Hamiltonian chaos and fractional dynamics by George M. Zaslavsky
By George M. Zaslavsky
The dynamics of practical Hamiltonian structures has strange microscopic beneficial properties which are direct results of its fractional space-time constitution and its section area topology. The publication bargains with the fractality of the chaotic dynamics and kinetics, and in addition comprises fabric on non-ergodic and non-well-mixing Hamiltonian dynamics. The booklet doesn't keep on with the normal scheme of so much of present day literature on chaos. The purpose of the writer has been to place jointly the most complicated and but open difficulties at the common idea of chaotic structures. the significance of the mentioned concerns and an realizing in their starting place may still motivate scholars and researchers to the touch upon a few of the private features of nonlinear dynamics. The booklet considers the elemental ideas of the Hamiltonian conception of chaos and a few purposes together with for instance, the cooling of debris and indications, keep an eye on and erasing of chaos, polynomial complexity, Maxwell's Demon, and others. It provides a brand new and sensible picture of the foundation of dynamical chaos and randomness. An figuring out of the starting place of the randomness in dynamical platforms, which can't be of an identical foundation as chaos, presents new insights within the different fields of physics, biology, chemistry and engineering.
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4) (∀j, k). 5) which mutually commute [Fj , Fk ] = 0, It is also assumed that the level set of Fj forms a smooth compact connected manifold M ⊂ IR2N , and the functions Fj (∀j) are linearly independent on M. 6) where ωj (I) are deﬁned below and the Hamiltonian equations integrated in quadratures. Without loss of generality, one can consider F1 = H. 3, as integrals of motion instead of Fj (∀j). Let the initial Hamiltonian be p = (p1 . . , pN ), H0 = H0 (p, q), q = (q1 . . , qN ). 7) Consider a generating function q S(I, q) = q(0) p(I, q) · dq, I = (I1 , .
2) where p(t), q(t) are solutions of the dynamical equations and δ is the Dirac function. 2) is singular and it does not have any information except the solutions p(t), q(t). There are diﬀerent ways how one can introduce smoothed distributions in phase space. Consider ﬁnite dynamics in the phase space Γ and let Π be a partitioning of Γ by hypercubes of the volume 2N . We can introduce the number M (Π ) which is a minimal number of the hypercubes that cover full Γ. Let us label all hypercubes by k and p(k) , q(k) are 37 38 CHAOTIC DYNAMICS coordinates of the centre of the k-hypercube.
36). 41) with Hamiltonian Ψ = Ψ(x, y) and (x, y) as a canonical pair. The same equations can be generalized for the case when Ψ = Ψ(x, y; t) where t is a real time. 41) together with Ψ = Ψ(x, y; t) is the Hamiltonian one. 42) with a free helicity parameter c that can be a function of r. This type of ﬁeld was considered much with respect to many diﬀerent physical problems. 42): B × curl B = 0 is the condition of the force-free ﬁeld. 43) HAMILTONIAN EQUATIONS FOR THE ABC-FLOW 21 In 1965 V. I. 44) vz = C sin y + B cos x would possess a non-trivial topology of streamlines since it satisﬁes the Beltrami condition (with c = 1).