Stochastic response of fractional-order van der Pol oscillator

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1 HEOREICAL & APPLIED MECHANICS LEERS 4, 3 4 Stochastic response of fractional-order van der Pol oscillator Lincong Chen,, a, Weiqiu Zhu b College of Civil Engineering, Huaqiao University, Xiamen 36, China Department of Mechanics, Zhejiang University, Hangzhou 37, China Received October 3; accepted 5 November 3 Abstract We studied the response of fractional-order van de Pol oscillator to Gaussian white noise excitation in this letter. An equivalent integral-order nonlinear stochastic system is obtained to replace the given system based on the principle of minimum mean-square error. hrough stochastic averaging, an averaged Itô equation is deduced. We obtained the Fokker Planck Kolmogorov equation connected to the averaged Itô equation and solved it to yield the approximate stationary response of the system. he analytical solution is confirmed by using Monte Carlo simulation. c 4 he Chinese Society of heoretical and Applied Mechanics. [doi:.63/.43] Keywords fractional-order van de Pol oscillator, Gaussian white noise, stationary response, equivalent nonlinear system method, stochastic averaging As a generalization of the classical calculus, fractional calculus has been applied to various fields in the past decades. he major advantage of the fractional calculus originates in the fact that it has been proven to be an excellent tool to describe the memory and hereditary properties of various materials and processes. Also, many practical systems e.g., electromagnetic waves in dielectrics can be described more adequately through the fractional-order differential equations. On the other hand, the van de Pol VDP oscillator, proposed originally as a model of vacuum tube circuits and later applied widely to various fields was one of the most famous self-excited systems. Recently, the model of classical VDP oscillator has been further developed, and the fractional-order VDP oscillator has been formulated and studied by means of different numerical methods. 3,4 In this letter, we investigate the stationary response of fractional-order VDP oscillator to external Gaussian white noise excitation. he critical procedure is to obtain an equivalent integral-order stochastic system and use the stochastic averaging technique to study the equivalent integral-order stochastic system. Consider a fractional-order VDP oscillator externally excited by a Gaussian white noise. he motion equation has the following form D α Xt β β Xt Ẋtω Xt=ξ t, where ω, β, and β are positive parameters. D α Xt is the fractional derivative under the defia chen4@6.com. b Corresponding author. wqzhu@yahoo.com.

2 3- L. C. Chen, W. Q. Zhu heor. Appl. Mech. Lett. 4, 3 4 nition of Riemann Liouville, and it reads as d D α Xt= Γ α dt Xt τ τ α dτ, < α, where Γ is Gamma function and α is the fractional order. ξ t is Gaussian white noise having intensity d. β, β, d are of the same order of ε, here ε is a small positive parameter. Since the fractional order satisfies < α, the fractional derivative term has contributions to both inertia and damping. Hence, introduce the following equivalent system maẍ [ β βaβ X ]Ẋ ω X = ξ t, 3 where ma and βa are the coefficients of equivalent inertia and damping forces, respectively, and X = Xt. he error between Eqs. and 3 is e = maẍ D α X βaẋ. 4 he necessary conditions for minimum mean square error are E[e ]/ m =, E[e ]/ β =. 5 Substituting Eq. 4 into Eq. 5 yields E[Ẍ ma ẌD α X βaẍẋ]= Ẍ ma ẌD α X βaẍẋdt =, E[ẌẊmA ẊD α X βaẋ ]= ẌẊmA ẊD α X βaẋ dt =. 6 where Assume that the solution of Eq. 3 is of the form 5 Ẋt = AtωA sinθt, Xt =At cosθt, 7 Θt=ΓtωAt, ωa=ω / ma, 8 with ωa being frequency of oscillator. Differentiating Eq. 7 with respect to t leads to Ẍt= sinθ d dt AωA AωA ωa dγ cosθ. 9 dt Since dat/dt and dγt/dt are approximate to and Θt τ is close to Θt τωa while τ

3 3-3 Stochastic response of fractional-order van der Pol oscillator doi:.63/.43 is small, we can obtain the following approximate expression Ẍt Aω AcosΘ. Substituting Eqs. 7 and into Eq. 6 yields maẍ t ẌtD α XtβAẌtẊtdt = maa ω 4 cos Θ Aω cosθd α XtβAA ω 3 cosθ sinθdt maa ω 4 Γ α maa ω 4 A ω 4 cosθ Γ α [ d dt Aω cosθ cosθ cosωτsinθ sinωτ τ α dτ maẍtẋt ẊtD α XtβAẊ tdt = ] Xt τ τ α dτ dt = dt =, maa ω 3 sinθ cosθ Aω sinθd α XtβAA ω sin Θdt βaa ω Γ α [ d Aω sinθ dt ] Xt τ τ α dτ dt = βaa ω Γ α A ω 3 cosθ cosωτsinθ sinωτ sinθ τ α dτ dt =. o simplify Eqs. and further, asymptotic integrals are introduced as follows cosωτ απ τ α dτ = ω α Γ αsin sinωt ωt α Oωt α, sinωτ απ τ α dτ = ω α Γ αcos coss ωt α Oωt α. 3 Substituting Eq. 3 into Eqs. and and averaging them regarding Θt produce the ultimate forms of ma and βa ma=ω α Asin[α π/], βa=ω α Acos[α π/], 4 in which ωa is determined by ω α Asin[α π/]=ω.

4 3-4 L. C. Chen, W. Q. Zhu heor. Appl. Mech. Lett. 4, 3 4 he equivalent system 3 can be rewritten as Ẍ ω {β ω α cos[α π/]β X }Ẋ ω ω X = ω ξ t ω. 5 he energy of the system 5isH = Ẋ / ω X / = ω A /. When we use the stochastic averaging technique of energy envelope 6 to study system 5, we can get averaged Itô equation of energy envelope H as dh = σhdbtuhdt, 6 in which the diffusion and drift coefficients have the form of σ H= ω4 dh ω 4, UH= ω {β 4ω α cos[α π/]}h β H / ω he Fokker Planck Kolmogorov FPK equation connected to Eq. 6is ω4 d ω 4. 7 p t = Up H σ p H, 8 where p = ph,t H represents the probability density of transition of H. After the reduced FPK equation 8 is solved, we can obtain the steady probability density for H as { ph=c exp H [ ] } d du σ u Uu σ u du, 9 where C is the normalization constant. We can get the other statistics of the steady response of system 5 from Eq. 9. For instance, the stationary probability densities px,x and px can be calculated as px,x = phω π, px = px,x dx. H=X /ω X / Numerical results as shown in Fig. for the stationary probability density of generalized displacement with different α s have been obtained with the following parameter values: β =., β =., ω =., D =., unless otherwise mentioned. It is observed from Fig. that the analytical results agree well with the Monte Carlo simulations. Furthermore, a phenomendogical bifurcation P-bifurcation as the fractional orders change can be found. In this paper, a technique for obtaining the stationary response of fractional-order VDP oscillator to Gaussian white noise has been developed. he technique consists of obtaining the equivalent integral-order stochastic system based on the principle of minimum mean-square error, and applying the stochastic averaging technique to the equivalent integral-order stochastic system. he numerical results have shown that the proposed technique works very well and the fractional-

5 3-5 Stochastic response of fractional-order van der Pol oscillator doi:.63/ α =.5.5 α =.8 PX.5 α =. α = X Fig.. he steady probability density of displacement of system for different fractional order. Solid lines represent the analytical results, symbols,,, denotes the Monte Carlo simulations. order s change may lead to P-bifurcation. It should be pointed out that the proposed technique can be extended in a general class of randomly excited nonlinear fractional-order dynamical systems, e.g., fractional-order Duffing oscillator. his work was supported by the National Natural Science Foundation of China 939, 7, 779, and 59, the Specialized Research Fund for the Doctoral Program of Higher Education 353, and the Fundamental Research Funds for Huaqiao University JB-SJ.. Q. Heaviside. Electromagnetic heory. Chelsea, New York 97.. R. S. M. Barbosa, J. A.. Machado, B. M. Vinagre, et al. Analysis of the van der Pol oscillator containing derivatives of fractional order. Journal of Vibration and Control 3, Z. J. Guo, A. Y.. Leung, H. X. Yang. Oscillatory region and asymptotic solution of fractional van der Pol oscillator via residue harmonic balance technique. Applied Mathematical Modeling 35, H. Jafari, C. M. Khalique, M. Nazari. An algorithm for the numerical solution of nonlinear fractional-order Van der Pol oscillator equation. Applied Mathematical Modeling 55, W. Q. Zhu, Z. L. Huang, Y. Suzuki. Response and stability of strongly non-linear oscillators under wide-band random excitation, International Journal of Non-Linear Mechanics 36, W. Q. Zhu, Y. K. Lin. Stochastic averaging of energy envelope. ASCE Journal of Engineering Mechanics. 7,

Correspondence should be addressed to Yongjun Wu, Received 29 March 2011; Revised 8 July 2011; Accepted 25 July 2011

Correspondence should be addressed to Yongjun Wu, Received 29 March 2011; Revised 8 July 2011; Accepted 25 July 2011 Mathematical Problems in Engineering Volume, Article ID 63583, pages doi:.55//63583 Research Article Optimal Bounded Control for Stationary Response of Strongly Nonlinear Oscillators under Combined Harmonic