By George M. Zaslavsky

The dynamics of sensible Hamiltonian structures has strange microscopic beneficial properties which are direct results of its fractional space-time constitution and its part area topology. The ebook bargains with the fractality of the chaotic dynamics and kinetics, and likewise comprises fabric on non-ergodic and non-well-mixing Hamiltonian dynamics. The e-book doesn't stick to the normal scheme of so much of latest literature on chaos. The goal of the writer has been to place jointly essentially the most complicated and but open difficulties at the normal concept of chaotic structures. the significance of the mentioned matters and an realizing in their starting place should still encourage scholars and researchers to the touch upon a few of the inner most features of nonlinear dynamics. The publication considers the elemental rules of the Hamiltonian concept of chaos and a few functions together with for instance, the cooling of debris and indications, regulate and erasing of chaos, polynomial complexity, Maxwell's Demon, and others. It offers a brand new and real looking photo of the starting place of dynamical chaos and randomness. An figuring out of the starting place of the randomness in dynamical structures, which can't be of an analogous starting place as chaos, offers new insights within the diversified fields of physics, biology, chemistry and engineering.

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23) LOCAL INSTABILITY AND LYAPUNOV EXPONENTS 43 d (t ) d (0) Fig. 2. Local instability of trajectories. A more frequently used characteristic is Lyapunov exponent. 17). The Jacobian matrix M ˆ = DTˆxn = M ∂xi;n+1 ∂xj;n , (i, j = 1, . . 24) where N is the number of degrees of freedom. According to the Liouville theorem ˆ | = 1 for any n. |M Consider the expression ˆ n = Dv Tˆn x, M where v denotes a vector along which the derivative is taken. 25) form an ‘ellipsoid’ of values λ with a centre at x and a fan of directions v.

15) and for the frequency we have ω = ω(H) = x˙ (2H)1/2 2π = = . 16), ω = ω(I) = I . 19) MAGNETIC MOMENTS 17 p V(x ) H – – a 2 0 a 2 a 2 0 a 2 x x Fig. 4. Particle oscillations between two walls (left) and the corresponding phase space (right). with θ2π as a periodic step-function and i I, 2an θ2π (t) = θ2π (t + T ) = ±1. 20) Since the trajectories are singular (see Fig. 4), the Fourier coeﬃcients pn do not have exponential decay and the ﬁnite N ∗ does not exist. 3 Magnetic moments Consider a particle with a magnetic moment M = (Mx , My , Mz ) and M2 = Mx2 + My2 + Mz2 .

E. 26) is considered in the vicinity of the resonance value of the action I0 . 33) where ω0 = ω(I0 ), ω = (iii) Neglect the second term of the order of for the angle. 36) assuming that ω > 0. In the case that ω < 0 a corresponding phase shift in the deﬁnition of ψ should be applied. 37) NON-LINEAR RESONANCE AND CHAIN OF ISLANDS 31 Fig. 5. Non-linear resonances on the phase plane (p, q). The dashed line is the unperturbed trajectory with I = I0 . The thick line is a new separatrix of phase oscillations: (a) k0 = 0 = 1; (b) k0 = 6, 0 = 1.