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By Roger C Jennison

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I. V. Smirnov ε 3/2 : 1 γ i∂τ0 ψ j,1 + i∂τ1 ψ j + ψ j + ∂ξ2 (ψ j − ψ¯ j ) − (ψ3− j − ψ¯ 3− j ) 2 2 3 − 4β (ψ j − ψ¯ j ) = 0, ψ j,1 = χ j,1 eiτ0 , (21) γ 1 i∂τ0 χ j,1 + i∂τ1 χ j + ∂ξ2 (χ j − χ¯ j e−2iτ0 ) − (χ3− j − χ¯ 3− j e−2iτ0 ) 2 2 − 4β (χ j eiτ0 − χ¯ j eiτ0 )3 e−iτ0 = 0. Integrating last equations (21) with respect to “fast” time τ0 , we get two coupled equations: γ 1 i∂τ1 χ j + ∂ξ 2 χ j − χ3− j + 12β |χ j |2 χ j = 0. (22) 2 2 First of all, we can see, that there are two symmetric solutions of Eqs.

Thus, Eqs. (6–9) are the same as for linear and nonlinear systems, but Eqs. (11) resulting from averaging are changed: γ2 1 3β i∂τ2 X1 + ∂ξ2 X1 − X1 + (3|X1 |2 X1 + 2|X2|2 X1 − X22X¯1 ) = 0, 2 8 8 (13) γ2 1 2 3β 2 2 2¯ (3|X2 | X2 + 2|X1| X2 − X1 X2 ) = 0. i∂τ2 X2 + ∂ξ X2 − X2 + 2 8 8 Equations (13) describe the pair of nonlinear oscillatory chains with nonlinear coupling contrary to the initial system with the linear coupling. It is very interesting that the structure of nonlinear terms is similar to the case of small FPU-system [7, 8].

6. Awrejcewicz J, Dzyubak L, Grebogi C (2004) A direct numerical method for quantifying regular and chaotic orbits, Chaos Solitons Fractals 19, 503–507. 7. Awrejcewicz J, Dzyubak L, Grebogi C (2005) Estimation of chaotic and regular (stick-slip and slip-slip) oscillations exhibited by coupled oscillators with dry friction, Nonlinear Dynamics 42, 383–394. Localized Nonlinear Excitations and Interchain Energy Exchange in the Case of Weak Coupling Leonid I. Manevich and Valeri V. Smirnov 1 Introduction The problem of energy exchange between weakly coupled nonlinear oscillators is actually far-reaching extension of classical beating problem in linear vibrations theory.