Dynamics of Nonlinear Stochastic Systems

Cover Dynamics of Nonlinear Stochastic Systems
Dynamics of Nonlinear Stochastic Systems
R H Kraichnan
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, (b^) -l[S(p)]-2[fi(p)]2j(3)(p) . (^. Is, - 37 - Combining our results (and suppressing the argument p) we have, finally, Eliminating H and J from (7. 6) and (7. I7), we obtain (b^> ^G^ ♦ p( ♦ ap2)G^ - ( ♦ 3ap^)G^ ♦ 3apO - a - . (7. 18) The relevant branch of (7. 18) is the one for which G(co) • n" Re {G(-iw)} reduces to (5. 6) when a ■ 0, and it is easily verified that such a branch exists. Let us examine the behavior of G((a) on this branch as a is varied. For sufficiently small a, it is p...lausible to assume that our assignment of values to the C's is realizable. As we increase a, we may plausibly anticipate that G(co) will begin to violate (2. U), or at least will exhibit some discontinuous change in its dependence on a, when a critical value is reached for which the C »s are unrealizable by any assignment of values to the Jjf's consistwit with (U. U). This argument is supported by the example of Section 6« The behavior of (7. 18) is accessible by standard techniques for quartic equations.

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