By Massimo Cencini
"Chaos: From uncomplicated types 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 study. the 1st half, meant for undergraduate and graduate scholars, is a gradual and self-contained advent to the innovations and major instruments for the characterization of deterministic chaotic platforms, with emphasis to statistical methods. the second one half can be utilized as a reference via researchers because it makes a speciality of extra complex subject matters together with the characterization of chaos with instruments of data conception and purposes encompassing fluid and celestial mechanics, chemistry and biology. The ebook is novel in devoting cognizance to some themes usually missed in introductory textbooks and that are often chanced on merely in complex surveys similar to: info and algorithmic complexity idea utilized to chaos and generalization of Lyapunov exponents to account for spatiotemporal and non-infinitesimal perturbations. the choice of subject matters, quite a few illustrations, routines and suggestions for machine experiments make the e-book excellent for either introductory and complex classes.
Read Online or Download Chaos: From Simple Models to Complex Systems (Series on Advances in Statistical Mechanics) PDF
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Extra resources for Chaos: From Simple Models to Complex Systems (Series on Advances in Statistical Mechanics)
E. the allowed values of the variables xi (i = 1, . . , d), deﬁnes the phase space of the system. The pendulum of Eq. 5in ChaosSimpleModels Chaos: From Simple Models to Complex Systems integer k) identify the same angle. The trajectories depicted in Fig. 1c represent the phase-space portrait of the pendulum. 1) which is a compact notation for dx1 = f1 (x1 (t), x2 (t), · · · , xd (t)) , dt .. dxd = fd (x1 (t), x2 (t), · · · , xd (t)) . dt More precisely, Eq. 1) deﬁnes an autonomous ODE as the functions fi ’s do not depend on time.
22) takes the form (we exclude the case of degenerate eigenvalues): δx(n) = c1 λn1 e1 + c2 λn2 e2 . 23) The above equation shows that, for discrete time systems, the stability properties depend on whether λ1 and λ2 are in modulus smaller or larger than unity. Using the notation λi = ρi eiθi , if all eigenvalues are inside the unit circle (ρi ≤ 1 for each i) the ﬁxed point is stable. As soon as, at least, one of them crosses the circle (ρj > 1 for some j) it becomes unstable. 1. For general d-dimensional maps, the classiﬁcation asymptotically stable/unstable remains the same but the boundary of stability/instability is now determined by ρi = 1.
Applying (AT )−1 to both sides of the above expression we ﬁnd 1 (AT )−1 e = e . 2) λ Finally, multiplying Eq. 2) by J and using Eq. 1), we end with A (J e) = 1 (J e) , λ meaning that J e is an eigenvector of A with eigenvalue 1/λ. As a consequence, a (d = 2N )dimensional symplectic map has 2N eigenvalues such that λi+N = 1 λi i = 1, . . , N . As we will see in Chapter 5 this symmetry has an important consequence for the Lyapunov exponents of chaotic Hamiltonian systems. 2 Nonlinear stability Linear stability, though very useful, is just a part of the history.