Ch. Skokos

Alignment indices: a new, simple method for determining the ordered or chaotic nature of orbits

We introduce a new, simple and efficient method for determining the ordered or chaotic nature of orbits in two-dimensional (2D), four-dimensional (4D) and six-dimensional (6D) symplectic maps: the computation of the alignment indices. For a given orbit we follow the evolution in time of two different initial deviation vectors computing the norms of the difference d− (parallel alignment index) and the addition d+ (antiparallel alignment index) of the two vectors. The time evolution of the smaller alignment index reflects the chaotic or ordered nature of the orbit. In 2D maps the smaller alignment index tends to zero for both ordered and chaotic orbits but with completely different time rates, which allows us to distinguish between the two cases. In 4D and 6D maps the smaller alignment index tends to zero in the case of chaotic orbits, while it tends to a positive non-zero value in the case of ordered orbits. The efficiency of the new method is also shown in a case of weak chaos and a comparison with other known methods that separate chaotic from regular orbits is presented.

Universal Spreading of Wave Packets in Disordered Nonlinear Systems

In the absence of nonlinearity all eigenmodes of a chain with disorder are spatially localized (Anderson localization). The width of the eigenvalue spectrum and the average eigenvalue spacing inside the localization volume set two frequency scales. An initially localized wave packet spreads in the presence of nonlinearity. Nonlinearity introduces frequency shifts, which define three different evolution outcomes: (i) localization as a transient, with subsequent subdiffusion; (ii) the absence of the transient and immediate subdiffusion; (iii) self-trapping of a part of the packet and subdiffusion of the remainder. The subdiffusive spreading is due to a finite number of packet modes being resonant. This number does not change on average and depends only on the disorder strength. Spreading is due to corresponding weak chaos inside the packet, which slowly heats the cold exterior. The second moment of the packet grows as t;{alpha}. We find alpha=1/3.

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 (GALIk) 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, GALIk tends exponentially to zero with exponents that involve the values of several Lyapunov exponents. In the case of regular orbits, GALIk 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: ∝ t −2(k−N)+m if 0 ≤ m < k − N, and ∝ t −(k−N) if m ≥ k − N. The GALIk is a generalization of the Smaller Alignment Index (SALI) (GALI2 ∝ SALI). However, GALIk 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.

Orbital dynamics of three-dimensional bars – I. The backbone of three-dimensional bars. A fiducial case

In this series of papers we investigate the orbital structure of three-dimensional (3D) models representing barred galaxies. In the present introductory paper we use a fiducial case to describe all families of periodic orbits that may play a role in the morphology of three-dimensional bars. We show that, in a 3D bar, the backbone of the orbital structure is not just the x1 family, as in two-dimensional (2D) models, but a tree of 2D and 3D families bifurcating from x1. Besides the main tree we have also found another group of families of lesser importance around the radial 3:1 resonance. The families of this group bifurcate from x1 and influence the dynamics of the system only locally. We also find that 3D orbits elongated along the bar minor axis can be formed by bifurcations of the planar x2 family. They can support 3D bar-like structures along the minor axis of the main bar. Banana-like orbits around the stable Lagrangian points build a forest of 2D and 3D families as well. The importance of the 3D x1-tree families at the outer parts of the bar depends critically on whether they are introduced in the system as bifurcations in z or in z˙.

Detecting order and chaos in Hamiltonian systems by the SALI method

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 behaviour 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 behaviour 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 σ1 ,σ 2 i.e. SALI ∝ e −(σ

Orbital dynamics of three-dimensional bars: III. Boxy/Peanut edge-on profiles

We present families, and sets of families, of periodic orbits that provide building blocks for boxy and peanut (hereafter b/p) edge-on profiles. We find cases where the b/p profile is confined to the central parts of the model and cases where a major fraction of the bar participates in this morphology. A b/p feature can be built either by 3D families associated with 3D bifurcations of the x1 family, or, in some models, even by families related with the z-axis orbits and existing over large energy intervals. The ‘X’ feature observed inside the boxy bulges of several edgeon galaxies can be attributed to the peaks of successive x1v1 orbits, provided their stability allows it. However in general, the x1v1 family has to overcome the obstacle of a S → � → S transition in order to support the structure of a b/p feature. Other families that can be the backbones of b/p features are x1v4 and z3.1s. The morphology and the size of the boxy or peanut-shaped structures we find in our models are determined by the presence and stability of the families that support b/p features. The present study favours the idea that the observed edge-on profiles are the imprints of families of periodic orbits that can be found in appropriately chosen Hamiltonian systems, describing the potential of the bar.

Delocalization of wave packets in disordered nonlinear chains

We consider the spatiotemporal evolution of a wave packet in disordered nonlinear Schrödinger and anharmonic oscillator chains. In the absence of nonlinearity all eigenstates are spatially localized with an upper bound on the localization length (Anderson localization). Nonlinear terms in the equations of motion destroy the Anderson localization due to nonintegrability and deterministic chaos. At least a finite part of an initially localized wave packet will subdiffusively spread without limits. We analyze the details of this spreading process. We compare the evolution of single-site, single-mode, and general finite-size excitations and study the statistics of detrapping times. We investigate the properties of mode-mode resonances, which are responsible for the incoherent delocalization process.

The crossover from strong to weak chaos for nonlinear waves in disordered systems

We observe a crossover from strong to weak chaos in the spatiotemporal evolution of multiple-site excitations within disordered chains with cubic nonlinearity. Recent studies have shown that Anderson localization is destroyed, and the wave packet spreading is characterized by an asymptotic divergence of the second moment m2 in time (as t 1/3 ), due to weak chaos. In the present paper, we observe the existence of a qualitatively new dynamical regime of strong chaos, in which the second moment spreads even faster (as t 1/2 ), with a crossover to the asymptotic law of weak chaos at larger times. We analyze the pecularities of these spreading regimes and perform extensive numerical simulations over large times with ensemble averaging. A technique of local derivatives on logarithmic scales is developed in order to quantitatively visualize the slow crossover processes. Copyright c EPLA, 2010

Orbital dynamics of three-dimensional bars - II. Investigation of the parameter space

We investigate the orbital structure in a class of three-dimensional (3D) models of barred galaxies. We consider different values of the pattern speed, of the strength of the bar and of the parameters of the central bulge of the galactic model. The morphology of the stable orbits in the bar region is associated with the degree of folding of the x1 characteristic. This folding is larger for lower values of the pattern speed. The elongation of rectangular-like orbits belonging to x1 and to x1-originated families depends mainly on the pattern speed. A detailed investigation of the trees of bifurcating families in the various models shows that major building blocks of 3D bars can be supplied by families initially introduced as unstable in the system, but becoming stable at another energy interval. In some models without radial and vertical 2:1 resonances we find, except for the x1 and x1-originated families, also families related to the z-axis orbits, which support the bar. Bifurcations of the x2 family can build a secondary 3D bar along the minor axis of the main bar. This is favoured in the slowly rotating bar case.

On the stability of periodic orbits of high dimensional autonomous Hamiltonian systems

We study the stability of periodic orbits of autonomous Hamiltonian systems with N + 1 degrees of freedom or equivalently of 2N -dimensional symplectic maps, with N ≥ 1. We classify the different stability types, introducing a new terminology which is perfectly suited for systems with many degrees of freedom, since it clearly reflects the configuration of the eigenvalues of the corresponding monodromy matrix, on the complex plane. The different stability types correspond to different regions of the N -dimensional parameter space S, defined by the coefficients of the characteristic polynomial of the monodromy matrix. All the possible direct transitions between different stability types are classified, and the corresponding transition hypersurface in S is determined. The dimension of the transition hypersurface is an indicator of how probable to happen is the corresponding transition. As an application of the general results we consider the well-known cases of Hamiltonian systems with two and three degrees of freedom. We also describe in detail the different stability regions in the three-dimensional parameter space S of a Hamiltonian system with four degrees of freedom or equivalently of a six-dimensional symplectic map.