This paper deals with the average principle for deterministic and stochastic perturbations. For a perturbed system, we suppose that the time change \(\varepsilon t\to t\) leads to the \((n+ m)\)-dimensional system,
\[ dX^\varepsilon(t)= \Biggl({1\over \varepsilon} B(X^\varepsilon(t))+ \delta(X^\varepsilon(t))\Biggr)\,dt+ \sigma(X^\varepsilon(t))\,dW(t), \]
where \(W\) is a Wiener process and the dynamics \({dX(t)\over dt}= B(X(t))\) means the nonperturbed part. Using the first integrals, \(z_1,\dots, z_n\), of system \(X\) as coordinates, the authors introduce the action-angle coordinates and reformulate the system as a diffusion,
\[ X^\varepsilon(t) =(Z^\varepsilon_1(t),\dots, Z^\varepsilon_n, \varphi^\varepsilon_1(t),\dots,\varphi^\varepsilon_m(t)) \quad (=:(Z^\varepsilon(t), \varphi^\varepsilon(t))) \]
on \([Z]\times{\mathbf T}^m\), described by stochastic differentials,
\[ \begin{aligned} dZ^\varepsilon_i(t) &= b_i(Z^\varepsilon, \varphi^\varepsilon(t))\,dt+ \sum_k \sigma^k_i(Z^\varepsilon(t), \varphi^\varepsilon(t))\,dW_k(t),\\ d\varphi^\varepsilon_j(t) &= \Biggl({1\over \varepsilon} \omega_j(Z^\varepsilon(t))+ c_j(Z^\varepsilon(t), \varphi^\varepsilon(t))\Biggr)\,dt+ \sum_k \tau^k_j(Z^\varepsilon(t), \varphi^\varepsilon(t))\,dW_k(t)\end{aligned} \]
with its starting point \((z,\varphi)\in Z\times F{\mathbf T}^m\), where \([Z]\) is the closure of the bounded set \(Z\subset\mathbb{R}^n\) and \({\mathbf T}^m\) the \(m\)-dimensional torus with one-cyclic coordinates, \(\varphi_1,\dots, \varphi_m\). \(\varphi^\varepsilon\) is the fast motion with speedvector \((\omega_1,\dots, \omega_m)\) and the problem is to characterize the limit of \(Z^\varepsilon\), as \(\varepsilon\to 0\).
Under the main assumptions that the Lebesgue measure of the set \(\{z\in Z;\omega_1(z),\dots, \omega_m(z)\) are rationally dependent} is \(0\) and some nondegeneracy on diffusion coefficients, the authors prove that the average principle of stochastic perturbations holds for any starting point, namely the distribution of \(Z^\varepsilon\) converges weakly, as \(\varepsilon\downarrow 0\), to the distribution of the diffusion process with the diffusion matrix \((\overline a_{ij})\) and drift coefficient \((\overline b_i)\), where
\[ \overline a_{ij}(z)= \int_{{\mathbf T}^m} \biggl(\sum_k \sigma^k_i(z, \varphi)\sigma^k_j(z, \varphi) \biggr)\,d\varphi \quad\text{and}\quad \overline b_i(z)= \int_{{\mathbf T}^m} b_i(z,\varphi)\,d\varphi. \]
Moreover, using this result, they derive new results for averaging of purely deterministic systems by means of small white noise type perturbations.