Efficient integration of the variational equations of multi-dimensional Hamiltonian systems: Application to the Fermi-Pasta-Ulam lattice

We study the problem of efficient integration of variational equations in multi-dimensional Hamiltonian systems. For this purpose, we consider a Runge-Kutta-type integrator, a Taylor series expansion method and the so-called `Tangent Map' (TM) technique based on symplectic integration schemes, and apply them to the Fermi-Pasta-Ulam $\beta$ (FPU-$\beta$) lattice of $N$ nonlinearly coupled oscillators, with $N$ ranging from 4 to 20. The fast and accurate reproduction of well-known behaviors of the Generalized Alignment Index (GALI) chaos detection technique is used as an indicator for the efficiency of the tested integration schemes. Implementing the TM technique--which shows the best performance among the tested algorithms--and exploiting the advantages of the GALI method, we successfully trace the location of low-dimensional tori.Comment: 14 pages, 6 figure

Low-dimensional q-Tori in FPU Lattices: Dynamics and Localization Properties

This is a continuation of our study concerning q-tori, i.e. tori of low dimensionality in the phase space of nonlinear lattice models like the Fermi-Pasta-Ulam (FPU) model. In our previous work we focused on the beta FPU system, and we showed that the dynamical features of the q-tori serve as an interpretational tool to understand phenomena of energy localization in the FPU space of linear normal modes. In the present paper i) we employ the method of Poincare - Lindstedt series, for a fixed set of frequencies, in order to compute an explicit quasi-periodic representation of the trajectories lying on q-tori in the alpha model, and ii) we consider more general types of initial excitations in both the alpha and beta models. Furthermore we turn into questions of physical interest related to the dynamical features of the q-tori. We focus on particular q-tori solutions describing low-frequency `packets' of modes, and excitations of a small set of modes with an arbitrary distribution in q-space. In the former case, we find formulae yielding an exponential profile of energy localization, following an analysis of the size of the leading order terms in the Poincare - Lindstedt series. In the latter case, we explain the observed localization patterns on the basis of a rigorous result concerning the propagation of non-zero terms in the Poincare - Lindstedt series from zeroth to subsequent orders. Finally, we discuss the extensive (i.e. independent of the number of degrees of freedom) properties of some q-tori solutions.Comment: To appear in Physica D, 34 pages, 9 figure

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

Energy localization on q-tori, long term stability and the interpretation of FPU recurrences

We focus on two approaches that have been proposed in recent years for the explanation of the so-called FPU paradox, i.e. the persistence of energy localization in the `low-q' Fourier modes of Fermi-Pasta-Ulam nonlinear lattices, preventing equipartition among all modes at low energies. In the first approach, a low-frequency fraction of the spectrum is initially excited leading to the formation of `natural packets' exhibiting exponential stability, while in the second, emphasis is placed on the existence of `q-breathers', i.e periodic continuations of the linear modes of the lattice, which are exponentially localized in Fourier space. Following ideas of the latter, we introduce in this paper the concept of `q-tori' representing exponentially localized solutions on low-dimensional tori and use their stability properties to reconcile these two approaches and provide a more complete explanation of the FPU paradox.Comment: 38 pages, 7 figure

Discrete Symmetry and Stability in Hamiltonian Dynamics

In this tutorial we address the existence and stability of periodic and quasiperiodic orbits in N degree of freedom Hamiltonian systems and their connection with discrete symmetries. Of primary importance in our study are the nonlinear normal modes (NNMs), i.e periodic solutions which represent continuations of the system's linear normal modes in the nonlinear regime. We examine the existence of such solutions and discuss different methods for constructing them and studying their stability under fixed and periodic boundary conditions. In the periodic case, we employ group theoretical concepts to identify a special type of NNMs called one-dimensional "bushes". We describe how to use linear combinations such NNMs to construct s(>1)-dimensional bushes of quasiperiodic orbits, for a wide variety of Hamiltonian systems and exploit the symmetries of the linearized equations to simplify the study of their destabilization. Applying this theory to the Fermi Pasta Ulam (FPU) chain, we review a number of interesting results, which have appeared in the recent literature. We then turn to an analytical and numerical construction of quasiperiodic orbits, which does not depend on the symmetries or boundary conditions. We demonstrate that the well-known "paradox" of FPU recurrences may be explained in terms of the exponential localization of the energies Eq of NNM's excited at the low part of the frequency spectrum, i.e. q=1,2,3,.... Thus, we show that the stability of these low-dimensional manifolds called q-tori is related to the persistence or FPU recurrences at low energies. Finally, we discuss a novel approach to the stability of orbits of conservative systems, the GALIk, k=2,...,2N, by means of which one can determine accurately and efficiently the destabilization of q-tori, leading to the breakdown of recurrences and the equipartition of energy, at high values of the total energy E.Comment: 50 pages, 13 figure

Complex Aspects in Hamiltonian Dynamics and Statistics

As we all know, and Marko Robnik has often emphasized in his work, many problems in theoretical physics are expressed in the form of Hamiltonian systems. Of these the first to be extensively studied were low-dimensional, possessing as few as two (or three) degrees of freedom. In the last 20 years, however, great attention has been devoted to Hamiltonian systems of high dimensionality. Among these perhaps the most famous are the ones that deal with the dynamics and statistics of a large number N of mass particles connected with nearest neighbor interactions. At low energies E, these typically execute quasiperiodic motions near some fundamental stable periodic orbits which represent nonlinear continuations of the N normal mode solutions of the corresponding linear system. However, as the energy is increased, these solutions destabilize causing the motion in their vicinity to drift into chaotic domains, thus giving rise to important questions concerning the system’s behavior in the thermodynamic limit where E and N diverge with E/N = constant. In this review, we start by discussing some very efficient techniques for identifying regular from chaotic domains in multi-degree of freedom Hamiltonian systems. Then we proceed to describe some highly complex features of the dynamics connected with the presence of unexpected ‘hierarchies’ of order and chaos in such systems. In particular, we will describe how these phenomena are manifested (a) in the form of low-dimensional tori responsible for the lack of energy equipartiton among normal modes and (b) in the presence of long lived quasi-stationary states whose weakly chaotic properties are related to Tsallis type and not Boltzmann-Gibbs thermodynamics. Finally, we will mention some recent results on the effect of long range interactions on these important dynamical and statistical phenomena. This paper is based on the lecture delivered by the first author at the Symposium ‘Quantum and Classical Chaos: What comes next?’ dedicated to Marko Robnik’s 60th birthday, Ljubljana, October 9 - 11 May, 2014

The effect of long–range interactions on the dynamics and statistics of 1D Hamiltonian lattices with on–site potential

We examine the role of long–range interactions on the dynamical and statistical properties of two 1D lattices with on–site potentials that are known to support discrete breathers: the Klein–Gordon (KG) lattice which includes linear dispersion and the Gorbach–Flach (GF) lattice, which shares the same on–site potential but its dispersion is purely nonlinear. In both models under the implementation of long–range interactions (LRI) we find that single–site excitations lead to special low–dimensional solutions, which are well described by the undamped Duffing oscillator. For random initial conditions we observe that the maximal Lyapunov exponent scales as N−0.12 in the KG model and as N−0.27 in the GF with LRI, suggesting in that case an approach to integrable behavior towards the thermodynamic limit. Furthermore, under LRI, their non-Gaussian momentum distributions are distinctly different from those of the FPU model

The effect of long-range interactions on the dynamics and statistics of 1D Hamiltonian lattices with on-site potential

We examine the role of long-range interactions on the dynamical and statistical properties of two 1D lattices with on-site potentials that are known to support discrete breathers: the Klein–Gordon (KG) lattice which includes linear dispersion and the Gorbach–Flach (GF) lattice, which shares the same on-site potential but its dispersion is purely nonlinear. In both models under the implementation of long-range interactions (LRI), we find that single-site excitations lead to special low-dimensional solutions, which are well described by the undamped Duffing oscillator. For random initial conditions, we observe that the maximal Lyapunov exponent λ scales as N^(−0.12) in the KG model and as N^(−0.27) in the GF with LRI, suggesting in that case an approach to integrable behavior towards the thermodynamic limit. Furthermore, under LRI, their non-Gaussian momentum distributions are distinctly different from those of the FPU model

Application of the GALI method to localization dynamics in nonlinear systems

We investigate localization phenomena and stability properties of quasiperiodic oscillations in degree of freedom Hamiltonian systems and coupled symplectic maps. In particular, we study an example of a parametrically driven Hamiltonian lattice with only quartic coupling terms and a system of coupled standard maps. We explore their dynamics using the Generalized Alignment Index (GALI), which constitutes a recently developed numerical method for detecting chaotic orbits in many dimensions, estimating the dimensionality of quasiperiodic tori and predicting slow diffusion in a way that is faster and more reliable than many other approaches known to date