We investigate the detailed dynamics of multidimensional Hamiltonian systems by studying the evolution of volume elements formed by unit deviation vectors about their orbits. The behavior of these volumes is strongly influenced by the regular or chaotic nature of the motion, the number of deviation vectors, their linear (in)dependence and the spectrum of Lyapunov exponents. The different time evolution of these volumes can be used to identify rapidly and efficiently the nature of the dynamics, leading to the introduction of quantities that clearly distinguish between chaotic behavior and quasiperiodic motion on $N$-dimensional tori. More specifically we introduce the Generalized Alignment Index of order $k$ (GALI$_k$) as the volume of a generalized parallelepiped, whose edges are $k$ initially linearly independent unit deviation vectors from the studied orbit whose magnitude is normalized to unity at every time step. The GALI$_k$ is a generalization of the Smaller Alignment Index (SALI) (GALI$_2$ $\propto$ SALI). However, GALI$_k$ provides significantly more detailed information on the local dynamics, allows for a faster and clearer distinction between order and chaos than SALI and works even in cases where the SALI method is inconclusive.
Deep Dive into Geometrical properties of local dynamics in Hamiltonian systems: the Generalized Alignment Index (GALI) method.
We investigate the detailed dynamics of multidimensional Hamiltonian systems by studying the evolution of volume elements formed by unit deviation vectors about their orbits. The behavior of these volumes is strongly influenced by the regular or chaotic nature of the motion, the number of deviation vectors, their linear (in)dependence and the spectrum of Lyapunov exponents. The different time evolution of these volumes can be used to identify rapidly and efficiently the nature of the dynamics, leading to the introduction of quantities that clearly distinguish between chaotic behavior and quasiperiodic motion on $N$-dimensional tori. More specifically we introduce the Generalized Alignment Index of order $k$ (GALI$_k$) as the volume of a generalized parallelepiped, whose edges are $k$ initially linearly independent unit deviation vectors from the studied orbit whose magnitude is normalized to unity at every time step. The GALI$_k$ is a generalization of the Smaller Alignment Index (SALI
Determining the chaotic or regular nature of orbits in conservative dynamical systems is a fundamental issue of nonlinear science. The difficulty with conservative systems, of course, is that regular and chaotic orbits are distributed throughout phase space in very complicated ways, in contrast with dissipative systems, where all orbits eventually fall on regular or chaotic attractors. Over the years, several methods distinguishing regular from chaotic motion in conservative systems have been proposed and applied, with varying degrees of success. These methods can be divided in two major categories: Some are based on the study of the evolution of small deviation vectors from a given orbit, while others rely on the analysis of the particular orbit itself.
The most commonly employed method for distinguishing between order and chaos, which belongs to the category related to the study of deviation vectors, is the evaluation of the maximal Lyapunov Characteristic Exponent (LCE) σ 1 ; if σ 1 > 0 the orbit is chaotic. The theory of Lyapunov exponents was applied to characterize chaotic orbits by Oseledec [1], while the connection between Lyapunov exponents and exponential divergence of nearby orbits was given in [2,3]. Benettin et al. [4] studied the problem of the computation of all LCEs theoretically and proposed in [5] an algorithm for their numerical computation. In particular, σ 1 is computed as the limit for t → ∞ of the quantity
where w(0), w(t) are deviation vectors from a given orbit, at times t = 0 and t > 0 respectively. It has been shown that the above limit is finite, independent of the choice of the metric for the phase space and converges to σ 1 for almost all initial vectors w(0) [1,4,5]. Similarly, all other LCEs, σ 2 , σ 3 etc. are computed as the limits for t → ∞ of some appropriate quantities, L 2 (t), L 3 (t) etc. (see [5] for more details). We note that throughout the present paper, whenever we need to compute the values of the maximal LCE or of several LCEs we apply respectively the algorithms proposed by Benettin et al. [2,5]. Since 1980, new methods have been introduced for the effective computation of LCEs (e. g. [6], see also [7] and references therein). The true power of these techniques is revealed in the study of multi-dimensional systems, when only a small number of LCE are of interest. In such cases, these methods are significantly more efficient than the method of [5], which computes the whole spectrum of LCEs. On the other hand, they are less or equally efficient when compared with the method of [2] for the computation of the maximal LCE, whose value is sufficient for the determination of the regular or chaotic nature of an orbit.
Among other chaoticity detectors, belonging to the same category with the evaluation of the maximal LCE, are the fast Lyapunov indicator (FLI) and its variants [8,9,10,11,12], the mean exponential growth of nearby orbits (MEGNO) [13,14], the smaller alignment index (SALI) [15,16,17], the relative Lyapunov indicator (RLI) [18], as well as methods based on the study of power spectra of deviation vectors [19], as well as spectra of quantities related to these vectors [20,21,22]. In the category of methods based on the analysis of a time series constructed by the coordinates of the orbit under study, one may list the frequency map analysis of Laskar [23,24,25,26,27,28], the method of the low frequency power (LFP) [29,30], the ‘0-1’ test [31], as well as some other more recently introduced techniques [32,33].
In the present paper, we generalize and improve considerably the SALI method mentioned above by introducing the Generalized ALignment Index (GALI). This index retains the advantages of the SALI -i.e. its simplicity and efficiency in distinguishing between regular and chaotic motion -but, in addition, is faster than the SALI, displays power law decays that depend on torus dimensionality and can also be applied successfully to cases where the SALI is inconclusive, like in the case of chaotic orbits whose two largest Lyapunov exponents are equal or almost equal.
For the computation of the GALI we use information from the evolution of more than two deviation vectors from the reference orbit, while SALI’s computation requires the evolution of only two such vectors. In particular, GALI k is proportional to ‘volume’ elements formed by k initially linearly independent unit deviation vectors whose magnitude is normalized to unity at every time step. If the orbit is chaotic, GALI k goes to zero exponentially fast by the law
If, on the other hand, the orbit lies in an N-dimensional torus, GALI k displays the following behaviors: Either
or, if N < k ≤ 2N, it decays with different power laws, depending on the number m of deviation vectors which initially lie in the tangent space of the torus, i. e. :
So, the GALI allows us to study more efficiently the geometrical properties of the dynamics in the neighborhood of an orbit, especially in higher dimensions, whe
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