By Gennady Leonov

The papers during this edited quantity goal to supply a greater realizing of the dynamics and keep watch over of a big classification of hybrid dynamical structures which are defined by way of various types in several country area domain names. They not just disguise very important features and instruments for hybrid structures research and keep watch over, but additionally a few experimental realizations. certain cognizance is given to synchronization - a common phenomenon in nonlinear technological know-how that received super importance due to the fact its discovery by way of Huygens within the seventeenth century. attainable functions of the implications brought within the ebook contain keep watch over of cellular robots, keep an eye on of CD/DVD avid gamers, versatile production strains, and complicated networks of interacting brokers. The publication is predicated at the fabric provided at a equally entitled minisymposium on the sixth eu Nonlinear Dynamics convention held in St Petersburg in 2008. it really is specific in that it includes result of a number of foreign and interdisciplinary collaborations within the box, and displays state of the art technological improvement within the region of hybrid mechanical structures on the vanguard of the twenty first century.

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**Extra resources for Dynamics and Control of Hybrid Mechanical Systems (World Scientific Series on Nonlinear Science, Series B) **

**Example text**

K! ∂ k η We shall say that the sums m m xhk (t)hk , yhm (t, h) = xhm (t, h) = k=1 yhk (t)hk k=1 are the mth approximation of solution of system with respect to h. 1). 8), we obtain dyh1 (t) dxh1 (t) = −yh1 (t), = xh1 (t). V. A. 9) for the first approximation of the solution (x(t, h), y(t, h)) with respect to h, we have xh1 (t, h) = xh1 (t)h = −h sin(t), yh1 (t, h) = yh1 (t)h = h cos(t). 13) Similarly, to obtain the second approximation (xh2 (t, h), yh2 (t, h)), we substitute the expressions x(t, h) = xh2 (t, h) + h3 ∂ 3 x(t, η) |η=hθx (t,h) , 3!

And Kudryashova, E. V. (2008). Limit cycles of two dimensional systems. Calculations, proofs, experiments. Petersburg University. Mathematics, 41(3), pp. 216–250. Leonov, G. A. (2009). Limit cycles of Lienard system with discontinuous coefficients. Doclady Akademii Nauk, 426(1), pp. 47–50. Leonov, G. A. (2009). Effective methods for investigation of limit cycles in dynamical systems (2009) Applied Mathematics and Mechanics, 4, [in print] Leonov, G. A. (2009). The criteria of four cycles existence in quadratic systems.

10) t and G(y(τ ) , Ky(τ ))dτ ≥ 0 , s ∀ s, t : 0 ≤ s < t , ∀ y ∈ L2loc (R+ ; Y1 ) . 11) With the superscript c we denote the complexification of spaces and operators and the extension of quadratic forms to Hermitian forms. 1. 8) with ϕ ∈ N (F, G) and ψ ∈ M(d). Suppose that for the operators Ac , B c the assumptions (F1) and (F2) are satisfied. 12) the frequency-domain condition F c ((iωI c − Ac )−1 B c ξ, ξ) + Gc ((iωI c − Ac )−1 B c ξ, ξ) ≤ −α χ(z) (iω)ξ ∀ ω ∈ R : iω ∈ σ(Ac ) , is satisfied and the functional 2 Zc ∀ ξ ∈ Ξc ∞ [F c (y(τ ), ξ(τ )) + Gc (y(τ ), ξ(τ )) J(y(·), ξ(·)) := 0 + α Dc y(τ ) + E c ξ(τ ) 2Z c dτ is bounded from above on any set My0 := {y(·), ξ(·) : y˙ = Ay + Bξ on R+ , y(0) = y0 , y(·) ∈ L∞ , ξ(·) ∈ L2 (0, ∞; Ξ)} .