The equivalence of the methods of stochastic averaging and stochastic normal forms is demonstrated for systems under the effect of linear multiplicative and additive noise. It is shown that both methods lead to reduced systems with the same Markovian approximation. The key result is that the second-order stochastic terms have to be retained in the normal form computation. Examples showing applications to systems undergoing divergence and flutter instability are provided. Furthermore, it is shown that unlike stochastic averaging, stochastic normal forms can be used in the analysis of nilpotent systems to eliminate the stable modes. Finally, some results pertaining to stochastic Lorenz equations are presented.

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