By Victor A. Sadovnichiy, Mikhail Z. Zgurovsky

Focused on fresh advances, this e-book covers theoretical foundations in addition to a number of functions. It provides sleek mathematical modeling techniques to the qualitative and numerical research of recommendations for complicated engineering difficulties in physics, mechanics, biochemistry, geophysics, biology and climatology. Contributions through a global workforce of revered authors bridge the space among summary mathematical techniques, resembling utilized tools of contemporary research, algebra, primary and computational mechanics, nonautonomous and stochastic dynamical structures at the one hand, and sensible functions in nonlinear mechanics, optimization, selection making conception and keep watch over idea at the different. As such, the booklet may be of curiosity to mathematicians and engineers operating on the interface of those fields.

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2 Billiard Systems as the Models for the Rigid Body Dynamics 31 Fig. 6 Billiard system and the Euler case of the rigid body dynamics. The motion of a rigid body, the appropriate settings in the shaded gray area on the bifurcation diagram, modeled billiards in the domain shaded in the same shade The Euler case is Liouville equivalent to the case of the geodesic flow on the ellipsoid [7]. This has been proven by the application of the theory of Fomenko– Zieschang—by calculating and comparing the invariants.

226]. 1). 1 (Zgurovsky and Kasyanov [4]) Let Assumptions (1)–(5) hold. Then, there exist positive constants c3 and c4 such that the following inequalities hold: 3 Uniform Global Attractors for Nonautonomous Evolution Inclusions y(t) 2 H ≤ y(s) 2 −c3 (t−s) He 39 + c4 , for each y ∈ K + , t ≥ s ≥ 0. 1 (Zgurovsky and Kasyanov [4]) Let Assumptions (1)–(5) hold. Let {yn }n≥1 ⊂ K + be a bounded in L∞ (R+ ; H) sequence. Then, there exist a subsequence {ynk }k≥1 ⊂ {yn }n≥1 and an element y ∈ K + such that max ynk (t) − y(t) t∈[τ,T ] H → 0, k → +∞, for any finite time interval [τ, T ] ⊂ (0, +∞).

Topology of Liouville foliations in the Steklov and the Sokolov integrable cases of Kirchhoff’s equations. Sb. Math. 195(3), 369–412 (2004) 23. : Topological classification of systems of Kovalevskaya-Yehia type. Sb. Math. 205(1), 101–155 (2014) 24. : (Russian) [Billiards] A genetic introduction to the dynamics of systems with impacts. Moskov Gos University, Moscow (1991) 25. : Integrable Billiards, Quadrics, and Multidymensional Poncelet Porisms. Regulyarnaya i Khaolichcskaya Dinamika, Izhevsk (2010).