By Charalampos (Haris) Skokos, Georg A. Gottwald, Jacques Laskar
Distinguishing chaoticity from regularity in deterministic dynamical platforms and specifying the subspace of the part area within which instabilities are anticipated to ensue is of maximum value in as disparate parts as astronomy, particle physics and weather dynamics.
To handle those concerns there exists a plethora of equipment for chaos detection and predictability. the main quite often hired approach for investigating chaotic dynamics, i.e. the computation of Lyapunov exponents, even though, may perhaps endure a few difficulties and downsides, for instance whilst utilized to noisy experimental data.
In the final 20 years, numerous novel equipment were built for the quick and trustworthy choice of the ordinary or chaotic nature of orbits, aimed toward overcoming the shortcomings of extra conventional recommendations. This set of lecture notes and educational reports serves as an creation to and review of contemporary chaos detection and predictability concepts for graduate scholars and non-specialists.
The publication covers theoretical and computational points of conventional tips on how to calculate Lyapunov exponents, in addition to of recent thoughts just like the quick (FLI), the Orthogonal (OFLI) and the Relative (RLI) Lyapunov signs, the suggest Exponential development issue of within reach Orbits (MEGNO), the Smaller (SALI) and the Generalized (GALI) Alignment Index and the ‘0-1’ try out for chaos.
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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  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. The detection of the Arnold’s web is therefore the first step to achieve if we are interested in studying the long term stability properties of a system. I; '/ are action angle variables.
At this purpose we provide in Sect. 2 the definition and use of the FLI on a simple 2-dimensional discrete model: the standard map. On this model we try to answer to some frequently asked questions about the implementation and use of the method. In Sect. 3 we show the use of the FLI for the computation of the stable and unstable manifolds. In Sect. 4 we provide an application on a generic Hamiltonian model. In Sect. 5 we show an application of the FLI for the detection of the resonances of a quasi-integrable Hamiltonian system with 3 degrees of freedom and we show how to use the FLI to follow the diffusion of orbits along resonant lines.
15 shows diagrams with reconstruction dimensions D D 6, D D 11, and D D 21 and corresponding lags L D 4, L D 2, and L D 1, respectively. e. a period of time of length 20 0:025 D 0:5 which is close to the period of the natural oscillations of the Lorenz-63 system. 5 3 4 0 1 2 kΔt Fig. n//g time series of the Lorenz-63 system of length Nd D 65;536 for different reconstruction dimensions D D 4, D D 12, and D D 21, all with a lag of L D 1. As reference points only those 20 % of the reconstructed states are used that possess the nearest neighbours.
Chaos Detection and Predictability by Charalampos (Haris) Skokos, Georg A. Gottwald, Jacques Laskar