By S L Sobolev

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L + ... -1~8A~ + .. -) +.... L = 0, the dependence of the frequency on the amplitude for a conservative system with the Duffing characteristics. 2. 43) mZ2 - C(ZI - Z2) - k(Zl - Z2) - k 1(Zl - Z2)3 = -(k3 - k o cos 2wt)Z2, where m and M are the masses, k, k 1 , k 3 and ko are the rigidities, and C is the damping coefficient. 43). They amount to a21 2 = ! , 2 3 ]. 44) Our consideration is limited to calculating the first simple parametric resOnanCe around the frequency al. 5 Nonstationary Nonlinear Systems 51 where 1 1/J = , 11 and the dot now denotes the differentiation with respect to r.

23) give the following estimation of the stability limits u(1) u(2) ~ ! _ tt ~ ! + tt. 24) 4 2 For n = 2 we have the following stability limits u(1) ~ 1 + ~H2 u(2) ~ 1 - -tt2 . 25) 2 In all these cases We introduced the small positive perturbation parameter tt, which characterizes the modulation depth of the parametric excitation. However, linear systems are an idealization of real systems which are nonlinear. Therefore, we consider the general form of the system of equations {±} = [[~o] + tt [~I(t)] + tt 2 [~2(t)] + ...

The resonance occurring in nonlinear systems can be classified as follows: 1. 2. 3. 4. Main resonance (m = n = 1). Subharmonic resonance (m = 1, n > 1). Ultraharmonic resonance (m > 1, n = 1). Ultrasubharmonic resonanCe (m > 1, n > 1). 4) where c is the perturbation parameter and the exciting force P(7]) = P(7] + 2'rr), where 7] = wt. 4). 7) Here we restrict ourselves to the O(c 2 ) approximation. 8) 28 2. 11) Because .. 4) takes the form • ! 4) into a power series in c, we have P = c [Q(y, y) + P(1])].

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Applications of functional analysis in mathematical physics by S L Sobolev
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