Lyapunov Characteristic Numbers Research Articles

Orbital stability of ensembles of particles in regions of magnetic reconnection in Earth's magneto-tail

We investigate the collective behavior of particle orbits in the vicinity of magnetic reconnection in Earth's magneto-tail. Various regions of different kinds of orbital stability of particle motions are found. We locate regimes of temporary capture of particle orbits in configuration space as well as locations, where strong particle accelerations take place. With this study, we are able to provide a detailed map, i.e., the topology, of high and low acceleration centers close to the reconnection site. Quasiregular and chaotic kinds of motions of elementary particles can be determined as well. The orbital stability of particle orbits is obtained by a statistical analysis of the outcome of the system of variational equations of particle orbits within the framework of particle-in-cell simulations. Using the concept of Lyapunov characteristic numbers to ensembles of particle orbits, we introduce Lyapunov ensemble averages to describe the response of particle orbits to local perturbations induced by the electromagnetic field.

Characteristic times in the standard map.

We study and compare three characteristic times of the standard map: the Lyapunov time t_{L}, the Poincaré recurrence time t_{r}, and the stickiness (or escape) time t_{st}. The Lyapunov time is the inverse of the Lyapunov characteristic number (L) and in general is quite small. We find empirical relations for the L as a function of the nonlinearity parameter K and of the chaotic area A. We also find empirical relations for the Poincaré recurrence time t_{r} as a function of the nonlinearity parameter K, of the chaotic area A, and of the size of the box of initial conditions ε. As a consequence, we find relations between t_{r} and L. We compare the distributions of the stickiness time and the Poincaré recurrence time. The stickiness time inside the sticky regions at the boundary of the islands of stability is orders of magnitude smaller than the Poincaré recurrence time t_{r} and this affects the diffusion exponent μ, which converges always to the value μ=1. This is shown in an extreme stickiness case. The diffusion is anomalous (ballistic motion) inside the accelerator mode islands of stability with μ=2 but it is normal everywhere outside the islands with μ=1. In a particular case of extreme stickiness, we find the hierarchy of islands around islands.

Whfast: a fast and unbiased implementation of a symplectic Wisdom–Holman integrator for long-term gravitational simulations

We present WHFast, a fast and accurate implementation of a Wisdom-Holman symplectic integrator for long-term orbit integrations of planetary systems. WHFast is significantly faster and conserves energy better than all other Wisdom-Holman integrators tested. We achieve this by significantly improving the Kepler-solver and ensuring numerical stability of coordinate transformations to and from Jacobi coordinates. These refinements allow us to remove the linear secular trend in the energy error that is present in other implementations. For small enough timesteps we achieve Brouwer's law, i.e. the energy error is dominated by an unbiased random walk due to floating-point round-off errors. We implement symplectic correctors up to order eleven that significantly reduce the energy error. We also implement a symplectic tangent map for the variational equations. This allows us to efficiently calculate two widely used chaos indicators the Lyapunov characteristic number (LCN) and the Mean Exponential Growth factor of Nearby Orbits (MEGNO). WHFast is freely available as a flexible C package, as a shared library, and as an easy-to-use python module.

Stability of librational motion in the spatial circular restricted three‐body problem for high inclinations and mass ratios

AbstractIn the spatial circular restricted three‐body problem librational motion around the Lagrangian points L4 and L5 exists up to high inclinations of the massless object. We report the results of numerical investigations on the stability of this librational motion for systematic variations in the inclination of the Trojan and the mass‐ratio μ of the two massive bodies. We show that stable motion prevails for inclinations below 60 degrees and mass‐ratios μ < 0.04 for typical integration times up to 106 revolution periods. At even higher mass‐ratios – beyond the critical mass‐ratio for the planar case – stable orbits were found to exist for up to 107 periods at moderate inclinations. We extracted the librational frequencies on a grid in the parameter space from the Fourier spectra and traced their variation. Several resonances between the short and long period librational frequency as well as the vertical frequency lie inside the investigated region. The application of the Lyapunov characteristic indicator and spectral number methods also reveals the chaotic regions. This simple model is equally applicable in the Solar system for low mass ratios for Trojans of the planets, as well as to Trojan‐type exoplanets in binary star systems at high mass‐ratios. (© 2013 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Orbit classification in the meridional plane of a disk galaxy model with a spherical nucleus

We investigate the regular or chaotic nature of star orbits moving in the meridional plane of an axially symmetric galactic model with a disk and a spherical nucleus. We study the influence of some important parameters of the dynamical system, such as the mass and the scale length of the nucleus, the angular momentum or the energy, by computing in each case the percentage of chaotic orbits, as well as the percentages of orbits of the main regular resonant families. Some heuristic arguments to explain and justify the numerically derived outcomes are also given. Furthermore, we present a new method to find the threshold between chaos and regularity for both Lyapunov Characteristic Numbers and SALI, by using them simultaneously.

Order in de Broglie–Bohm quantum mechanics

A usual assumption in the so-called de Broglie–Bohm approach to quantum dynamics is that the quantum trajectories subject to typical ‘guiding’ wavefunctions turn to be quite irregular, i.e. chaotic (in the dynamical systems’ sense). In this paper, we consider mainly cases in which the quantum trajectories are ordered, i.e. they have zero Lyapunov characteristic numbers. We use perturbative methods to establish the existence of such trajectories from a theoretical point of view, while we analyze their properties via numerical experiments. Using a 2D harmonic oscillator system, we first establish conditions under which a trajectory can be shown to avoid close encounters with a moving nodal point, thus avoiding the source of chaos in this system. We then consider series expansions for trajectories both in the interior and the exterior of the domain covered by nodal lines, probing the domain of convergence as well as how successful the series are in comparison with numerical computations or regular trajectories. We then examine a Hénon–Heiles system possessing regular trajectories, thus generalizing previous results. Finally, we explore a key issue of physical interest in the context of the de Broglie–Bohm formalism, namely the influence of order on the so-called quantum relaxation effect. We show that the existence of regular trajectories poses restrictions on the quantum relaxation process and give examples in which the relaxation is suppressed even when we consider initial ensembles of only chaotic trajectories, provided, however, that the system as a whole is characterized by a certain degree of order.

Examining the Chaotic Behavior in Dynamical Systems by Means of the 0‐1 Test

We perform the stability analysis and we study the chaotic behavior of dynamical systems, which depict the 3‐particle Toda lattice truncations through the lens of the 0‐1 test, proposed by Gottwald and Melbourne. We prove that the new test applies successfully and with good accuracy in most of the cases we investigated. We perform some comparisons of the well‐known maximum Lyapunov characteristic number method with the 0‐1 method, and we claim that 0‐1 test can be subsidiary to the LCN method. The 0‐1 test is a very efficient method for studying highly chaotic Hamiltonian systems of the kind we study in our paper and is particularly useful in characterizing the transition from regularity to chaos.

A REVIEW STUDY OF THE 3-PARTICLE TODA LATTICE AND HIGHER ORDER TRUNCATIONS: THE EVEN-ORDER CASES — PART II

We complete the study of the numerical behavior of the truncated 3-particle Toda lattice (3pTL) with even truncations at orders n = 2k, k = 2, …, 10. We use (as in Part I): (a) the method of Poincaré surface of section, (b) the maximum Lyapunov characteristic number and (c) the ratio of the families of ordered periodic orbits. We derived some similarities and quite many differences between the odd and even order expansions.

A REVIEW STUDY OF THE 3-PARTICLE TODA LATTICE AND HIGHER-ORDER TRUNCATIONS: THE ODD-ORDER CASES — PART I

The numerical behavior of the truncated 3-particle Toda lattice (3pTL) is reviewed and studied in more detail (than in previous papers) and at higher energies (at odd-orders n ≤ 9). We further extended our study to higher truncations at odd-orders, n = 2k + 1, k = 1, …, 9. We have located the majority of the families of periodic orbits along with their main bifurcations. By using: (a) the method of Poincaré surface of section, (b) the maximum Lyapunov characteristic number and (c) the ratio of the families of ordered periodic orbits, we studied the topology of the nine odd-order Hamiltonians with respect to their order of truncation.

Regions of stability in rotational dynamics

We investigate the rotational dynamics of a triaxial planet moving on a Keplerian orbit around its star. The dynamics is ruled by several parameters, like the eccentricity, the obliquity, the non-principal rotation, the angular momentum, etc. We consider two specific cases in which the planet is symmetric or asymmetric, according to whether two moments of inertia coincide or differs from each other. We study the dynamics by constructing maps of dynamical stability based on the computation of the maximum Lyapunov characteristic number versus some typical parameters. The results show that only specific resonances appear in the symmetric case, while the asymmetric case shows a much richer phenomenology.

Analysis on the stability of triangular points in the perturbed photogravitational restricted three-body problem with variable masses

This paper examines the effect of a constant κ of a particular integral of the Gylden-Meshcherskii problem on the stability of the triangular points in the restricted three-body problem under the influence of small perturbations in the Coriolis and centrifugal forces, together with the effects of radiation pressure of the bigger primary, when the masses of the primaries vary in accordance with the unified Meshcherskii law. The triangular points of the autonomized system are found to be conditionally stable due to κ. We observed further that the stabilizing or destabilizing tendency of the Coriolis and centrifugal forces is controlled by κ, while the destabilizing effects of the radiation pressure remain unchanged but can be made strong or weak due to κ. The condition that the region of stability is increasing, decreasing or does not exist depend on this constant. The motion around the triangular points L4,5 varying with time is studied using the Lyapunov Characteristic Numbers, and are found to be generally unstable.

RUELLE'S INEQUALITY FOR ISOTROPIC ORNSTEIN–UHLENBECK FLOWS

Ruelle's inequality asserts that the entropy of a dynamical system is bounded from above by the Lyapunov characteristic numbers counted with their multiplicities. We show that this inequality holds true in the case of a random dynamical system deduced from an isotropic Ornstein–Uhlenbeck-flow (IOUF).

Stability of the photogravitational restricted three-body problem with variable masses

This paper investigates the stability of equilibrium points in the restricted three-body problem, in which the masses of the luminous primaries vary isotropically in accordance with the unified Meshcherskii law, and their motion takes place within the framework of the Gylden–Meshcherskii problem. For the autonomized system, it is found that collinear and coplanar points are unstable, while the triangular points are conditionally stable. It is also observed that, in the triangular case, the presence of a constant κ, of a particular integral of the Gylden–Meshcherskii problem, makes the destabilizing tendency of the radiation pressures strong. The stability of equilibrium points varying with time is tested using the Lyapunov Characteristic Numbers (LCN). It is seen that the range of stability or instability depends on the parameter κ. The motion around the equilibrium points L i (i=1,2,…,7) for the restricted three-body problem with variable masses is in general unstable.

Origin of chaos near critical points of quantum flow

The general theory of motion in the vicinity of a moving quantum nodal point (vortex) is studied in the framework of the de Broglie-Bohm trajectory method of quantum mechanics. Using an adiabatic approximation, we find that near any nodal point of an arbitrary wave function psi there is an unstable point (called the X point) in a frame of reference moving with the nodal point. The local phase portrait forms always a characteristic pattern called the "nodal-point- X -point complex." We find general formulas for this complex as well as necessary and sufficient conditions of validity of the adiabatic approximation. We demonstrate that chaos emerges from the consecutive scattering events of the orbits with nodal-point- X -point complexes. The scattering events are of two types (called type I and type II). A theoretical model is constructed yielding the local value of the Lyapunov characteristic numbers in scattering events of both types. The local Lyapunov characteristic number scales as an inverse power of the speed of the nodal point in the rest frame, implying that it scales proportionally to the size of the nodal-point- X -point complex. It is also an inverse power of the distance of a trajectory from the X point's stable manifold far from the complex. This distance plays the role of an effective "impact parameter." The results of detailed numerical experiments with different wave functions, possessing one, two, or three moving nodal points, are reported. Examples are given of regular and chaotic trajectories, and the statistics of the Lyapunov characteristic numbers of the orbits are found and compared to the number of encounter events of each orbit with the nodal-point- X -point complexes. The numerical results are in agreement with the theory, and various phenomena appearing at first as counterintuitive find a straightforward explanation.

STICKINESS IN CHAOS

We distinguish two types of stickiness in systems of two degrees of freedom: (a) stickiness around an island of stability, and (b) stickiness in chaos, along the unstable asymptotic curves of unstable periodic orbits. In fact, there are asymptotic curves of unstable orbits near the outer boundary of an island that remain close to the island for some time, and then extend to large distances into the surrounding chaotic sea. But later the asymptotic curves return close to the island and contribute to the overall stickiness that produces dark regions around the islands and dark lines extending far from the islands. We have studied these effects in the standard map with a rather large nonlinearity K = 5, and we emphasized the role of the asymptotic curves U , S from the central orbit O (x = 0.5, y = 0), that surround two large islands O 1 and O ′1, and the asymptotic curves U + U - S + S - from the simplest unstable orbit around the island O 1. This is the orbit 4/9 that has 9 points around the island O 1 and 9 more points around the symmetric island O ′1. The asymptotic curves produce stickiness in the positive time direction ( U , U +, U -) and in the negative time direction ( S , S +, S -). The asymptotic curves U +, S + are closer to the island O 1 and make many oscillations before reaching the chaotic sea. The curves U -, S - are further away from the island O 1 and escape faster. Nevertheless all curves return many times close to O 1 and contribute to the stickiness near this island. The overall stickiness effects of U +, U - are very similar and the stickiness effects along S +, S - are also very similar. However, the stickiness in the forward time direction, along U +, U -, is very different from the stickiness in the opposite time direction along S +, S -. We calculated the finite time LCN (Lyapunov characteristic number) χ( t ), which is initially smaller for U +, S + than for U -, S -. However, after a long time all the values of χ( t ) in the chaotic zone approach the same final value LCN = lim t → ∞ χ(t). The stretching number (LCN for one iteration only) varies along an asymptotic curve going through minima at the turning points of the asymptotic curve. We calculated the escape times (initial stickiness times) for many initial points outside but close to the island O 1. The lines that separate the regions of the fast from the slow escape time follow the shape of the asymptotic curves S +, S -. We explained this phenomenon by noting that lines close to S + on its inner side (closer to O 1) approach a point of the orbit 4/9, say P 1, and then follow the oscillations of the asymptotic curve U +, and escape after a rather long time, while the curves outside S + after their approach to P 1 follow the shape of the asymptotic curves U - and escape fast into the chaotic sea. All these curves return near the original arcs of U +, U - and contribute to the overall stickiness close to U +, U -. The isodensity curves follow the shape of the curves U +, U - and the maxima of density are along U +, U -. For a rather long time, the stickiness effects along U +, U - are very pronounced. However, after much longer times (about 1000 iterations) the overall stickiness effects are reduced and the distribution of points in the chaotic sea outside the islands tends to be uniform. The stickiness along the asymptotic curve U of the orbit O is very similar to the stickiness along the asymptotic curves U +, U - of the orbit 4/9. This is related to the fact that the asymptotic curves of O and 4/9 are connected by heteroclinic orbits. However, the main reason for this similarity is the fact that the asymptotic curves U , U +, U - cannot intersect but follow each other.

Chaotic and regular motion around generalized Kalnajs discs

The motion of test particles in the gravitational fields generated by the first four members of the infinite family of generalized Kalnajs discs is studied. In first instance, we analyse the stability of circular orbits under radial and vertical perturbations and describe the behaviour of general equatorial orbits and so we find that radial stability and vertical instability dominate such disc models. Then we study bounded axially symmetric orbits by using the Poincaré surfaces of section and Lyapunov characteristic numbers and find chaos in the case of disc-crossing orbits and completely regular motion in other cases.

Analysis of the Chaotic Behaviour of Orbits Diffusing along the Arnold Web

In a previous work [Guzzo et al. DCDS B 5, 687–698 (2005)] we have provided numerical evidence of global diffusion occurring in slightly perturbed integrable Hamiltonian systems and symplectic maps. We have shown that even if a system is sufficiently close to be integrable, global diffusion occurs on a set with peculiar topology, the so-called Arnold web, and is qualitatively different from Chirikov diffusion, occurring in more perturbed systems. In the present work we study in more detail the chaotic behaviour of a set of 90 orbits which diffuse on the Arnold web. We find that the largest Lyapunov exponent does not seem to converge for the individual orbits while the mean Lyapunov exponent on the set of 90 orbits does converge. In other words, a kind of average mixing characterizes the diffusion. Moreover, the Local Lyapunov Characteristic Numbers (LLCNs), on individual orbits appear to reflect the different zones of the Arnold web revealed by the Fast Lyapunov Indicator. Finally, using the LLCNs we study the ergodicity of the chaotic part of the Arnold web.

Stability and instability in the anisotropic Kepler problem

We study the degree of order and chaos in the anisotropic Kepler problem. The Lyapunov characteristic number (LCN) is constant for all orbits in the chaotic domain with a fixed mass ratio µ. It increases from zero as the mass ratio µ increases from µ = 1, reaches a maximum for µ = 4 and then decreases slowly to zero for a very large µ. We found the characteristics of many families of the periodic orbits of multiplicities up to 17. Several families have a stable part for µ close to µ = 1 (namely, µ < 1.748). We conjecture that there are stable orbits of high multiplicities for larger values of µ. The stable orbits are surrounded by islands of stability consisting of closed invariant curves. The rotation number along these invariant curves decreases from the periodic orbits outwards. Higher-order islands are enclosed in every main island around a low-order periodic orbit. In the case µ = 1.1, we find a large number of islands of multiplicities from 3 up to 17. Then we calculate the asymptotic curves from unstable periodic orbits. These curves go through the collision orbits but continue beyond these points. The intersections of the stable and unstable asymptotic curves are homoclinic and heteroclinic points. The corresponding orbits are the main characteristics of chaos.

Local chaotic behaviour in the Fermi-Pasta-Ulam system

We discuss the use of the maximal Lyapunov Characteristic Number as a stochasticity indicator in connection with the persistence of the FPU paradox in the thermodynamic limit. We show that the positiveness of the LCN does not imply that the dynamic is ergodic in statistical sense. On the other hand, our numerical exploration suggests that the energy surface may be separated into different chaotic regions that may trap the orbit for a long time. This is compatible with the existence of exponentially long times of relaxation to statistical equilibrium in the sense of Nekhoroshev's theory. Thus, the relevance of the FPU phenomenon for large systems remains a still open problem.

Phase space structure of multi-dimensional systems by means of the mean exponential growth factor of nearby orbits

In this paper we deal with an alternative technique to study global dynamics in Hamiltonian systems, the mean exponential growth factor of nearby orbits (MEGNO), that proves to be efficient to investigate both regular and stochastic components of phase space. It provides a clear picture of resonance structures, location of stable and unstable periodic orbits as well as a measure of hyperbolicity in chaotic domains which coincides with that given by the Lyapunov characteristic number. Here the MEGNO is applied to a rather simple model, the 3D perturbed quartic oscillator, in order to visualize the structure of its phase space and obtain a quite clear picture of its resonance structure. Examples of application to multi-dimensional canonical maps are also included.