oðe2 Þ
assumed
ð3Þ
3.1. Trivial solution and the largest Lyapunov exponent Apparently, Eq. (7) possesses a trivial steady-state response, viz. a ¼0, whose stability would be analyzed in detail in the current subsection. Linearizing Eq. (7) at a¼0 and setting v ¼ ln a, one gets the dominated equations of v and Z in a sense of Itˆo differential equations. 8 > > dv ¼ a þ B sinðo0 t1 ÞC cosðo0 t2 Þ þ F sin Z dT1 > < 2 2o0 2 4o0 B F > > > : dZ ¼ so cosðo0 t1 ÞC sinðo0 t2 Þ þ 2o cos Z dT1 þ g dWðT1 Þ 0 0 ð9Þ It is clear that the diffused Markov process ZðT1 Þ is ergodic [18]. The time-independent stationary probability density pðZÞ of ZðT1 Þ satisfies the reduced Fokker–Plank equation d2 p d ½ðE1 E2 cos ZÞP ¼ 0 dZ2 dZ
Substituting Eqs. (2) and (3) into system (1) and equating coefficients of e and e2 to zero, it is easy to obtain
in which
8 2 D u þ o20 u0 ¼ 0 > > < 0 0 D20 u1 þ o20 u1 ¼ 2D0 D1 u0 þ að1u20 ÞD0 u0 Bu0 ðT0 t1 ,T1 ÞCD0 u0 ðT0 t2 ,T1 Þ > > : þ u0 F cosðOT0 þ g WðT1 ÞÞ
E0 ¼ s
ð4Þ To reveal the principle parametric resonance of the controlled system (1), the remain sections is confined to the case of O ¼ 2o0 þ es, where s is detuning frequency. Considering the first linear equation in Eq. (4), the general solution can be
B
o0
cosðo0 t1 ÞC sinðo0 t2 Þ,
ð10Þ
E1 ¼
2
g2
E0 ,
E2 ¼
F
o0 g 2
:
Being complemented with the periodicity condition pðZ þ2pÞ ¼ pðZÞ R 2p and the normality condition 0 pðZÞ dZ ¼ 1, the unique solution of Eq. (10) is then obtained 3 exp Z þ p E1 E2 sin Z Z Z þ 2p p 2 þ E2 sin y dy pðZÞ ¼ exp E1 yþ 2 4p2 IiE1 ðE2 ÞIiE1 ðE2 Þ Z
ð11Þ
ARTICLE IN PRESS X.L. Yang, Z.K. Sun / International Journal of Non-Linear Mechanics 45 (2010) 621–627
where In ðxÞ is the Bessel function of the first kind and n can be any real and complex number. According to the Oseledec multiplicative ergodic theorem [19], the largest Lyapunov exponent, characterized the exponential growth rate of the solution aðT1 Þ of Eq. (7) for any non-trivial initial values ða0 , Z0 Þ, is defined as l ¼ lim
1
T1 -1 T1
lnjaðT1 Þjw:p:1
B C F sinðo0 t1 Þ cosðo0 t2 Þ þ sin Zðt 0 Þ dt 0 2 2 4 o o 0 0 0 Z 2p a B C F ¼ þ sinðo0 t1 Þ cosðo0 t2 Þ þ sin ZpðZÞ dZ 2 4o0 0 2 2o0 E3 F I1iE1 ðE2 Þ I1 þ iE1 ðE2 Þ þ þ ¼ ð12Þ 8o0 IiE1 ðE2 Þ IiE1 ðE2 Þ 2 ¼ lim
1
T1 -1 T1
Z að0Þ þ
T1
a
2
þ
where E3 ¼ a
B
o0
sinðo0 t1 Þ þC cosðo0 t2 Þ:
Clearly, the stability of the trivial response is definitely dependent on the sign of the largest Lyapunov exponent l, namely the trivial solution is almost-sure asymptotic stable if and only if l o 0. Several cases of the variations of l versus noise amplitude, tuning frequency, feedback gains and time delays are described to reveal how these key parameters influence the stability of trivial solution. In the case of o0 ¼ 1:0, a ¼ 0:1, e ¼ 0:1, g ¼ 0:1 (these four parameters are fixed in this section), B ¼ 0:5, C ¼ 0:3, t1 ¼ 4:0, t2 ¼ 1:0, the dependences of the largest Lyapunov exponent l on noise amplitude F and tuning frequency s are illustrated in Fig. 1. The three-dimensional plots of l over the parameter ranges 6 r s r6 and 0 rF r 6 are depicted in Fig. 1(a), through which one can find that there are two different solution ranges for the largest Lyapunov exponent. Near s ¼0, i.e., the parameter resonance at excitation frequency O ¼2o0, the largest Lyapunov exponent increases and reaches its maximum value at the center of the instability region. Out of this mountain, there is a plane in which the largest Lyapunov exponent maintains a certain negative value. To give a vivid description of the largest Lyapunov exponent imposed by tuning frequency, the evolution of l with s are portrayed in Fig. 1(b), which says as the tuning frequency increases, the sign of largest Lyapunov exponent is firstly negative, then positive, and once again negative, then system (1) is correspondingly stable, unstable and stable in succession. Fig. 1 may imply that, when system (1) is far away parameter resonance, the stability of trivial solution is more robust against the disturbance of noise excitation than that when system (1) is near parameter resonance. Moreover, the nearer system (1) approaches parameter resonance, the more unstable the system is.
Fig. 1. The evolutions of the largest Lyapunov exponent: (a) l versus s and F; (b) l versus s with fixed F.
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In addition, the noise excitation of large amplitude easily brings stability in resonance region, which is consistent with physical intuition. The influences of feedback gains on the largest Lyapunov exponent are demonstrated in Fig. 2 when t1 ¼ 4:0, t2 ¼ 1:0, s ¼0.2, F¼ 1.4. Fig. 2(a) displays the three-dimensional plots of g over the parameter ranges 3 rB r 3 and 3 r C r 3, from which we can see clearly that l decreases from positive to negative on the whole as both the displacement feedback gain and velocity feedback gain increases. This is further confirmed by the (l,C) curves depicted in Fig. 2(b). Fig. 2 may indicate that large feedback gains enable us easily to stabilize the trivial solution of system (1). When the parameters are fixed as s ¼0.2, F¼1.4, B ¼0.5, C¼ 0.3, Fig. 3(a) depicts the three-dimensional figures of l versus the modulation of two time delays, together with Fig. 3(b) showing the stability boundary of trivial response in t1 2t2 plane. Obviously, the three-dimensional figure presents a variety of l over the parameter ranges 0 r t1 r 6 and 0 r t2 r 6. The t1 2t2 plane is divided into three parts by equating l to zero: region I corresponding to l 4 0 (which consists of two separate parts in the parameter plane) and region II corresponding to l o0. That is to say, if the two time delays fall into region II, trivial solution of system (1) is asymptotically stable, otherwise, the trivial solution is unstable. This may imply proper choices of time delays shall provide a better vibration control.
3.2. The case of non-trivial steady-state solution When g is sufficiently small, the classical perturbation method and the stochastic moment technique [20] are utilized to analyze further the non-trivial steady-state solution of system (1). For
Fig. 2. The evolutions of the largest Lyapunov exponent: (a) l versus B and C; (b) l versus C with fixed B.
Fig. 3. (a) The variations of the largest Lyapunov exponent with t1 and t2; (b) stability boundary of trivial solution.
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non-zero response, Eq. (7) can be recast as 8 a a 3 B C F > 0 > > < a ¼ 2 a8 a þ 2o a sinðo0 t1 Þ2 a cosðo0 t2 Þ þ 4o a sin Z 0 0 > > Z0 ¼ s B cosðo0 t1 ÞC sinðo0 t2 Þ þ F cos Z þ g W 0 ðT1 Þ > : o0 2o0 ð13Þ The non-trivial steady-state response of Eq. (13) is firstly determined in the case of g ¼0. Setting a0 ¼ 0, Z0 ¼ 0, a set of algebraic equations, governing the amplitude a0 and phase Z0 of the steady-state parametric resonance, are obtained 8 B F > > > < so cosðo0 t1 ÞC sinðo0 t2 Þ þ 2o cos Z0 ¼ 0 0 0 ð14Þ a 2 a 2 B C F > > a a sinðo0 t1 Þ cosðo0 t2 Þ þ sin Z0 ¼ 0 > : 0 0þ 2 8 2o0 2 4o0 Then frequency response relation can be expressed 8 > a 2 F2 > > E3 a20 þE20 ¼ < 4 4o20 > a 2 > > : tan Z0 ¼ 2 E3 þ a20 =g E1 4
a ¼ a0 þ a1 , ð15Þ
According to the Routh–Hurwitz criterion [20], the steadystate solution of Eq. (13) is asymptotically stable if the following two inequalities hold simultaneously 8 2 < aa0 4 0 a ð16Þ : E3 þ a20 4 0 4 For typical cases, existence regions and stability analysis for the non-trivial steady-state solution are illustrated in (F, s) plane for the uncontrolled system (Fig. 4(a)) and the controlled system (Fig. 4(b)). In Fig. 4, Region I stands for a region where there is no non-trivial steady-state response for system (1), while region II represents a region in which there is an asymptotical stable nontrivial steady-state solution. In the remanent region, there are two non-trivial steady-state solutions with the large solution stable and the small one unstable. Drawing a comparison between these two figures, the region for two non-trivial steady-state solutions is distinctly enlarged when the time delayed feedback control is present in system (1). Setting @a0 =@t1 ¼ 0,@a0 =@t2 ¼ 0 in Eq. (14), one can readily find the time delays ðt 1 , t 2 Þ for the possible maximum or minimum amplitude of the steady-state response. These two time delays, viz. ðt 1 , t 2 Þ, correspond to a maximum value of the response if the second partial derivatives satisfy @2 a0 @2 a0 @ 2 a0 40 2 2 @t1 @t2 @t1 @t2
and
Fig. 5 exhibits the dependences of the amplitude a0 of the steadystate response on tuning frequency s for the uncontrolled system (the line of A1), a controlled system with no time delay (the line of A2), a controlled system with a certain time delays (the line of A3) and a controlled system with different time delays corresponding to the maximum amplitude for the response (the line of A4). The dashed lines symbolize unstable motion while the solid lines stable motion. Numerical integral of the original system for the maximum response is performed and the results are depicted by asterisks ‘‘ ’’ in Fig. 5. Obviously, numerical results agree well with theoretical ones. Moreover, Fig. 5 reveals that an appropriate modulation between the two time delays in feedback control not only can reduce the response amplitude but also can enlarge the response amplitude. Then time delays appear to be acted as dual roles from the view point of vibration control. In the following part, the steady-state response of system (1) is discussed in a mean square sense when g a0 but is sufficiently small. Let
Z ¼ Z0 þ Z1
ð17Þ
in which a0 and Z0 are steady-state solutions governing by Eq. (15), a1 and Z1 are small perturbations. Linearizing Eq. (13) at (a0, Z0) with respect to a and Z, one can obtain 8 a F > > a0 Z1 cos Z0 dT1 > da1 ¼ a20 a1 þ < 4 4o0 ð18Þ F > > > : dZ1 ¼ 2o Z1 sin Z0 dT1 þ g dWðT1 Þ 0 Using the moment method, the steady-state moments of first order are given Ea1 ¼ EZ1 ¼ 0
ð19Þ
According to the Itˆo differential formulas, the second moments of (a1, Z1) satisfy 8 dEa21 1 2 2 F > > > > dT 2 ¼ 2 aa0 Ea1 þ 2o a0 cos Z0 Ea1 Z1 > > 0 1 > > > < dEZ2 F 1 2 ¼ sin Z0 EZ1 þ g 2 ð20Þ o0 > dT12 > > > > > dEa1 Z1 1 1 2 F > 2 > > : dT 2 ¼ 4o0 Fa0 cos Z0 EZ1 4 aa0 þ 2o0 sin Z0 Ea1 Z1 1
@2 a0 o0 @t21
Fig. 4. Existence regions for the non-trivial steady-state solution in the (F, s) plane: (a) the uncontrolled system, (b) the time-delay controlled system with o0 ¼1.0, a ¼ 0.1, B¼ 0.5, C¼0.3, t1 ¼ 0.5, t2 ¼ 2.5.
Fig. 5. Frequency-amplitude relation of the parametric resonance for the uncontrolled system (the line of A1), a controlled system with B¼ 0.5, C¼ 0.3, t1 ¼ 0.0, t2 ¼0.0. (the line of A2), a controlled system with B¼ 0.5, C¼ 0.3, t1 ¼ 0.5, t2 ¼ 2.5 (A3) and a controlled system with B ¼0.5, C¼0.3 and different time delays (the line of A4).
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Then the second steady-state moments are expressed 8 > ða0 F cos Z0 =2o0 Þ2 g 2 > 2 > g2 > Ea1 ¼ 2 > > aa0 F sin Z0 ðaa20 =4þ F sin Z0 =2o0 Þ=o0 > > < o0 2 EZ21 ¼ g F sin Z0 > > > > > a0 cos Z0 2 > > > : Ea1 Z1 ¼ ðaa2 þ 2F sin Z =o0 Þsin Z g 0 0 0
4. Numerical calculations
ð21Þ
To analyze the stability of the second steady-state moments, the Jacobian matrix can be obtained by performing linearization of Eq. (20) 2 3 0 Fa0 cos Z0 =2o0 aa20 =2 6 7 0 F sin Z0 =o0 0 ð22Þ 4 5 2 0 Fa0 cos Z0 =4o0 ðaa0 =4 þ F sin Z0 =2o0 Þ The second steady-state moments are asymptotic stable when the real parts of the eigenvalues in characteristic equation are all negative. According to the Routh–Hurwitz criterion [20], the necessary and sufficient condition for all the eigenvalues having negative parts is the two inequalities hold simultaneously 8 2 < aa0 4 0 a ð23Þ : E3 þ a20 4 0 4 which accords with the stable condition of Eq. (14). Combining Eqs. (19) and (21), the mean square steady-state response of system (1) is Ea2 ¼ a20 þ
ða0 F cos Z0 =2o0 Þ2 g 2 g2 aa20 F sin Z0 ðaa20 =4þ F sin Z0 =2o0 Þ=o0
625
ð24Þ
In this section, numerical simulations are performed by numerically integrating the time delayed van der Pol oscillator (1) using 4th Runge–Kutta algorithm. For the simulations of narrowband noise, readers can refer to Shinozuka [21] and Zhu [22]. Firstly, simulations of the stochastic response for the controlled system (1) are employed to verify the analytical results derived from the largest Lyapunov exponent. Fig. 1(b) shows when freezing noise amplitude the largest Lyapunov exponent is firstly negative, then positive, and once again negative as tuning frequency increases. For instance, when F¼1.4 l is less than zero for s ¼ 0.8 and s ¼0.8, while it is more than zero for s ¼ 0.0. The change of the largest Lyapunov exponent from minus to plus definitely denotes a switching of the stability, i.e. from stable to unstable, of the trivial response caused only by the frequency. Figs. 6(a) and (c) depict the time histories of system (1) in the case of s ¼ 0.8 and s ¼0.8, respectively, showing the controlled van der Pol oscillator is stabilized asymptotically to the stable trivial solution after a short transition, namely the trivial solution of system (1) is asymptotically stable. While the time history of controlled van der Pol oscillator in the case of s ¼ 0.0, displayed in Fig. 6(b), is oscillatory regularly, namely the trivial solution of system (1) is not stable. Fig. 3(b) indicates that modulation of the two time delays in displacement feedback and velocity feedback can adjust the stability property of the trivial solution. When the two time delays ðt1 , t2 Þ fall into the stable region, for example t1 ¼ 4, t2 ¼ 0:2 and t1 ¼ 4, t2 ¼ 5, the stochastic response of system (1) undergoes a transition stage and then asymptotically converges to the stable trivial solution, which are displayed in Figs. 7(a) and (c),
Fig. 6. The time histories of system (1) corresponding to Fig. 1(b): (a) F ¼1.4, s ¼ 0.8; (b) F ¼1.4, s ¼ 0.0; (c) F¼ 1.4, s ¼0.8.
Fig. 7. The time histories of system (1) corresponding to Fig. 3(b): (a) t1 ¼4, t2 ¼ 0.2; (b) t1 ¼4, t2 ¼ 3; (c) t1 ¼ 4, t2 ¼5.
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Fig. 8. (a) The deterministic steady-state amplitude a0 and (b) steady-state mean square response of system (1): — theoretical results;
numerical results.
Fig. 9. Influences of bandwidth of noise excitation on the response are depicted by the time history, phase portrait and power spectrum: the left line for g ¼ 0; the middle line for g ¼0.1; the right line for g ¼1.0.
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respectively. In the case of t1 ¼ 4 and t2 ¼ 3, the two time delays ðt1 , t2 Þ situate in the unstable region of Fig. 3(b), the corresponding oscillatory response is demonstrated in Fig. 7(b). Obviously, numerical calculations for the stability of the trivial solution are consistent with theoretical analysis. Without loss of generality, when the parameters of system (1) are selected as o0 ¼ 1:0, a ¼ 0:1, B ¼ 0:5, C ¼ 0:3, e ¼ 0:1, g ¼ 0:01, s ¼ 0:1, F ¼ 1:0, t1 ¼ 4, t2 ¼ 1, Fig. 8 depicts the variations of steady-state response of the controlled van der Pol oscillator (1). The deterministic steady-state amplitude a0 as a function of noise strength F is showed in Fig. 8(a), and the mean square response Ea2 versus tuning frequency is displayed in Fig. 8(b). The solid lines stand for approximate analytical results obtained in Section 3.2, and the asterisks represent numerical results. The well coincidence of the analytical results and numerical ones readily confirms that the present analysis is effective. Secondly, further numerical calculations are employed to reveal how the bandwidth of narrow-band noise influences the stochastic response of system (1). In the case of o0 ¼ 1:0, a ¼ 0:1, B ¼ 0:5, C ¼ 0:3, e ¼ 0:1, s ¼ 0:1, F ¼ 1:5, t1 ¼ 0:5, t2 ¼ 2:5, the results of time history, phase portrait as well as power spectrum are illustrated in Fig. 9 for different bandwidth of stochastic noise. When g ¼ 0, i.e., the narrow-band noise is degenerated into a deterministic harmonic excitation, the time history along with the phase portrait and the power spectrum of system (1) is portrayed in Figs. 9(a)–(c), respectively. The corresponding results for g ¼ 0.1 and g ¼1.0 are respectively illustrated in Figs. 9(d)–(f) and (h)–(j). These results may indicate that the random noise gW(t) can change the periodic motion to a quasiperiodic motion, and change a limit cycle to a diffused limit cycle. Moreover, the width of the diffused limit cycle increases as the intensity of the random excitation increases. The remarkable influences are further confirmed by the variations of power spectrum, in which it changes from a spectrum with a peak, a spectrum with many discrete peaks to a continuous, broad-band spectrum as the noise intensity increases.
5. Conclusions The present paper studies the principal parametric resonance of a self-excited van der Pol oscillator in the presence of random narrow-band excitation and multiple time delayed state feedback control. The narrow-band noise is a more realistic model in engineering applications than white noise. The state feedback term involves both displacement and velocity feedback. Using a cooperation of analytical methods and numerical analysis, the steady-state trivial and non-trivial responses as well as their dynamic stability have been analyzed in various regimes characterized by tuning frequency, random noise and time delayed feedback. The principal results are presented in these parameter spaces revealing a rich variety of dynamic behaviors including the phenomenon of multiple solutions and time delays induced stabilization or unstabilization. Moreover, an appropriate modulation between the two time delays in feedback control can enhance or reduce the response amplitude. Thus the multiple
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time delayed feedback may be acted as a simple and efficient switch to adjust control performance from the viewpoint of vibration control. In addition, the random noise can change a limit cycle to a diffused limit cycle, the width of which increases as the intensity of random excitation increases.
Acknowledgements This work is partially supported by the National Natural Science Foundation of China (Grant nos. 10902062, 10871123 and 10902085), the NSF of Shaanxi Province (Grant no. 2009JQ1002), and the Outstanding Science and Technology Pre-Research Foundation of Shaanxi Normal University, China. References [1] W. Wischert, A. Wunderlin, A. Pelster, M. Olivier, J. Groslambert, Delayinduced instabilities in nonlinear feedback systems, Phys. Rev. E 49 (1994) 203–219. [2] K. Miyakawa, K. Yamada, Entrainment in coupled salt-water oscillators, Physica D 127 (1999) 177–186. [3] P. Tass, J. Kurths, M.G. Rosenblum, G. Guasti, H. Hefter, Delayinduced transitions in visually guided movements, Phys. Rev. E 54 (1996) R2224–R2227. [4] A. Destexhe, Stability of periodic oscillations in a network of neurons with time delay, Phys. Lett. A 187 (1994) 309–316. [5] Z.K. Sun, W. Xu, X.L. Yang, T. Fang, Inducing or suppressing chaos in a doublewell Duffing oscillator by time delay feedback, Chaos, Solitons Fractals 27 (3) (2006) 705–714. [6] A. Maccari, The response of a parametrically excited van der Pol oscillator to a time delay state feedback, Nonlinear Dyn. 26 (2001) 105–119. [7] A. Maccari, Vibration control for the primary resonance of the van der Pol oscillator by a time delay state feedback, Int. J. Nonlinear Mech. 38 (2003) 123–131. [8] J. Xu, K.W. Chung, Effects of time delayed position feedback on a van der Pol– Duffing oscillator, Physica D 180 (2003) 17–39. [9] J.C. Ji, C.H. Hansen, Stability and dynamics of a controlled van der Pol–Duffing oscillator, Chaos, Solitons Fractals 28 (2006) 555–570. [10] T.M. Morrison, R.H. Rand, 2:1 Resonance in the delayed nonlinear Mathieu equation, Nonlinear Dyn. 50 (2007) 341–352. _ ¼ AxðtÞ þ BxðttÞ, IEEE Trans. Autom. [11] T. Mori, H. Kokame, Stability of xðtÞ Control 34 (1989) 460–462. [12] H.Y. Hu, Z.H. Wang, Stability of analysis of damped SDof systems with two time delays in state feedback, J. Sound Vib. 214 (1998) 213–225. [13] Z.K. Sun, W. Xu, X.L. Yang, Response of nonlinear system to random narrowband excitation with time delay state feedback, J. Vib. Eng. 19 (2006) 57–64. [14] Y.F. Jin, H.Y. Hu, Principal resonance of a Duffing oscillator with delayed state feedback under narrow-band random parametric excitation, Nonlinear Dyn. 50 (2007) 213–227. [15] W.V. Weding, Invariant measures and Lyapunov exponents for generalized parameter fluctuations, Struct. Saf. 8 (1990) 13–25. [16] A.H. Hayfeh, B. Blachandran, Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods, John Wiley & Sons, Inc., New York, 1995. [17] H.Y. Hu, Z.H. Wang, Dynamics of Controlled Mechanical Systems with Delayed Feedback, Springer-Verlag, Berlin Heidelberg, 2002. [18] R.Z. Khasminskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen and den Rijn, 1980. [19] V.I. Oseledec, A multiplicative ergodic theorem Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (2) (1968) 197–231. [20] G. Schmidt, A. Tondl, Nonlinear Vibrations, Cambridge University Press, Cambridge, 1986. [21] M. Shinozuka, Digital simulation of random processes and its applications, J. Sound Vib. 25 (1972) 111–128. [22] W.Q. Zhu, Random Vibration, Beijing Science Press, China, 1992.
Contents lists available at ScienceDirect
International Journal of Non-Linear Mechanics journal homepage: www.elsevier.com/locate/nlm
Research on parametric resonance in a stochastic van der Pol oscillator under multiple time delayed feedback control X.L. Yang a,, Z.K. Sun b a b
College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, PR China Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, PR China
a r t i c l e in fo
abstract
Article history: Received 7 June 2008 Received in revised form 16 February 2010 Accepted 14 March 2010
Analytical derivations and numerical calculations are employed to gain insight into the parametric resonance of a stochastically driven van der Pol oscillator with delayed feedback. This model is the prototype of a self-excited system operating with a combination of narrow-band noise excitation and two time delayed feedback control. A slow dynamical system describing the amplitude and phase of resonance, as well as the lowest-order approximate solution of this oscillator is firstly obtained by the technique of multiple scales. Then the explicit asymptotic formula for the largest Lyapunov exponent is derived. The influences of system parameters, such as magnitude of random excitation, tuning frequency, gains of feedback and time delays, on the almost-sure stability of the steady-state trivial solution are discussed under the direction of the signal of largest Lyanupov exponent. The non-trivial steady-state solution of mean square response of this system is studied by moment method. The results reveal the phenomenon of multiple solutions and time delays induced stabilization or unstabilization, moreover, an appropriate modulation between the two time delays in feedback control may be acted as a simple and efficient switch to adjust control performance from the viewpoint of vibration control. Finally, theoretical analysis turns to a validation through numerical calculations, and good agreements can be found between the numerical results and the analytical ones. & 2010 Elsevier Ltd. All rights reserved.
Keywords: Parametric resonance Multiple time delayed feedback Random noise
1. Introduction As well known actual systems are inevitably fluctuated by various uncertain factors usually modeled by some kind of stochastic noise. On the other hand, time delay is an intrinsic feature of many mechanical, neural and natural systems [1–5] commonly in consequence of finite propagation speeds of signals, finite processing times in synapses, and so on. Thus, investigation on the interaction influences of random noise excitation and time delayed feedback on non-linear systems is a practical and interesting topic. Up to now, notable attentions among such problems focused on the study of system dynamics in time delay controlled dynamical systems without stochastic forcing, which has fascinated a great deal of researchers in various fields. For instance, Maccari [6,7] examined the resonance response of a parametrically or externally forced van der Pol oscillator with delayed state feedback. Using an asymptotic perturbation method, he pointed out that proper choices of feedback gains and time delay are possible to exclude quasi-periodic motion and reduce the response amplitude. By means of the center manifold technique
Corresponding author.
E-mail address: [email protected] (X.L. Yang). 0020-7462/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijnonlinmec.2010.03.005
and averaging theorem, Xu and Chung [8] described the effect of linear and non-linear time delayed position feedbacks on a Duffing–van der Pol oscillator. The system considered in [8] demonstrated rich dynamical behaviors under time delayed feedbacks, including phase shifting solution, chaos, period doubling and quasi-periodic motion. Through stability analysis and normal forms, Ji and Hansen [9] considered the effect of time delay on the stability of trivial equilibrium. The result indicated that the trivial equilibrium of the controlled system may lose its stability via a subcritical or supercritical Hopf bifurcation and regain its stability via a reverse subcritical or supercritical Hopf bifurcation as the time delay increases. Using the averaging method, Morrison and Rand [10] reported the dynamics of the time delayed Mathieu system in the neighborhood of 2:1 resonance. Mori and Kokame [11] obtained the delay-independent stability criterion in a system with single time delay, then Hu and Wang [12] proposed a practical delay-independent stability criterion for linear damped single degree of freedom system with two time delays. Comparing with the above abundant results provided by deterministic non-linear systems with time delayed feedback, to the author’s most knowledge, little literature [13,14] is available for time delayed non-linear systems subject to stochastic forcing. In Refs. [13,14], stability, bifurcation and jump of the steady-state response in a stochastic additionally or parametrically forced
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Duffing oscillator under a time delay feedback is studied by means of qualitative analysis. In order to reveal potential dynamics in stochastic non-linear systems with time delayed feedback, especially in the case of multiple time delayed feedback, the present work attempts to explore the non-linear dynamics of a self-excited oscillator in the neighborhood of parametric resonance under a combination of two time delayed feedback control and narrow-band noise excitation. As the first study, a stochastic parametrically excited van der Pol system under two time delayed linear feedback control is introduced. The oscillator of van der Pol originally arose as model in electric circuit theory. Now, many applications consisting of vehicle dynamics, oil-film journal bearings, flutter of plates and shells are all related with this system. The motion of the van der Pol oscillator is dominated by the following equation 2 0 u
u€ þ o
2
_ eað1u Þu_ þ eBuðtt1 Þ þ eC uðt t2 Þ ¼ euxðtÞ
The technique of multiple scales has been well-established by Hayfeh [16] and the references cited therein. Moreover, it has been extended to deal with the dynamics in non-linear systems with time delay (s) [13,14,17]. By the extended technique, the approximate asymptotic solution of system (1) is firstly derived in this section. A two scales expansion of the solution is sought in the form of ð2Þ
in which Ti ¼ ei t,i ¼ 0,1 are different time scales. The slow scale T0 characterizes the modulation in the amplitude and phase caused by the non-linearity and damping. The fast scale T1 is associated with the relatively fast changes in the response. Denoting Dn ¼ @=@Tn , the first and second derivatives with respect to t are given by d2 ¼ D20 þ 2eD0 D1 þOðe2 Þ dt 2
ð5Þ
The quantities a and j are arbitrary to this order of approximation and they will be determined by imposing solvability conditions at the next order of approximation. Substituting Eq. (5) into the second equation of Eq. (4) yields D20 u1 þ o20 u1 ¼ ð2a0 o0 aao0 þ 14a3 ao0 Ba sinðo0 t1 Þ þ Cao0 cosðo0 t2 Þ12Fa sin ZÞ sinðo0 T0 þ jðT1 ÞÞ þ ð2ao0 j0 Ba cosðo0 t1 ÞCao0 sinðo0 t2 Þ þ 12Fa cos ZÞ cosðo0 T0 þ jðT1 ÞÞ þ 14aa3 o0 sinð3ðo0 T0 þ jðT1 ÞÞÞ þ 12Fa cosð3ðo0 T0 þ jðT1 ÞÞÞ cos Z12Fa sinð3ðo0 T0 þ jðT1 ÞÞÞsin Z
ð6Þ
where prime denotes differentiation with respect to T1 and pffiffiffi Z ¼ sT1 þ g WðT1 Þ2jðT1 Þ, g ¼ g= e. In order to eliminate the secular producing terms in Eq. (6), it is required a and Z vary in the slow time scale 8 a a 3 B C F > 0 > > < a ¼ 2 a8 a þ 2o a sinðo0 t1 Þ2 a cosðo0 t2 Þ þ 4o a sin Z 0 0 > > aZ0 ¼ sa B a cosðo0 t1 ÞCa sinðo0 t2 Þ þ F a cos Z þag W 0 ðT1 Þ > : o0 2o0 ð7Þ Once solving a and Z, the lowest-order uniform expansion of the solution can be expressed 1 g ð8Þ ðOtZðetÞ þ pffiffiffi WðetÞÞ þ OðeÞ uðt, eÞ ¼ aðetÞcos 2 2 e
3. Steady-state response and stability analysis
2. Analysis of the lowest-order approximate solution
d ¼ D0 þ eD1 þOðe2 Þ, dt
u0 ðT0 ,T1 Þ ¼ aðT1 Þcosðo0 T0 þ jðT1 ÞÞ
ð1Þ
where u is the displacement of this oscillator, the dot denotes differentiation with respect to time t, 0 o e 5 1 is a small parameter, o0 is natural frequency, a is damped coefficient, B and C are feedback gains, t1 and t2 are the time delays in the paths of displacement feedback and velocity feedback, respectively. The random excitation xðtÞ [15], having the form xðtÞ ¼ F cosðOt þ gWðtÞÞ, is an ergodic narrow-band stochastic process with zero mean, in which W(t) stands for a standard Wiener process with the intensity of g, F and O are amplitude and frequency of the noise, respectively. The next section will discuss the lowest-order approximate solution by virtue of the technique of multiple scales. Presented in Section 3 are the steady-state response and its stability analysis. The comparisons between theoretical results and numerical ones are sketched in Section 4.
uðt, eÞ ¼ u0 ðT0 ,T1 Þ þ eu1 ðT0 ,T1 Þ þ Oðe2 Þ
assumed
ð3Þ
3.1. Trivial solution and the largest Lyapunov exponent Apparently, Eq. (7) possesses a trivial steady-state response, viz. a ¼0, whose stability would be analyzed in detail in the current subsection. Linearizing Eq. (7) at a¼0 and setting v ¼ ln a, one gets the dominated equations of v and Z in a sense of Itˆo differential equations. 8 > > dv ¼ a þ B sinðo0 t1 ÞC cosðo0 t2 Þ þ F sin Z dT1 > < 2 2o0 2 4o0 B F > > > : dZ ¼ so cosðo0 t1 ÞC sinðo0 t2 Þ þ 2o cos Z dT1 þ g dWðT1 Þ 0 0 ð9Þ It is clear that the diffused Markov process ZðT1 Þ is ergodic [18]. The time-independent stationary probability density pðZÞ of ZðT1 Þ satisfies the reduced Fokker–Plank equation d2 p d ½ðE1 E2 cos ZÞP ¼ 0 dZ2 dZ
Substituting Eqs. (2) and (3) into system (1) and equating coefficients of e and e2 to zero, it is easy to obtain
in which
8 2 D u þ o20 u0 ¼ 0 > > < 0 0 D20 u1 þ o20 u1 ¼ 2D0 D1 u0 þ að1u20 ÞD0 u0 Bu0 ðT0 t1 ,T1 ÞCD0 u0 ðT0 t2 ,T1 Þ > > : þ u0 F cosðOT0 þ g WðT1 ÞÞ
E0 ¼ s
ð4Þ To reveal the principle parametric resonance of the controlled system (1), the remain sections is confined to the case of O ¼ 2o0 þ es, where s is detuning frequency. Considering the first linear equation in Eq. (4), the general solution can be
B
o0
cosðo0 t1 ÞC sinðo0 t2 Þ,
ð10Þ
E1 ¼
2
g2
E0 ,
E2 ¼
F
o0 g 2
:
Being complemented with the periodicity condition pðZ þ2pÞ ¼ pðZÞ R 2p and the normality condition 0 pðZÞ dZ ¼ 1, the unique solution of Eq. (10) is then obtained 3 exp Z þ p E1 E2 sin Z Z Z þ 2p p 2 þ E2 sin y dy pðZÞ ¼ exp E1 yþ 2 4p2 IiE1 ðE2 ÞIiE1 ðE2 Þ Z
ð11Þ
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where In ðxÞ is the Bessel function of the first kind and n can be any real and complex number. According to the Oseledec multiplicative ergodic theorem [19], the largest Lyapunov exponent, characterized the exponential growth rate of the solution aðT1 Þ of Eq. (7) for any non-trivial initial values ða0 , Z0 Þ, is defined as l ¼ lim
1
T1 -1 T1
lnjaðT1 Þjw:p:1
B C F sinðo0 t1 Þ cosðo0 t2 Þ þ sin Zðt 0 Þ dt 0 2 2 4 o o 0 0 0 Z 2p a B C F ¼ þ sinðo0 t1 Þ cosðo0 t2 Þ þ sin ZpðZÞ dZ 2 4o0 0 2 2o0 E3 F I1iE1 ðE2 Þ I1 þ iE1 ðE2 Þ þ þ ¼ ð12Þ 8o0 IiE1 ðE2 Þ IiE1 ðE2 Þ 2 ¼ lim
1
T1 -1 T1
Z að0Þ þ
T1
a
2
þ
where E3 ¼ a
B
o0
sinðo0 t1 Þ þC cosðo0 t2 Þ:
Clearly, the stability of the trivial response is definitely dependent on the sign of the largest Lyapunov exponent l, namely the trivial solution is almost-sure asymptotic stable if and only if l o 0. Several cases of the variations of l versus noise amplitude, tuning frequency, feedback gains and time delays are described to reveal how these key parameters influence the stability of trivial solution. In the case of o0 ¼ 1:0, a ¼ 0:1, e ¼ 0:1, g ¼ 0:1 (these four parameters are fixed in this section), B ¼ 0:5, C ¼ 0:3, t1 ¼ 4:0, t2 ¼ 1:0, the dependences of the largest Lyapunov exponent l on noise amplitude F and tuning frequency s are illustrated in Fig. 1. The three-dimensional plots of l over the parameter ranges 6 r s r6 and 0 rF r 6 are depicted in Fig. 1(a), through which one can find that there are two different solution ranges for the largest Lyapunov exponent. Near s ¼0, i.e., the parameter resonance at excitation frequency O ¼2o0, the largest Lyapunov exponent increases and reaches its maximum value at the center of the instability region. Out of this mountain, there is a plane in which the largest Lyapunov exponent maintains a certain negative value. To give a vivid description of the largest Lyapunov exponent imposed by tuning frequency, the evolution of l with s are portrayed in Fig. 1(b), which says as the tuning frequency increases, the sign of largest Lyapunov exponent is firstly negative, then positive, and once again negative, then system (1) is correspondingly stable, unstable and stable in succession. Fig. 1 may imply that, when system (1) is far away parameter resonance, the stability of trivial solution is more robust against the disturbance of noise excitation than that when system (1) is near parameter resonance. Moreover, the nearer system (1) approaches parameter resonance, the more unstable the system is.
Fig. 1. The evolutions of the largest Lyapunov exponent: (a) l versus s and F; (b) l versus s with fixed F.
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In addition, the noise excitation of large amplitude easily brings stability in resonance region, which is consistent with physical intuition. The influences of feedback gains on the largest Lyapunov exponent are demonstrated in Fig. 2 when t1 ¼ 4:0, t2 ¼ 1:0, s ¼0.2, F¼ 1.4. Fig. 2(a) displays the three-dimensional plots of g over the parameter ranges 3 rB r 3 and 3 r C r 3, from which we can see clearly that l decreases from positive to negative on the whole as both the displacement feedback gain and velocity feedback gain increases. This is further confirmed by the (l,C) curves depicted in Fig. 2(b). Fig. 2 may indicate that large feedback gains enable us easily to stabilize the trivial solution of system (1). When the parameters are fixed as s ¼0.2, F¼1.4, B ¼0.5, C¼ 0.3, Fig. 3(a) depicts the three-dimensional figures of l versus the modulation of two time delays, together with Fig. 3(b) showing the stability boundary of trivial response in t1 2t2 plane. Obviously, the three-dimensional figure presents a variety of l over the parameter ranges 0 r t1 r 6 and 0 r t2 r 6. The t1 2t2 plane is divided into three parts by equating l to zero: region I corresponding to l 4 0 (which consists of two separate parts in the parameter plane) and region II corresponding to l o0. That is to say, if the two time delays fall into region II, trivial solution of system (1) is asymptotically stable, otherwise, the trivial solution is unstable. This may imply proper choices of time delays shall provide a better vibration control.
3.2. The case of non-trivial steady-state solution When g is sufficiently small, the classical perturbation method and the stochastic moment technique [20] are utilized to analyze further the non-trivial steady-state solution of system (1). For
Fig. 2. The evolutions of the largest Lyapunov exponent: (a) l versus B and C; (b) l versus C with fixed B.
Fig. 3. (a) The variations of the largest Lyapunov exponent with t1 and t2; (b) stability boundary of trivial solution.
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non-zero response, Eq. (7) can be recast as 8 a a 3 B C F > 0 > > < a ¼ 2 a8 a þ 2o a sinðo0 t1 Þ2 a cosðo0 t2 Þ þ 4o a sin Z 0 0 > > Z0 ¼ s B cosðo0 t1 ÞC sinðo0 t2 Þ þ F cos Z þ g W 0 ðT1 Þ > : o0 2o0 ð13Þ The non-trivial steady-state response of Eq. (13) is firstly determined in the case of g ¼0. Setting a0 ¼ 0, Z0 ¼ 0, a set of algebraic equations, governing the amplitude a0 and phase Z0 of the steady-state parametric resonance, are obtained 8 B F > > > < so cosðo0 t1 ÞC sinðo0 t2 Þ þ 2o cos Z0 ¼ 0 0 0 ð14Þ a 2 a 2 B C F > > a a sinðo0 t1 Þ cosðo0 t2 Þ þ sin Z0 ¼ 0 > : 0 0þ 2 8 2o0 2 4o0 Then frequency response relation can be expressed 8 > a 2 F2 > > E3 a20 þE20 ¼ < 4 4o20 > a 2 > > : tan Z0 ¼ 2 E3 þ a20 =g E1 4
a ¼ a0 þ a1 , ð15Þ
According to the Routh–Hurwitz criterion [20], the steadystate solution of Eq. (13) is asymptotically stable if the following two inequalities hold simultaneously 8 2 < aa0 4 0 a ð16Þ : E3 þ a20 4 0 4 For typical cases, existence regions and stability analysis for the non-trivial steady-state solution are illustrated in (F, s) plane for the uncontrolled system (Fig. 4(a)) and the controlled system (Fig. 4(b)). In Fig. 4, Region I stands for a region where there is no non-trivial steady-state response for system (1), while region II represents a region in which there is an asymptotical stable nontrivial steady-state solution. In the remanent region, there are two non-trivial steady-state solutions with the large solution stable and the small one unstable. Drawing a comparison between these two figures, the region for two non-trivial steady-state solutions is distinctly enlarged when the time delayed feedback control is present in system (1). Setting @a0 =@t1 ¼ 0,@a0 =@t2 ¼ 0 in Eq. (14), one can readily find the time delays ðt 1 , t 2 Þ for the possible maximum or minimum amplitude of the steady-state response. These two time delays, viz. ðt 1 , t 2 Þ, correspond to a maximum value of the response if the second partial derivatives satisfy @2 a0 @2 a0 @ 2 a0 40 2 2 @t1 @t2 @t1 @t2
and
Fig. 5 exhibits the dependences of the amplitude a0 of the steadystate response on tuning frequency s for the uncontrolled system (the line of A1), a controlled system with no time delay (the line of A2), a controlled system with a certain time delays (the line of A3) and a controlled system with different time delays corresponding to the maximum amplitude for the response (the line of A4). The dashed lines symbolize unstable motion while the solid lines stable motion. Numerical integral of the original system for the maximum response is performed and the results are depicted by asterisks ‘‘ ’’ in Fig. 5. Obviously, numerical results agree well with theoretical ones. Moreover, Fig. 5 reveals that an appropriate modulation between the two time delays in feedback control not only can reduce the response amplitude but also can enlarge the response amplitude. Then time delays appear to be acted as dual roles from the view point of vibration control. In the following part, the steady-state response of system (1) is discussed in a mean square sense when g a0 but is sufficiently small. Let
Z ¼ Z0 þ Z1
ð17Þ
in which a0 and Z0 are steady-state solutions governing by Eq. (15), a1 and Z1 are small perturbations. Linearizing Eq. (13) at (a0, Z0) with respect to a and Z, one can obtain 8 a F > > a0 Z1 cos Z0 dT1 > da1 ¼ a20 a1 þ < 4 4o0 ð18Þ F > > > : dZ1 ¼ 2o Z1 sin Z0 dT1 þ g dWðT1 Þ 0 Using the moment method, the steady-state moments of first order are given Ea1 ¼ EZ1 ¼ 0
ð19Þ
According to the Itˆo differential formulas, the second moments of (a1, Z1) satisfy 8 dEa21 1 2 2 F > > > > dT 2 ¼ 2 aa0 Ea1 þ 2o a0 cos Z0 Ea1 Z1 > > 0 1 > > > < dEZ2 F 1 2 ¼ sin Z0 EZ1 þ g 2 ð20Þ o0 > dT12 > > > > > dEa1 Z1 1 1 2 F > 2 > > : dT 2 ¼ 4o0 Fa0 cos Z0 EZ1 4 aa0 þ 2o0 sin Z0 Ea1 Z1 1
@2 a0 o0 @t21
Fig. 4. Existence regions for the non-trivial steady-state solution in the (F, s) plane: (a) the uncontrolled system, (b) the time-delay controlled system with o0 ¼1.0, a ¼ 0.1, B¼ 0.5, C¼0.3, t1 ¼ 0.5, t2 ¼ 2.5.
Fig. 5. Frequency-amplitude relation of the parametric resonance for the uncontrolled system (the line of A1), a controlled system with B¼ 0.5, C¼ 0.3, t1 ¼ 0.0, t2 ¼0.0. (the line of A2), a controlled system with B¼ 0.5, C¼ 0.3, t1 ¼ 0.5, t2 ¼ 2.5 (A3) and a controlled system with B ¼0.5, C¼0.3 and different time delays (the line of A4).
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Then the second steady-state moments are expressed 8 > ða0 F cos Z0 =2o0 Þ2 g 2 > 2 > g2 > Ea1 ¼ 2 > > aa0 F sin Z0 ðaa20 =4þ F sin Z0 =2o0 Þ=o0 > > < o0 2 EZ21 ¼ g F sin Z0 > > > > > a0 cos Z0 2 > > > : Ea1 Z1 ¼ ðaa2 þ 2F sin Z =o0 Þsin Z g 0 0 0
4. Numerical calculations
ð21Þ
To analyze the stability of the second steady-state moments, the Jacobian matrix can be obtained by performing linearization of Eq. (20) 2 3 0 Fa0 cos Z0 =2o0 aa20 =2 6 7 0 F sin Z0 =o0 0 ð22Þ 4 5 2 0 Fa0 cos Z0 =4o0 ðaa0 =4 þ F sin Z0 =2o0 Þ The second steady-state moments are asymptotic stable when the real parts of the eigenvalues in characteristic equation are all negative. According to the Routh–Hurwitz criterion [20], the necessary and sufficient condition for all the eigenvalues having negative parts is the two inequalities hold simultaneously 8 2 < aa0 4 0 a ð23Þ : E3 þ a20 4 0 4 which accords with the stable condition of Eq. (14). Combining Eqs. (19) and (21), the mean square steady-state response of system (1) is Ea2 ¼ a20 þ
ða0 F cos Z0 =2o0 Þ2 g 2 g2 aa20 F sin Z0 ðaa20 =4þ F sin Z0 =2o0 Þ=o0
625
ð24Þ
In this section, numerical simulations are performed by numerically integrating the time delayed van der Pol oscillator (1) using 4th Runge–Kutta algorithm. For the simulations of narrowband noise, readers can refer to Shinozuka [21] and Zhu [22]. Firstly, simulations of the stochastic response for the controlled system (1) are employed to verify the analytical results derived from the largest Lyapunov exponent. Fig. 1(b) shows when freezing noise amplitude the largest Lyapunov exponent is firstly negative, then positive, and once again negative as tuning frequency increases. For instance, when F¼1.4 l is less than zero for s ¼ 0.8 and s ¼0.8, while it is more than zero for s ¼ 0.0. The change of the largest Lyapunov exponent from minus to plus definitely denotes a switching of the stability, i.e. from stable to unstable, of the trivial response caused only by the frequency. Figs. 6(a) and (c) depict the time histories of system (1) in the case of s ¼ 0.8 and s ¼0.8, respectively, showing the controlled van der Pol oscillator is stabilized asymptotically to the stable trivial solution after a short transition, namely the trivial solution of system (1) is asymptotically stable. While the time history of controlled van der Pol oscillator in the case of s ¼ 0.0, displayed in Fig. 6(b), is oscillatory regularly, namely the trivial solution of system (1) is not stable. Fig. 3(b) indicates that modulation of the two time delays in displacement feedback and velocity feedback can adjust the stability property of the trivial solution. When the two time delays ðt1 , t2 Þ fall into the stable region, for example t1 ¼ 4, t2 ¼ 0:2 and t1 ¼ 4, t2 ¼ 5, the stochastic response of system (1) undergoes a transition stage and then asymptotically converges to the stable trivial solution, which are displayed in Figs. 7(a) and (c),
Fig. 6. The time histories of system (1) corresponding to Fig. 1(b): (a) F ¼1.4, s ¼ 0.8; (b) F ¼1.4, s ¼ 0.0; (c) F¼ 1.4, s ¼0.8.
Fig. 7. The time histories of system (1) corresponding to Fig. 3(b): (a) t1 ¼4, t2 ¼ 0.2; (b) t1 ¼4, t2 ¼ 3; (c) t1 ¼ 4, t2 ¼5.
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Fig. 8. (a) The deterministic steady-state amplitude a0 and (b) steady-state mean square response of system (1): — theoretical results;
numerical results.
Fig. 9. Influences of bandwidth of noise excitation on the response are depicted by the time history, phase portrait and power spectrum: the left line for g ¼ 0; the middle line for g ¼0.1; the right line for g ¼1.0.
ARTICLE IN PRESS X.L. Yang, Z.K. Sun / International Journal of Non-Linear Mechanics 45 (2010) 621–627
respectively. In the case of t1 ¼ 4 and t2 ¼ 3, the two time delays ðt1 , t2 Þ situate in the unstable region of Fig. 3(b), the corresponding oscillatory response is demonstrated in Fig. 7(b). Obviously, numerical calculations for the stability of the trivial solution are consistent with theoretical analysis. Without loss of generality, when the parameters of system (1) are selected as o0 ¼ 1:0, a ¼ 0:1, B ¼ 0:5, C ¼ 0:3, e ¼ 0:1, g ¼ 0:01, s ¼ 0:1, F ¼ 1:0, t1 ¼ 4, t2 ¼ 1, Fig. 8 depicts the variations of steady-state response of the controlled van der Pol oscillator (1). The deterministic steady-state amplitude a0 as a function of noise strength F is showed in Fig. 8(a), and the mean square response Ea2 versus tuning frequency is displayed in Fig. 8(b). The solid lines stand for approximate analytical results obtained in Section 3.2, and the asterisks represent numerical results. The well coincidence of the analytical results and numerical ones readily confirms that the present analysis is effective. Secondly, further numerical calculations are employed to reveal how the bandwidth of narrow-band noise influences the stochastic response of system (1). In the case of o0 ¼ 1:0, a ¼ 0:1, B ¼ 0:5, C ¼ 0:3, e ¼ 0:1, s ¼ 0:1, F ¼ 1:5, t1 ¼ 0:5, t2 ¼ 2:5, the results of time history, phase portrait as well as power spectrum are illustrated in Fig. 9 for different bandwidth of stochastic noise. When g ¼ 0, i.e., the narrow-band noise is degenerated into a deterministic harmonic excitation, the time history along with the phase portrait and the power spectrum of system (1) is portrayed in Figs. 9(a)–(c), respectively. The corresponding results for g ¼ 0.1 and g ¼1.0 are respectively illustrated in Figs. 9(d)–(f) and (h)–(j). These results may indicate that the random noise gW(t) can change the periodic motion to a quasiperiodic motion, and change a limit cycle to a diffused limit cycle. Moreover, the width of the diffused limit cycle increases as the intensity of the random excitation increases. The remarkable influences are further confirmed by the variations of power spectrum, in which it changes from a spectrum with a peak, a spectrum with many discrete peaks to a continuous, broad-band spectrum as the noise intensity increases.
5. Conclusions The present paper studies the principal parametric resonance of a self-excited van der Pol oscillator in the presence of random narrow-band excitation and multiple time delayed state feedback control. The narrow-band noise is a more realistic model in engineering applications than white noise. The state feedback term involves both displacement and velocity feedback. Using a cooperation of analytical methods and numerical analysis, the steady-state trivial and non-trivial responses as well as their dynamic stability have been analyzed in various regimes characterized by tuning frequency, random noise and time delayed feedback. The principal results are presented in these parameter spaces revealing a rich variety of dynamic behaviors including the phenomenon of multiple solutions and time delays induced stabilization or unstabilization. Moreover, an appropriate modulation between the two time delays in feedback control can enhance or reduce the response amplitude. Thus the multiple
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time delayed feedback may be acted as a simple and efficient switch to adjust control performance from the viewpoint of vibration control. In addition, the random noise can change a limit cycle to a diffused limit cycle, the width of which increases as the intensity of random excitation increases.
Acknowledgements This work is partially supported by the National Natural Science Foundation of China (Grant nos. 10902062, 10871123 and 10902085), the NSF of Shaanxi Province (Grant no. 2009JQ1002), and the Outstanding Science and Technology Pre-Research Foundation of Shaanxi Normal University, China. References [1] W. Wischert, A. Wunderlin, A. Pelster, M. Olivier, J. Groslambert, Delayinduced instabilities in nonlinear feedback systems, Phys. Rev. E 49 (1994) 203–219. [2] K. Miyakawa, K. Yamada, Entrainment in coupled salt-water oscillators, Physica D 127 (1999) 177–186. [3] P. Tass, J. Kurths, M.G. Rosenblum, G. Guasti, H. Hefter, Delayinduced transitions in visually guided movements, Phys. Rev. E 54 (1996) R2224–R2227. [4] A. Destexhe, Stability of periodic oscillations in a network of neurons with time delay, Phys. Lett. A 187 (1994) 309–316. [5] Z.K. Sun, W. Xu, X.L. Yang, T. Fang, Inducing or suppressing chaos in a doublewell Duffing oscillator by time delay feedback, Chaos, Solitons Fractals 27 (3) (2006) 705–714. [6] A. Maccari, The response of a parametrically excited van der Pol oscillator to a time delay state feedback, Nonlinear Dyn. 26 (2001) 105–119. [7] A. Maccari, Vibration control for the primary resonance of the van der Pol oscillator by a time delay state feedback, Int. J. Nonlinear Mech. 38 (2003) 123–131. [8] J. Xu, K.W. Chung, Effects of time delayed position feedback on a van der Pol– Duffing oscillator, Physica D 180 (2003) 17–39. [9] J.C. Ji, C.H. Hansen, Stability and dynamics of a controlled van der Pol–Duffing oscillator, Chaos, Solitons Fractals 28 (2006) 555–570. [10] T.M. Morrison, R.H. Rand, 2:1 Resonance in the delayed nonlinear Mathieu equation, Nonlinear Dyn. 50 (2007) 341–352. _ ¼ AxðtÞ þ BxðttÞ, IEEE Trans. Autom. [11] T. Mori, H. Kokame, Stability of xðtÞ Control 34 (1989) 460–462. [12] H.Y. Hu, Z.H. Wang, Stability of analysis of damped SDof systems with two time delays in state feedback, J. Sound Vib. 214 (1998) 213–225. [13] Z.K. Sun, W. Xu, X.L. Yang, Response of nonlinear system to random narrowband excitation with time delay state feedback, J. Vib. Eng. 19 (2006) 57–64. [14] Y.F. Jin, H.Y. Hu, Principal resonance of a Duffing oscillator with delayed state feedback under narrow-band random parametric excitation, Nonlinear Dyn. 50 (2007) 213–227. [15] W.V. Weding, Invariant measures and Lyapunov exponents for generalized parameter fluctuations, Struct. Saf. 8 (1990) 13–25. [16] A.H. Hayfeh, B. Blachandran, Applied Nonlinear Dynamics: Analytical, Computational, and Experimental Methods, John Wiley & Sons, Inc., New York, 1995. [17] H.Y. Hu, Z.H. Wang, Dynamics of Controlled Mechanical Systems with Delayed Feedback, Springer-Verlag, Berlin Heidelberg, 2002. [18] R.Z. Khasminskii, Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, Alphen and den Rijn, 1980. [19] V.I. Oseledec, A multiplicative ergodic theorem Lyapunov characteristic numbers for dynamical systems, Trans. Moscow Math. Soc. 19 (2) (1968) 197–231. [20] G. Schmidt, A. Tondl, Nonlinear Vibrations, Cambridge University Press, Cambridge, 1986. [21] M. Shinozuka, Digital simulation of random processes and its applications, J. Sound Vib. 25 (1972) 111–128. [22] W.Q. Zhu, Random Vibration, Beijing Science Press, China, 1992.