Chaos: From simple models to complex systems by Massimo Cencini

By Massimo Cencini

Chaos: From uncomplicated versions to complicated structures goals to lead technological know-how and engineering scholars via chaos and nonlinear dynamics from classical examples to the latest fields of analysis. the 1st half, meant for undergraduate and graduate scholars, is a steady and self-contained advent to the options and major instruments for the characterization of deterministic chaotic platforms, with emphasis to statistical ways. the second one half can be utilized as a reference by means of researchers because it makes a speciality of extra complex issues together with the characterization of chaos with instruments of data thought and purposes encompassing fluid and celestial mechanics, chemistry and biology. The booklet is novel in devoting cognizance to some issues usually ignored in introductory textbooks and that are frequently chanced on in simple terms in complicated surveys resembling: info and algorithmic complexity conception utilized to chaos and generalization of Lyapunov exponents to account for spatiotemporal and non-infinitesimal perturbations. the choice of subject matters, a number of illustrations, workouts and suggestions for laptop experiments make the booklet excellent for either introductory and complex classes.

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G. a center with pure imaginary eigenvalues, while when the fixed point is an attractor, repeller or a saddle the flow topology around it remains locally unchanged. Anyway nonlinear terms may also give rise to other kinds of motion, not permitted in linear systems, as limit cycles. 25) with fixed point x∗ = (0, 0) of eigenvalues λ1,2 = 1 ± iω, corresponding to an unstable spiral. For any x(0) in a neighborhood of 0, the distance from the origin of the resulting trajectory x(t) grows in time so that the nonlinear terms soon becomes dominant.

2) of a Hamiltonian system is symplectic. Finally, we observe that the numerical integration of a Hamiltonian flow amounts to build up a map (time is always discretized), therefore it is very important to use algorithms preserving the symplectic structure — symplectic integrators — (see also Sec. 1 and Lichtenberg and Lieberman (1992)). It is worth remarking that the Hamiltonian/Symplectic structure is very “fragile” as it is destroyed by arbitrary transformations or perturbations of Hamilton equations.

N . , the Lorenz (1963) model: dx1 = −σx1 + σx2 dt dx2 = −x2 − x1 x3 + r x1 dt dx3 = −bx3 + x1 x2 , dt where σ, r, b are control parameters, and xi ’s are variables related to the state of fluid in an idealized Rayleigh-B´enard cell (see Sec. 2). 1 Conservative and dissipative dynamical systems We can identify two general classes of dynamical systems. To introduce them, let’s imagine to have N pendulums as that in Fig. 1a and to choose a slightly different initial state for any of them. e. a spot of points, occupying a Γ-volume, whose distribution is described by a probability density function (pdf) ρ(x, t = 0) normalized in such a way that Γ dx ρ(x, 0) = 1.

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