By Professor Jan Awrejcewicz (auth.), Professor Jan Awrejcewicz (eds.)
Bifurcation and Chaos provides a suite of in particular written articles describing the speculation and alertness of nonlinear dynamics to a wide selection of difficulties encountered in physics and engineering. every one bankruptcy is self-contained and contains an trouble-free advent, an exposition of the current state-of-the-art, and information of modern theoretical, computational and experimental effects. incorporated one of the sensible structures analysed are: hysteretic circuits, Josephson circuits, magnetic structures, railway dynamics, rotor dynamics and nonlinear dynamics of speech. This booklet includes vital info and ideas for all mathematicians, physicists and engineers whose paintings in R&D or academia comprises the sensible end result of chaotic dynamics.
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Additional resources for Bifurcation and Chaos: Theory and Applications
In this case, the attractor is called asymptotically self-similar. If the scaling exponents of a system are not identical with respect to the different directions in the phase-space, the attractor is called self-affine . For completeness, let us point out that with the identification: -nc = energy = E; nco: = volume/temperature = V/kBT; -(3 = inverse temperature = l/kBT; q = pressure = p, the analogy with statistical mechanics can be made explicit [7, 19] (kB denotes Boltzmann's constant). In this isothermisobar partition, n can be identified with the number of spins with as many possible states as given by the associated symbolic dynamics.
For the first point of view, the self similar structures in the phase-space can be described with the help of a logarithmic scaling exponent of the measure. The local crowding index a(xo, f) is then introduced  as R. -_ _ _ _ _ _ _--' ,I I II BBB BAA --1- _ ~ 0 o o Fig. 1. Supercritical tent map (a), generated Cantor set (b), temporal sequence of points (c) ( a xo, E ) = logP(B(xo,E)) IOgE (3) ' where P(B(xo, E)) denotes the probability of the ball B(xo, E) of radius E and center Xo with respect to the natural measure P(B(xo, E)) = fB(xQ,E)dp(x).
Much of the success of this approach, however, depends on the fact that the alphabet associated with our model can be reformulated to R. 4 Fig. 8. Asymptotic form of the generalized entropy function, calculated from the zeta-function approach. , the region in the (a, e)-plane for which SG(a, e) is a nonzero. The dashed line indicates the support of SG(e) = SG(a, e)lq=O constitute a complete one. As a consequence, for an experimental setting or a model with a complex grammar, the application of this powerful tool may not be straightforward.
Bifurcation and Chaos: Theory and Applications by Professor Jan Awrejcewicz (auth.), Professor Jan Awrejcewicz (eds.)