An asymptotic method for quasi-integrable Hamiltonian system with multi-time-delayed feedback controls under combined Gaussian and Poisson white noises

Wantao Jia; Yong Xu; Zhonghua Liu; Weiqiu Zhu

doi:10.1007/s11071-017-3832-3

Nonlinear Dynamics

December 2017, Volume 90, Issue 4, pp 2711–2727 | Cite as

An asymptotic method for quasi-integrable Hamiltonian system with multi-time-delayed feedback controls under combined Gaussian and Poisson white noises

Authors
Authors and affiliations

Wantao JiaEmail author
Yong XuEmail author
Zhonghua Liu
Weiqiu Zhu

Original Paper

First Online: 06 October 2017

Received: 04 February 2017

Accepted: 23 September 2017

187 Downloads
1 Citations

Abstract

In the present paper, we consider an approximate approach for predicting the responses of the quasi-integrable Hamiltonian system with multi-time-delayed feedback control under combined Gaussian and Poisson white noise excitations. Two-step approximation is taken here to obtain the responses of such system. First, based on the property of the system solution, the time-delayed system state variables are approximated by using the system state variables without time delay. After this approximation, the system is converted to the one without time delay but with delay time as parameters. Then, stochastic averaging method for quasi-integrable Hamiltonian system under combined Gaussian and Poisson white noises is applied to simplify the converted system to obtain the averaged stochastic integro-differential equations and generalized Fokker–Planck–Kolmogorov equations for both non-resonant and resonant cases. Finally, two examples are worked out to show the detailed procedure of proposed method for the illustrative purpose. And the influences of the time delay on the responses of the systems are also discussed. In addition, the validity of the results obtained by present method is verified by Monte Carlo simulation.

Keywords

Quasi-integrable Hamiltonian system Multi-time-delayed feedback control Combined Gaussian and Poisson white noise excitations Stochastic averaging method

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Notes

Acknowledgements

This study was supported by the National Natural Science Foundation of China under Grants Nos. 11502199, 11372262, 11572247 and the Fundamental Research Funds for Central Universities.

Appendix A

The coefficients of averaged GFPK equation for the non-resonant case:

$$\begin{aligned}&\bar{{A}}_{r_1 } \left( \mathbf{I} \right) =\frac{\varepsilon ^{2}}{\left( {2\pi } \right) ^{n}}\int _0^{2\pi } \left( -\sum _{i=1}^n \sum _{j=1}^n \frac{\partial H}{\partial p_j }\frac{\partial I_{r_1 } }{\partial p_i }\right. \nonumber \\&\quad \left. +\frac{1}{2}\sum _{i=1}^n {\sum _{j=1}^n {\sum _{k=1}^{n_g } {\sigma _{i,k} \sigma _{j,k} } } } \frac{\partial ^{2}I_{r_1 } }{\partial p_i \partial p_j } \right) \mathrm{d}{\varvec{\uptheta }} \nonumber \\&\quad +\sum _{k=2}^u {\varepsilon ^{k}\sum _{l=1}^{n_p } {\frac{\lambda _l E[Y_l^k ]}{\left( {2\pi } \right) ^{n}}\int _0^{2\pi } {A_{r;k;l} \mathrm{d}{\varvec{\uptheta }}} } } , \end{aligned}$$

(47)

$$\begin{aligned}&\bar{{A}}_{r_1 ,r_2 } \left( \mathbf{I} \right) =\frac{\varepsilon ^{2}}{\left( {2\pi } \right) ^{n}}\int _0^{2\pi } \nonumber \\&\quad {\left( {\sum _{i=1}^n {\sum _{j=1}^n {\sum _{k=1}^{n_g } {\frac{\partial I_{r_1 } }{\partial p_i }\frac{\partial I_{r_2 } }{\partial p_j }\sigma _{i,k} \sigma _{j,k} } } } } \right) } \mathrm{d}{\varvec{\uptheta }} \nonumber \\&\quad +\sum _{k=2}^u {\varepsilon ^{k}\sum _{l=1}^{n_p } {\frac{\lambda _l E[Y_l^k ]}{\left( {2\pi } \right) ^{n}}} } \int _0^{2\pi }\nonumber \\&\quad {\left( {\sum _{k_1 +k_2 =k} {A_{r_1 ;k_1 ;l} A_{r_2 ;k_2 ;l} } } \right) \mathrm{d}{\varvec{\uptheta }}} , \end{aligned}$$

(48)

$$\begin{aligned}&\bar{{A}}_{r_1 ,r_2 ,\ldots ,r_j } \left( \mathbf{I} \right) =\sum _{k=j}^u \varepsilon ^{k}\sum _{l=1}^{n_p } \frac{\lambda _l E[Y_l^k ]}{\left( {2\pi } \right) ^{n}}\int _0^{2\pi }\nonumber \\&\quad {\left( {\sum _{k_1 +k_2 +\cdots +k_j =k} {A_{r_1 ;k_1 ;l} A_{r_2 ;k_2 ;l} } \cdots A _{r_j ;k_j ;l} } \right) \mathrm{d}{\varvec{\uptheta }}} , \end{aligned}$$

(49)

where$A_{r;k;l} =A_{r;k;l} \left( {\mathbf{q},\mathbf{p}} \right) $ given in Ref. [23], $\mathbf{I}=[I_1 ,I_2 ,\ldots ,I_n ]^{T}, {\varvec{\uptheta }}=[\theta _1 ,\theta _2 ,\ldots ,\theta _n ]^{T}$ and $\int _0^{2\pi } {\left[ {\cdot } \right] \mathrm{d}{\varvec{\uptheta }}} =\int _0^{2\pi } {\int _0^{2\pi } {\cdots \int _0^{2\pi } {\left[ {\cdot } \right] \mathrm{d}\theta _1 \mathrm{d}\theta _2 \cdots \mathrm{d}\theta _n } } } $ denotes the n-fold integral.

Appendix B

The coefficients of averaged GFPK equation for resonant case :

$$\begin{aligned} \bar{{A}}_{r_1 } \left( {\mathbf{I},{\varvec{\uppsi }}} \right)= & {} \frac{\varepsilon ^{2}}{\left( {2\pi } \right) ^{n-\alpha }}\int _0^{2\pi } \left( -\sum _{i,j=1}^n {m_{ij} \frac{\partial H}{\partial p_j }\frac{\partial I_{r_1 } }{\partial p_i }}\right. \nonumber \\&\left. +\,\frac{1}{2}\sum _{i,j=1}^n {\sum _{k=1}^{n_g } {\sigma _{ik} \sigma _{jk} \frac{\partial ^{2}I_{r_1 } }{\partial p_i \partial p_j }} } \right) \mathrm{d}{\varvec{\uptheta }}_1 \nonumber \\&+\,\sum _{k=1}^u {\sum _{l=1}^{n_p } {\frac{\varepsilon ^{k}\lambda _l E[Y_l^k ]}{\left( {2\pi } \right) ^{n-\alpha }}\int _0^{2\pi } {A_{r_1 ;k;l} \mathrm{d}{\varvec{\uptheta }}_1 } } } \nonumber \\ \end{aligned}$$

(50)

$$\begin{aligned}&\bar{{A}}_{n+v_1 } \left( {\mathbf{I},{\varvec{\uppsi }}} \right) =\frac{1}{\left( {2\pi } \right) ^{n-\alpha }}\int _0^{2\pi } \left[ \mathrm{O}\left( {\varepsilon ^{2}} \right) \right. \nonumber \\&\quad \left. +\,\varepsilon ^{2}\left( -\sum _{i,j=1}^n {m_{ij} \frac{\partial H}{\partial p_j }\frac{\partial \psi _{v_1 } }{\partial p_i }}\right. \right. \nonumber \\&\quad \left. \left. +\,\frac{1}{2}\sum _{i,j=1}^n {\sum _{k=1}^{n_g } {\sigma _{ik} \sigma _{jk} \frac{\partial ^{2}\psi _{v_1 } }{\partial p_i \partial p_j }} } \right) \right] \mathrm{d}{\varvec{\uptheta }}_1 \nonumber \\&\quad +\,\sum _{k=1}^u {\sum _{l=1}^{n_p } {\frac{\varepsilon ^{k}\lambda _l E[Y_l^k ]}{\left( {2\pi } \right) ^{n-\alpha }}\int _0^{2\pi } {C_{v_1 ;k;l} \mathrm{d}{\varvec{\uptheta }}_1 } } } \end{aligned}$$

(51)

$$\begin{aligned}&\bar{{A}}_{r_1 ,r_2 } \left( {\mathbf{I},{\varvec{\uppsi }}} \right) \nonumber \\&\quad =\frac{\varepsilon ^{2}}{\left( {2\pi } \right) ^{n-\alpha }}\int _0^{2\pi } {\left( {\sum _{i,j=1}^n {\sum _{k=1}^{n_g } {\frac{\partial I_{r_1 } }{\partial p_i }\frac{\partial I_{r_2 } }{\partial p_j }\sigma _{i,k} \sigma _{j,k} } } } \right) } d{\varvec{\uptheta }}_1\nonumber \\&\quad +\,\sum _{k=2}^u {\sum _{l=1}^{n_p } {\frac{\varepsilon ^{k}\lambda _l E[Y_l^k ]}{\left( {2\pi } \right) ^{n-\alpha }}} } \int _0^{2\pi } {\left( {\sum _{k_1 +k_2 =k} {A_{r_1 ;k_1 ;l} A_{r_2 ;k_2 ;l} } } \right) \mathrm{d}{\varvec{\uptheta }}_1 }\nonumber \\ \end{aligned}$$

(52)

$$\begin{aligned}&{\bar{A}}_{r_1 ,n+v_1 } \left( {\mathbf{I},{\varvec{\uppsi }}} \right) =\frac{\varepsilon ^{2}}{\left( {2\pi } \right) ^{n-\alpha }}\int _0^{2\pi }\nonumber \\&\quad {\left( {\sum _{i,j=1}^n {\sum _{k=1}^{n_g } {\frac{\partial I_{r_1 } }{\partial p_i }\frac{\partial \psi _{v_1 } }{\partial p_j }\sigma _{i,k} \sigma _{j,k} } } } \right) } \mathrm{d}{\varvec{\uptheta }}_1 \nonumber \\&\quad +\,\sum _{k=2}^u {\sum _{l=1}^{n_p } {\frac{\varepsilon ^{k}\lambda _l E[Y_l^k ]}{\left( {2\pi } \right) ^{n-\alpha }}} } \int _0^{2\pi } {\left( {\sum _{k_1 +k_2 =k} {A_{r_1 ;k_1 ;l} C_{v_1 ;k_2 ;l} } } \right) \mathrm{d}{\varvec{\uptheta }}_1 }\nonumber \\ \end{aligned}$$

(53)

$$\begin{aligned}&\bar{{A}}_{n+v_1 ,n+v_2 } \left( {\mathbf{I},{\varvec{\uppsi }}} \right) =\frac{\varepsilon ^{2}}{\left( {2\pi } \right) ^{n-\alpha }}\int _0^{2\pi }\nonumber \\&\quad {\left( {\sum _{i,j=1}^n {\sum _{k=1}^{n_g } {\sigma _{i,k} \sigma _{j,k} } } \frac{\partial \psi _{v_1 } }{\partial p_i }\frac{\partial \psi _{v_2 } }{\partial p_j }} \right) } \mathrm{d}{\varvec{\uptheta }}_1 \nonumber \\&\quad +\,\sum _{k=2}^u {\sum _{l=1}^{n_p } {\frac{\varepsilon ^{k}\lambda _l E[Y_l^k ]}{\left( {2\pi } \right) ^{n-\alpha }}} } \int _0^{2\pi }\nonumber \\&\quad {\left( {\sum _{k_1 +k_2 =k} {C_{v_1 ;k_1 ;l} C_{v_2 ;k_2 ;l} } } \right) \mathrm{d}{\varvec{\uptheta }}_1 } \end{aligned}$$

(54)

$$\begin{aligned}&\bar{{A}}_{r_1 ,r_2 ,r_3 } \left( {\mathbf{I},{\varvec{\uppsi }}} \right) =\sum _{k=3}^u {\sum _{l=1}^{n_p } {\frac{\varepsilon ^{k}\lambda _l E[Y_l^k ]}{\left( {2\pi } \right) ^{n-\alpha }}} } \int _0^{2\pi } \nonumber \\&\quad {\left( {\sum _{k_1 +k_2 +k_3 =k} {A_{r_1 ;k_1 ;l} A_{r_2 ;k_2 ;l} A_{r_3 ;k_3 ;l} } } \right) \mathrm{d}{\varvec{\uptheta }}_1} \end{aligned}$$

(55)

$$\begin{aligned}&\bar{{A}}_{r_1 ,r_2 ,n+v_1 } \left( {\mathbf{I},{\varvec{\uppsi }}} \right) \nonumber \\&\quad =\sum _{k=3}^u {\sum _{l=1}^{n_p } {\frac{\varepsilon ^{k}\lambda _l E[Y_l^k ]}{\left( {2\pi } \right) ^{n-\alpha }}} } \int _0^{2\pi }\nonumber \\&\quad {\left( {\sum _{k_1 +k_2 +k_3=k} {A_{r_1 ;k_1 ;l} A_{r_2 ;k_2 ;l} C_{v_1 ;k_3 ;l} } } \right) \mathrm{d}{\varvec{\uptheta }}_1 } \end{aligned}$$

(56)

$$\begin{aligned}&\bar{{A}}_{r_1 ,n+v_1 ,n+v_2 } \left( {\mathbf{I},{\varvec{\uppsi }}} \right) \nonumber \\&\quad =\sum _{k=3}^u {\sum _{l=1}^{n_p } {\frac{\varepsilon ^{k}\lambda _l E[Y_l^k ]}{\left( {2\pi } \right) ^{n-\alpha }}} } \int _0^{2\pi } \nonumber \\&\quad {\left( {\sum _{k_1 +k_2 +k_3 =k} {A_{r_1 ;k_1 ;l} C_{v_1 ;k_2 ;l} C_{v_2 ;k_3 ;l} } } \right) \mathrm{d}{\varvec{\uptheta }}_1 } \end{aligned}$$

(57)

$$\begin{aligned}&\bar{{A}}_{n+v_1 ,n+v_2 ,n+v_3 } \left( {\mathbf{I},{\varvec{\uppsi }}} \right) \nonumber \\&\quad =\sum _{k=3}^u {\sum _{l=1}^{n_p } {\frac{\varepsilon ^{k}\lambda _l E[Y_l^k ]}{\left( {2\pi } \right) ^{n-\alpha }}} } \int _0^{2\pi } \nonumber \\&\quad {\left( {\sum _{k_1 +k_2 +k_3 =k} {C_{v_1 ;k_1 ;l} C_{v_2 ;k_2 ;l} C_{v_3 ;k_3 ;l} } } \right) \mathrm{d}{\varvec{\uptheta }}_1 } \end{aligned}$$

(58)

$$\begin{aligned}&\bar{{A}}_{r_1 ,\ldots r_{j-s} ,n+v_1 ,\ldots ,n+v_s } \left( {\mathbf{I},{\varvec{\uppsi }}} \right) \nonumber \\&\quad =\sum _{k=j}^u {\sum _{l=1}^{n_p } {\frac{\varepsilon ^{k}\lambda _l E[Y_l^k ]}{\left( {2\pi } \right) ^{n-\alpha }}} } \int _0^{2\pi } \nonumber \\&\quad {\left( {\sum _{k_1 +k_2 +\cdots +k_j =k} {A_{r_1 ;k_1 ;l} \cdots A_{r_s ;k_s ;l} C_{v_1 ;k_{s+1} ;l} \cdots C_{v_{j-s} ;k_j ;l} } } \right) \mathrm{d}{\varvec{\uptheta }}_1 }\nonumber \\ \end{aligned}$$

(59)

$$\begin{aligned} s=0,\ldots ,j, j=4,\ldots ,u, r_i =1,\ldots ,n; v_i =1,\ldots ,\alpha . \end{aligned}$$

where $\mathbf{I}=\left[ {I_1 ,\ldots ,I_n } \right] ^{T}, \psi =\left[ {\psi _1 ,\ldots ,\psi _\alpha } \right] ^{T}$and $\int _0^{2\pi } {\left[ {\cdot } \right] } \mathrm{d}{\varvec{\uptheta }}_1 =\int _0^{2\pi } {\int _0^{2\pi } {\cdots \int _0^{2\pi } {\left[ {\cdot } \right] } } } \mathrm{d}\theta _1 \mathrm{d}\theta _2 \cdots \mathrm{d}\theta _\alpha $ is the $n-\alpha $-fold integral notation. The terms $A_{r;k;l} =A_{r;k;l} \left( {\mathbf{q},\mathbf{p}} \right) $ and $C_{r;k;l} =C_{r;k;l} \left( {\mathbf{q},\mathbf{p}} \right) $ are given in Ref. [24].

Appendix C

Substituting this solution (27) to Eq. (25) and collecting the terms of same order of $\varepsilon $, the equations that $p_0 , p_1 $ and $p_2 $ satisfy are

$$\begin{aligned}&\varepsilon ^{2}:\quad 0=-\frac{\partial }{\partial I}\left( {\bar{{A}}_1 \left( I \right) p_0 } \right) +\frac{1}{2}\frac{\partial }{\partial I^{2}}\left( {\bar{{A}}_2^{(1)} \left( I \right) p_0 } \right) \end{aligned}$$

(60)

$$\begin{aligned}&\varepsilon ^{3}:\quad 0=-\frac{\partial }{\partial I}\left( {\varepsilon \bar{{A}}_1 \left( I \right) p_1 } \right) +\frac{1}{2}\frac{\partial ^{2}}{\partial I^{2}}\left( {\varepsilon \bar{{A}}_2^{(1)} \left( I \right) p_1 } \right) \end{aligned}$$

(61)

$$\begin{aligned}&\varepsilon ^{4}:\quad 0=-\frac{\partial }{\partial I}\left( {\varepsilon ^{2}\bar{{A}}_1 \left( I \right) p_2 } \right) +\frac{1}{2}\frac{\partial ^{2}}{\partial I^{2}}\left( {\varepsilon ^{2}\bar{{A}}_2^{(2)} \left( I \right) p_2 } \right) \nonumber \\&\quad \qquad \quad \quad -\frac{1}{3!}\frac{\partial ^{3}}{\partial I^{3}}\left( {\bar{{A}}_3 \left( I \right) p_0 } \right) +\frac{1}{4!}\frac{\partial ^{4}}{\partial I^{4}}\left( {\bar{{A}}_4 \left( I \right) p_0 } \right) \end{aligned}$$

(62)

$$\begin{aligned} \vdots \end{aligned}$$

where

$$\begin{aligned} \bar{{A}}_2^{(1)} \left( I \right)= & {} \varepsilon ^{2}\frac{2D+\lambda E\left[ {Y^{2}} \right] }{\omega }I;\\ \bar{{A}}_2^{(2)} \left( I \right)= & {} \varepsilon ^{4}\frac{\lambda E\left[ {Y^{4}} \right] }{4\omega ^{2}}\hbox { and }\\ \bar{{A}}_2 \left( I \right)= & {} \bar{{A}}_2^{\left( 1 \right) } \left( I \right) +\bar{{A}}_2^{\left( 2 \right) } \left( I \right) \end{aligned}$$

One can get $p_0 , p_1 , p_2, \ldots $ by solving Eqs. (60)–(62) step by step. And, the solution (27) can be obtained.

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Authors and Affiliations

Wantao Jia
- 1
Email author
Yong Xu
- 1
Email author
Zhonghua Liu
- 2
Weiqiu Zhu
- 3

1.Department of Applied MathematicsNorthwestern Polytechnical UniversityXi’anChina
2.Department of Civil EngineeringXiamen UniversityXiamenChina
3.Department of Mechanics, State Key Laboratory of Fluid Power Transmission and ControlZhejiang UniversityHangzhouChina

An asymptotic method for quasi-integrable Hamiltonian system with multi-time-delayed feedback controls under combined Gaussian and Poisson white noises

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