Probing the local dynamics of periodic orbits by the generalized alignment index (GALI) method

As originally formulated, the Generalized Alignment Index (GALI) method of chaos detection has so far been applied to distinguish quasiperiodic from chaotic motion in conservative nonlinear dynamical systems. In this paper we extend its realm of applicability by using it to investigate the local dynamics of periodic orbits. We show theoretically and verify numerically that for stable periodic orbits the GALIs tend to zero following particular power laws for Hamiltonian flows, while they fluctuate around non-zero values for symplectic maps. By comparison, the GALIs of unstable periodic orbits tend exponentially to zero, both for flows and maps. We also apply the GALIs for investigating the dynamics in the neighborhood of periodic orbits, and show that for chaotic solutions influenced by the homoclinic tangle of unstable periodic orbits, the GALIs can exhibit a remarkable oscillatory behavior during which their amplitudes change by many orders of magnitude. Finally, we use the GALI method to elucidate further the connection between the dynamics of Hamiltonian flows and symplectic maps. In particular, we show that, using for the computation of GALIs the components of deviation vectors orthogonal to the direction of motion, the indices of stable periodic orbits behave for flows as they do for maps.Comment: 17 pages, 9 figures (accepted for publication in Int. J. of Bifurcation and Chaos

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

Coupled symplectic maps as models for subdiffusive processes in disordered Hamiltonian lattices

© 2015 IMACS We investigate dynamically and statistically diffusive motion in a chain of linearly coupled 2-dimensional symplectic McMillan maps and find evidence of subdiffusion in weakly and strongly chaotic regimes when all maps of the chain possess a saddle point at the origin and the central map is initially excited. In the case of weak coupling, there is either absence of diffusion or subdiffusion with q > 1-Gaussian probability distributions, characterizing weak chaos. However, for large enough coupling and already moderate number of maps, the system exhibits strongly chaotic (q≈1) subdiffusive behavior, reminiscent of the subdiffusive energy spreading observed in a disordered Klein–Gordon Hamiltonian. Our results provide evidence that coupled symplectic maps can exhibit physical properties similar to those of disordered Hamiltonian systems, even though the local dynamics in the two cases is significantly different

Quasi-stationary chaotic states in multi-dimensional Hamiltonian systems

We study numerically statistical distributions of sums of chaotic orbit coordinates, viewed as independent random variables, in weakly chaotic regimes of three multi-dimensional Hamiltonian systems: Two Fermi-Pasta-Ulam (FPU-$\beta$) oscillator chains with different boundary conditions and numbers of particles and a microplasma of identical ions confined in a Penning trap and repelled by mutual Coulomb interactions. For the FPU systems we show that, when chaos is limited within "small size" phase space regions, statistical distributions of sums of chaotic variables are well approximated for surprisingly long times (typically up to $t\approx10^6$) by a $q$-Gaussian ($1<q<3$) distribution and tend to a Gaussian ($q=1$) for longer times, as the orbits eventually enter into "large size" chaotic domains. However, in agreement with other studies, we find in certain cases that the $q$-Gaussian is not the only possible distribution that can fit the data, as our sums may be better approximated by a different so-called "crossover" function attributed to finite-size effects. In the case of the microplasma Hamiltonian, we make use of these $q$-Gaussian distributions to identify two energy regimes of "weak chaos"-one where the system melts and one where it transforms from liquid to a gas state-by observing where the $q$-index of the distribution increases significantly above the $q=1$ value of strong chaos.Comment: 32 pages, 13 figures, Submitted for publication to Physica

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

Cellular O-Glycome Reporter/Amplification to explore O-glycans of living cells

Protein O-glycosylation has key roles in many biological processes, but the repertoire of O-glycans synthesized by cells is difficult to determine. Here we describe an approach termed Cellular O-Glycome Reporter/Amplification (CORA), a sensitive method used to amplify and profile mucin-type O-glycans synthesized by living cells. Cells convert added peracetylated benzyl-α-N-acetylgalactosamine to a large variety of modified O-glycan derivatives that are secreted from cells, allowing for easy purification for analysis by HPLC and mass spectrometry (MS). Relative to conventional O-glycan analyses, CORA resulted in an ∼100-1,000-fold increase in sensitivity and identified a more complex repertoire of O-glycans in more than a dozen cell types from Homo sapiens and Mus musculus. Furthermore, when coupled with computational modeling, CORA can be used for predictions about the diversity of the human O-glycome and offers new opportunities to identify novel glycan biomarkers for human diseases