By Jan Awrejcewicz, Jan Awrejcewicz

This quantity includes the invited papers awarded on the ninth foreign convention Dynamical platforms conception and purposes held in LÃ³dz, Poland, December 17-20, 2007, facing nonlinear dynamical platforms. The convention introduced jointly a wide team of remarkable scientists and engineers, who take care of a variety of difficulties of dynamics encountered either in engineering and in day-by-day life.Topics coated comprise, between others, bifurcations and chaos in mechanical platforms; regulate in dynamical structures; asymptotic equipment in nonlinear dynamics; balance of dynamical structures; lumped and non-stop structures vibrations; unique numerical tools of vibration research; and man-machine interactions.Thus, the reader is given an summary of the latest advancements of dynamical platforms and will stick with the most recent developments during this box of technological know-how. This e-book could be of curiosity to to natural and utilized scientists operating within the box of nonlinear dynamics.

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**Example text**

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.