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

Distinguishing chaoticity from regularity in deterministic dynamical platforms and specifying the subspace of the section house during which instabilities are anticipated to happen is of extreme significance in as disparate parts as astronomy, particle physics and weather dynamics.


To handle those matters there exists a plethora of equipment for chaos detection and predictability. the main usually hired procedure for investigating chaotic dynamics, i.e. the computation of Lyapunov exponents, even though, could endure a couple of difficulties and downsides, for instance while utilized to noisy experimental data.


In the final twenty years, a number of novel equipment were built for the short and trustworthy selection of the typical or chaotic nature of orbits, aimed toward overcoming the shortcomings of extra conventional strategies. This set of lecture notes and educational studies serves as an creation to and evaluation of recent chaos detection and predictability strategies for graduate scholars and non-specialists.


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

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4 Δ EE/ Δt 15 −3 −4 Δ EL/ Δt (b) 1 0 E EE (a) U. 3 0 15 5 kΔt 10 15 kΔt Fig. 10 Direct estimation of the largest Lyapunov exponent from a noisy Hénon time series (SNR 30 dB) for different reconstruction dimensions D D 2, D D 4, D D 6 using a lag of L D 1. The diagrams (a), (c), and (e) show EE , EL and EF vs. 30). In (b), (d), and (f) the corresponding slopes E= t vs. k t (Eq. 48)) are shown. 2 D=2 D=4 D=6 0 10 kΔt 15 5 10 15 kΔt Fig. n/g time series of the folded-towel map of length Nd D 65;536 for different embedding dimensions D D 2, D D 4, D D 6 using a lag of L D 1.

In Fig.

1) is Hamiltonian and if the motion is regular (except for some peculiar hyperbolic structures, such as whiskered tori) then the largest Lyapunov exponent is zero, otherwise it is positive. This property has been largely used to discriminate between chaotic and ordered motions. However, among regular motions the Lyapunov exponent does not distinguish between circulation and libration orbits. In contrast, the FLI distinguishes between them ([13, 34], see Sect. 1). e. circulation and libration orbits) in relatively short CPU times [11, 13].

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