Stochastic averaging of quasi-integrable and non-resonant Hamiltonian systems under combined Gaussian and Poisson white noise excitations

Abstract

A stochastic averaging method for predicting the response of quasi-integrable and non-resonant Hamiltonian systems to combined Gaussian and Poisson white noise excitations is proposed. First, the motion equations of a quasi-integrable and non-resonant Hamiltonian system subject to combined Gaussian and Poisson white noise excitations is transformed into stochastic integro-differential equations (SIDEs). Then $n$-dimensional averaged SIDEs and generalized Fokker–Plank–Kolmogrov (GFPK) equations for the transition probability densities of $n$ action variables and $n$- independent integrals of motion are derived by using stochastic jump–diffusion chain rule and stochastic averaging principle. The probability density of the stationary response is obtained by solving the averaged GFPK equation using the perturbation method. Finally, as an example, two coupled non-linear damping oscillators under both external and parametric excitations of combined Gaussian and Poisson white noises are worked out in detail to illustrate the application and validity of the proposed stochastic averaging method.

Appendices

Appendix 1

The Taylor expansions of $I_r ({\mathbf {Q}},{\mathbf {P}}+{\hat{\gamma }}_l )-I_r ({\mathbf {Q}},{\mathbf {P}})$ $\Theta _r ({\mathbf {Q}},{\mathbf {P}}+{\hat{\gamma }}_l )-\Theta _r ({\mathbf {Q}},{\mathbf {P}})$, and $H_r ({\mathbf {Q}},{\mathbf {P}}+{\hat{\gamma }}_l )-H_r ({\mathbf {Q}},{\mathbf {P}})$ ($r$ = 1, 2, $\ldots$, $n$; $l$ = 1, 2, $\ldots$, $n_{p})$ are

$$\begin{aligned}&I_r ( {{\mathbf {Q,P+}}\widehat{\gamma }_l })-I_r ( {{\mathbf {Q,P}}}) \nonumber \\&=\sum \limits _{s=1}^\infty {\frac{1}{s!}} \sum \limits _{i_1 =1}^n {\cdots \sum \limits _{i_s =1}^n {\frac{\partial ^sI_r }{\partial P_{i_1 } \cdots \partial P_{i_s } }} } \nonumber \\&\qquad \times \sum \limits _{j_1 =1}^\infty {\frac{\varepsilon ^{j_1 }}{j_1 !}} f_{i_1 l}^{( {j_1 })} Y_l^{j_1 } \cdots \sum \limits _{j_r =1}^\infty {\frac{\varepsilon ^{j_s }}{j_r !}} f_{i_s l}^{( {j_s })} Y_l^{j_s } \nonumber \\&=\sum \limits _{s=1}^\infty {\frac{1}{s!}} \sum \limits _{i_1 =1}^n {\cdots \sum \limits _{i_s =1}^n {\frac{\partial ^sI_r }{\partial P_{i_1 } \cdots \partial P_{i_s } }} } \sum \limits _{j_1 =1}^\infty \cdots \nonumber \\&\quad \times {\sum \limits _{j_s =1}^\infty {\frac{\varepsilon ^{j_1 +\cdots +j_r }}{j_1 !\cdots j_s !}} } f_{i_1 l}^{( {j_1 })} \cdots f_{i_s l}^{( {j_s })} Y_l^{j_1 +\cdots +j_s } \nonumber \\&=\sum \limits _{k=1}^\infty {\varepsilon ^kY_l^k A_{r;k;l} ( {Q_1 ,\cdots ,Q_n ,P_1 ,\cdots ,P_n })} \nonumber \\&=\sum \limits _{k=1}^\infty {\varepsilon ^kY_l^k A_{r;k;l} ( {{\mathbf {Q}},{\mathbf {P}}})} \end{aligned}$$

(88)

$$\begin{aligned}&\Theta _r ( {{\mathbf {Q,P+}}\hat{\gamma }_l })-\Theta _r ( {\mathbf {Q,P}}) \nonumber \\&=\sum \limits _{s=1}^\infty {\frac{1}{s!}} \sum \limits _{i_1 =1}^n {\cdots \sum \limits _{i_s =1}^n {\frac{\partial ^s\Theta _r }{\partial P_{i_1 } \cdots \partial P_{i_s } }} }\nonumber \\&\quad \times \sum \limits _{j_1 =1}^\infty {\frac{\varepsilon ^{j_1 }}{j_1 !}} f_{i_1 l}^{( {j_1 })} Y_l^{j_1 } \cdots \sum \limits _{j_r =1}^\infty {\frac{\varepsilon ^{j_s }}{j_r !}} f_{i_s l}^{( {j_s })} Y_l^{j_s } \nonumber \\&=\sum \limits _{s=1}^\infty {\frac{1}{s!}} \sum \limits _{i_1 =1}^n {\cdots \sum \limits _{i_s =1}^n {\frac{\partial ^s\Theta _r }{\partial P_{i_1 } \cdots \partial P_{i_s } }} } \sum \limits _{j_1 =1}^\infty \cdots \nonumber \\&\quad \times { \sum \limits _{j_s =1}^\infty {\frac{\varepsilon ^{j_1 +\cdots +j_r }}{j_1 !\cdots j_s !}} } f_{i_1 l}^{( {j_1 })} \cdots f_{i_s l}^{( {j_s })} Y_l^{j_1 +\cdots +j_s } \nonumber \\&=\sum \limits _{k=1}^\infty {\varepsilon ^kY_l^k B_{r;k;l} ( {{\mathbf {Q}},{\mathbf {P}}})} \end{aligned}$$

(89)

and

$$\begin{aligned}&H_r ( {{\mathbf {Q,P+}}\hat{\gamma }_l })-H_r ( {\mathbf {Q,P}}) \nonumber \\&=\sum \limits _{s=1}^\infty {\frac{1}{s!}} \sum \limits _{i_1 =1}^n {\cdots \sum \limits _{i_s =1}^n {\frac{\partial ^sH_r }{\partial P_{i_1 } \cdots \partial P_{i_s } }} }\nonumber \\&\quad \times \sum \limits _{j_1 =1}^\infty {\frac{\varepsilon ^{j_1 }}{j_1 !}} f_{i_1 l}^{( {j_1 })} Y_l^{j_1 } \cdots \sum \limits _{j_r =1}^\infty {\frac{\varepsilon ^{j_s }}{j_r !}} f_{i_s l}^{( {j_s })} Y_l^{j_s } \nonumber \\&=\sum \limits _{s=1}^\infty {\frac{1}{s!}} \sum \limits _{i_1 =1}^n {\cdots \sum \limits _{i_s =1}^n {\frac{\partial ^sH_r }{\partial P_{i_1 } \cdots \partial P_{i_s } }} } \sum \limits _{j_1 =1}^\infty \cdots \nonumber \\&\quad \times { \sum \limits _{j_s =1}^\infty {\frac{\varepsilon ^{j_1 +\cdots +j_r }}{j_1 !\cdots j_s !}} } f_{i_1 l}^{( {j_1 })} \cdots f_{i_s l}^{( {j_s })} Y_l^{j_1 +\cdots +j_s } \nonumber \\&=\sum \limits _{k=1}^\infty {\varepsilon ^kY_l^k C_{r;k;l} ( {{\mathbf {Q}},{\mathbf {P}}})} \end{aligned}$$

(90)

Appendix 2

Following the jump–diffusion rule in Ref. [27], the SIDE of action variables $I_{\eta }$ is

$$\begin{aligned} \mathrm{d}I_r&= \left[ \frac{\partial I_r }{\partial t}+\sum \limits _{i=1}^n \frac{\partial I_r }{\partial Q_i }\frac{\partial H}{\partial P_i }-\sum \limits _{i=1}^n \frac{\partial I_r }{\partial P_i }\right. \nonumber \\&\left. \times \left( {\frac{\partial H}{\partial Q_i }+\varepsilon ^2\sum \limits _{j=1}^n {m_{ij} \frac{\partial H}{\partial P_j }} }\right) \right. \nonumber \\&\left. +\frac{\varepsilon ^2}{2}\sum \limits _{j=1}^n {\sum \limits _{k=1}^n {\sum \limits _{l=1}^{n_g } {\sigma _{jl} \sigma _{kl} \frac{\partial ^2I_r }{\partial P_j \partial P_k }} } } \right] \mathrm{d}t \nonumber \\&+\varepsilon \sum \limits _{i=1}^n {\sum \limits _{j=1}^{n_g } {\frac{\partial I_r }{\partial P_i }\sigma _{ij} \mathrm{d}B_j (t)} } \nonumber \\&+\sum \limits _{l=1}^{n_p } {\int _{\mathcal{Q}_l } {\left[ {I_r ({\mathbf {Q}},{\mathbf {P}}\!+\!{\hat{\gamma }}_l )\!-\!I_r ({\mathbf {Q}},{\mathbf {P}})} \right] {\mathcal {P}}_l (\mathrm{d}t,\mathrm{d}Y_l ).} }\nonumber \\ \end{aligned}$$

(91)

Consider the first part (drift part) of $\mathrm{d}I_{r}$ equation (91):

$$\begin{aligned}&\frac{\partial I_r }{\partial t}+\sum \limits _{i=1}^n {\frac{\partial I_r }{\partial Q_i }\frac{\partial H}{\partial P_i }}\nonumber \\&\qquad -\sum \limits _{i=1}^n {\frac{\partial I_r }{\partial P_i }\left( {\frac{\partial H}{\partial Q_i }+\varepsilon ^2\sum \limits _{j=1}^n {m_{ij} \frac{\partial H}{\partial P_j }} }\right) } \nonumber \\&\qquad +\frac{\varepsilon ^2}{2}\sum \limits _{j=1}^n {\sum \limits _{k=1}^n {\sum \limits _{l=1}^{n_g } {\sigma _{jl} \sigma _{il} \frac{\partial ^2I_r }{\partial P_j \partial P_k }} } } \nonumber \\&\quad =\frac{\partial I_r }{\partial t}+\sum \limits _{i=1}^n \left( \frac{\partial I_r }{\partial Q_i }\frac{\partial H}{\partial P_i }-\frac{\partial I_r }{\partial P_i }\frac{\partial H}{\partial Q_i }\right) \nonumber \\&\qquad -\sum \limits _{i=1}^n {\frac{\partial I_r }{\partial P_i }\varepsilon ^2\sum \limits _{j=1}^n {m_{ij} \frac{\partial H}{\partial P_j }} }\nonumber \\&\qquad +\frac{\varepsilon ^2}{2}\sum \limits _{j=1}^n {\sum \limits _{k=1}^n {\sum \limits _{l=1}^{n_g } {\sigma _{jl} \sigma _{il} \frac{\partial ^2I_r }{\partial P_j \partial P_k }} } } , \end{aligned}$$

(92)

According to Eq. (6), in Eq. (92), we have

$$\begin{aligned} \frac{\partial I_r }{\partial t}=0, \end{aligned}$$

(93)

because $I_{\eta }$ and $H$ are all integrals of motion of the Hamiltonian system, the vanishing Poisson brackets (2) can be used here. The following equation can be obained:

$$\begin{aligned} \sum \limits _{i=1}^n {\left( {\frac{\partial I_r }{\partial Q_i }\frac{\partial H}{\partial P_i }-\frac{\partial I_r }{\partial P_i }\frac{\partial H}{\partial Q_i }}\right) } =0. \end{aligned}$$

(94)

Substituting Eqs. (93) and (94) in Eq. (92) leads to the last line of Eq. (92) as

$$\begin{aligned} -\varepsilon ^2\sum \limits _{i=1}^n {\sum \limits _{j=1}^n {m_{ij} \frac{\partial I_r }{\partial P_i }\frac{\partial H}{\partial P_j }} } +\frac{\varepsilon ^2}{2}\sum \limits _{j=1}^n {\sum \limits _{k=1}^n {\sum \limits _{l=1}^{n_g } {\sigma _{jl} \sigma _{kl} \frac{\partial ^2I_r }{\partial P_j \partial P_k }} } } .\nonumber \\ \end{aligned}$$

(95)

Replacing the drift part of Eq. (91) by Eq. (95), Eq. (90) can be rewritten as

$$\begin{aligned} \mathrm{d}I_r&= \left[ -\varepsilon ^2\sum \limits _{i=1}^n {\sum \limits _{j=1}^n {m_{ij} \frac{\partial I_r }{\partial P_i }\frac{\partial H}{\partial P_j }} }\right. \nonumber \\&\left. +\frac{\varepsilon ^2}{2}\sum \limits _{j=1}^n {\sum \limits _{k=1}^n {\sum \limits _{l=1}^{n_g } {\sigma _{jl} \sigma _{kl} \frac{\partial ^2I_r }{\partial P_j \partial P_k }} } } \right] dt \nonumber \\&+\varepsilon \sum \limits _{i=1}^n {\sum \limits _{j=1}^{n_g } {\frac{\partial I_r }{\partial P_i }\sigma _{ij} \mathrm{d}B_j (t)} }\nonumber \\&+\!\sum \limits _{l=1}^{n_p } {\int _{\mathcal{Q}_l } {\left[ {I_r ({\mathbf {Q}},{\mathbf {P}}\!+\!{\hat{\gamma }}_l )\!-\!I_r ({\mathbf {Q}},{\mathbf {P}})} \right] _l (\mathrm{d}t,\mathrm{d}Y_l ).} }\nonumber \\ \end{aligned}$$

(96)

Finally, the SIDEs for $I_{r}$ in Eq. (27) can be derived. Similar derivation of that for (96), the SIDEs for $\Theta _{r}$ in Eq. (27) and Eq. (47) can be derived.