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

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.

Appendices

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.