One should note that Eqs. ͑22a͒ and ͑22b͒ also can be found by naïvely adding the stochastic terms to the reduced system of evolution equations for the deterministic problem ͓Eqs. ͑18a͒ and ͑18b͔͒. This type of stochastic center manifold reduction has been done for the case of discrete noise. 27 Additionally, in many other fields ͑e.g., oceanography, solid mechanics, fluid mechanics͒, researchers have performed stochastic model reduction using a Karhunen-Loève expansion ͑principal component analysis, proper orthogonal decomposition͒. 33, 34 However, this linear projection does not properly capture the nonlinear effects. Furthermore, one must subjectively choose the number of modes needed for the expansion. Therefore, even though the solution to the reduced model found using this technique may have the correct statistics, individual solution realizations will not agree with the original, complete solution.