By Charalampos (Haris) Skokos, Georg A. Gottwald, Jacques Laskar

ISBN-10: 3662484080

ISBN-13: 9783662484081

ISBN-10: 3662484102

ISBN-13: 9783662484104

Distinguishing chaoticity from regularity in deterministic dynamical platforms and specifying the subspace of the part area during which instabilities are anticipated to take place is of extreme value in as disparate components as astronomy, particle physics and weather dynamics.

To deal with those matters there exists a plethora of tools for chaos detection and predictability. the main often hired method for investigating chaotic dynamics, i.e. the computation of Lyapunov exponents, in spite of the fact that, may well endure a couple of difficulties and disadvantages, for instance whilst utilized to noisy experimental data.

In the final twenty years, a number of novel equipment were constructed for the short and trustworthy choice of the common or chaotic nature of orbits, geared toward overcoming the shortcomings of extra conventional strategies. This set of lecture notes and educational experiences serves as an advent to and evaluate of recent chaos detection and predictability strategies for graduate scholars and non-specialists.

The ebook covers theoretical and computational elements of conventional how to calculate Lyapunov exponents, in addition to of contemporary thoughts just like the speedy (FLI), the Orthogonal (OFLI) and the Relative (RLI) Lyapunov signs, the suggest Exponential development issue of close by Orbits (MEGNO), the Smaller (SALI) and the Generalized (GALI) Alignment Index and the ‘0-1’ try out for chaos.

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ECMWF, Reading (1996) 33. : A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19, 197–231 (1968) 34. : Chaos in Dynamical Systems. Cambridge University Press, Cambridge (1993) 35. : Identification of true and spurious Lyapunov exponents from time series. Int. J. Bifurcat. Chaos 2, 155–165 (1992) 36. : Lyapunov exponents from Chua’s circuit. J. Circuits Syst. Comput. 3, 507–523 (1993) 37. : Numerical Recipes: The Art of Scientific Computing, 3rd edn.

X/ can be used to model the (linearized) flow (very) close to the reference points xn along the orbits. To approximate the flow in a larger neighbourhood of xn or even globally, nonlinear basis functions are required, like multidimensional polynomials [2, 4–7], or radial basis functions [23, 24, 35]. To estimate the coefficient matrix C in Eq. 50) or the coefficient vector c in Eq. zj / is known. n/ 2 Un g of the reference point xn where Un defines the chosen neighbourhood that can be of fixed mass (a fixed number K of nearest neighbours of xn ) or of fixed size (all points with distance smaller than a given bound ).

25) we recognize the typical lobes related to the hyperbolic manifold of hyperbolic periodic orbits. 2 Theory and Applications of the Fast Lyapunov Indicator (FLI) Method 47 Fig. 9 Zoom of Fig. 7. 5 The FLI for Detecting the Geography of Resonances The problem of long term stability of an Hamiltonian system is strongly related to the famous KAM [1, 28, 43] and Nekhoroshev [48] theorems which leave the possibility of a slow drift of the orbits on a peculiar subset of the phase space, the so called Arnold’s web.

### Chaos Detection and Predictability by Charalampos (Haris) Skokos, Georg A. Gottwald, Jacques Laskar

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