How does the Smaller Alignment Index (SALI) distinguish order from chaos?

The ability of the Smaller Alignment Index (SALI) to distinguish chaotic from ordered motion, has been demonstrated recently in several publications.\cite{Sk01,GRACM} Basically it is observed that in chaotic regions the SALI goes to zero very rapidly, while it fluctuates around a nonzero value in ordered regions. In this paper, we make a first step forward explaining these results by studying in detail the evolution of small deviations from regular orbits lying on the invariant tori of an {\bf integrable} 2D Hamiltonian system. We show that, in general, any two initial deviation vectors will eventually fall on the ``tangent space'' of the torus, pointing in different directions due to the different dynamics of the 2 integrals of motion, which means that the SALI (or the smaller angle between these vectors) will oscillate away from zero for all time.Comment: To appear in Progress of Theoretical Physics Supplemen

Unfolding the procedure of characterizing recorded ultra low frequency, kHZ and MHz electromagetic anomalies prior to the L'Aquila earthquake as pre-seismic ones. Part I

Ultra low frequency, kHz and MHz electromagnetic anomalies were recorded prior to the L'Aquila catastrophic earthquake that occurred on April 6, 2009. The main aims of this contribution are: (i) To suggest a procedure for the designation of detected EM anomalies as seismogenic ones. We do not expect to be possible to provide a succinct and solid definition of a pre-seismic EM emission. Instead, we attempt, through a multidisciplinary analysis, to provide elements of a definition. (ii) To link the detected MHz and kHz EM anomalies with equivalent last stages of the L'Aquila earthquake preparation process. (iii) To put forward physically meaningful arguments to support a way of quantifying the time to global failure and the identification of distinguishing features beyond which the evolution towards global failure becomes irreversible. The whole effort is unfolded in two consecutive parts. We clarify we try to specify not only whether or not a single EM anomaly is pre-seismic in itself, but mainly whether a combination of kHz, MHz, and ULF EM anomalies can be characterized as pre-seismic one

Time--Evolving Statistics of Chaotic Orbits of Conservative Maps in the Context of the Central Limit Theorem

We study chaotic orbits of conservative low--dimensional maps and present numerical results showing that the probability density functions (pdfs) of the sum of $N$ iterates in the large $N$ limit exhibit very interesting time-evolving statistics. In some cases where the chaotic layers are thin and the (positive) maximal Lyapunov exponent is small, long--lasting quasi--stationary states (QSS) are found, whose pdfs appear to converge to $q$--Gaussians associated with nonextensive statistical mechanics. More generally, however, as $N$ increases, the pdfs describe a sequence of QSS that pass from a $q$--Gaussian to an exponential shape and ultimately tend to a true Gaussian, as orbits diffuse to larger chaotic domains and the phase space dynamics becomes more uniformly ergodic.Comment: 15 pages, 14 figures, accepted for publication as a Regular Paper in the International Journal of Bifurcation and Chaos, on Jun 21, 201