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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. We show analytically and verify numerically on particular examples of N degree of freedom Hamiltonian systems that, for chaotic orbits, GALI k tends exponentially to zero with exponents that involve the values of several Lyapunov exponents. In the case of regular orbits, GALI k fluctuates around non-zero values for 2 ≤ k ≤ N and goes to zero for N < k ≤ 2N following power laws that depend on the dimension of the torus and the number m of deviation vectors initially tangent to the torus:The GALI k is a generalization of the Smaller Alignment Index (SALI) (GALI 2 ∝ 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.

Chimera states, namely the coexistence of coherent and incoherent behavior, were previously analyzed in complex networks. However, they have not been extensively studied in modular networks. Here, we consider a neural network inspired by the connectome of the C. elegans soil worm, organized into six interconnected communities, where neurons obey chaotic bursting dynamics. Neurons are assumed to be connected with electrical synapses within their communities and with chemical synapses across them. As our numerical simulations reveal, the coaction of these two types of coupling can shape the dynamics in such a way that chimera-like states can happen. They consist of a fraction of synchronized neurons which belong to the larger communities, and a fraction of desynchronized neurons which are part of smaller communities. In addition to the Kuramoto order parameter ρ, we also employ other measures of coherence, such as the chimera-like χ and metastability λ indices, which quantify the degree of synchronization among communities and along time, respectively. We perform the same analysis for networks that share common features with the C. elegans neural network. Similar results suggest that under certain assumptions, chimera-like states are prominent phenomena in modular networks, and might provide insight for the behavior of more complex modular networks.

Abstract. We use the Smaller Alignment Index (SALI) to distinguish rapidly and with certainty between ordered and chaotic motion in Hamiltonian flows. This distinction is based on the different behavior of the SALI for the two cases: the index fluctuates around non-zero values for ordered orbits, while it tends rapidly to zero for chaotic orbits. We present a detailed study of SALI's behavior for chaotic orbits and show that in this case the SALI exponentially converges to zero, following a time rate depending on the difference of the two largest Lyapunov exponents σ 1 , σ 2 i.e. SALI ∝ e −(σ1−σ2)t . Exploiting the advantages of the SALI method, we demonstrate how one can rapidly identify even tiny regions of order or chaos in the phase space of Hamiltonian systems of 2 and 3 degrees of freedom.

This work uses an information-based methodology to infer the connectivity of complex systems from observed time-series data. We first derive analytically an expression for the Mutual Information Rate (MIR), namely, the amount of information exchanged per unit of time, that can be used to estimate the MIR between two finite-length low-resolution noisy time-series, and then apply it after a proper normalization for the identification of the connectivity structure of small networks of interacting dynamical systems. In particular, we show that our methodology successfully infers the connectivity for heterogeneous networks, different time-series lengths or coupling strengths, and even in the presence of additive noise. Finally, we show that our methodology based on MIR successfully infers the connectivity of networks composed of nodes with different time-scale dynamics, where inference based on Mutual Information fails.The Mutual Information Rate (MIR) measures the time rate of information exchanged between two non-random and correlated variables. Since variables in complex systems are not purely random, MIR is an appropriate quantity to access the amount of information exchanged in complex systems. However, its calculation requires infinitely long measurements with arbitrary resolution. Having in mind that it is impossible to perform infinitely long measurements with perfect accuracy, this work shows how to estimate MIR taking into consideration this fundamental limitation and how to use it for the characterization and understanding of dynamical and complex systems. Moreover, we introduce a novel normalized form of MIR that successfully infers the structure of small networks of interacting dynamical systems. The proposed inference methodology is robust in the presence of additive noise, different time-series lengths, and heterogeneous node dynamics and coupling strengths. Moreover, it also outperforms inference methods based on Mutual Information when analysing networks formed by nodes possessing different time-scales.

The recently introduced GALI method is used for rapidly detecting chaos, determining the dimensionality of regular motion and predicting slow diffusion in multi-dimensional Hamiltonian systems. We propose an efficient computation of the GALI k indices, which represent volume elements of k randomly chosen deviation vectors from a given orbit, based on the Singular Value Decomposition (SVD) algorithm. We obtain theoretically and verify numerically asymptotic estimates of GALIs long-time behavior in the case of regular orbits lying on low-dimensional tori. The GALI k indices are applied to rapidly detect chaotic oscillations, identify low-dimensional tori of Fermi-Pasta-Ulam (FPU) lattices at low energies and predict weak diffusion away from quasiperiodic motion, long before it is actually observed in the oscillations.

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The ability of the Smaller Alignment Index (SALI) to distinguish chaotic from ordered motion, has been demonstrated recently in several publications.1), 2) 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 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. §1. IntroductionThe evaluation of the Smaller Alignment Index (SALI) is an efficient and simple method to determine the ordered or chaotic nature of orbits in dynamical systems. The SALI was proposed in Ref. 1) and it has been successfully applied to distinguish between ordered and chaotic motion both in symplectic maps 1) as well as in Hamiltonian flows. 2) In order to compute the SALI for a given orbit one has to follow the time evolution of the orbit itself and two deviation vectors which initially point in two different directions. The evolution of these vectors is given by the variational equations for a flow and by the tangent map for a discrete-time system. At every time step the two vectors − → v 1 (t), − → v 2 (t) are normalized and the SALI is computed as:where t is the continuous or the discrete time and · denotes the Euclidean norm. * )