Local and global diffusion in the Arnold web of a priori unstable systems

Using the numerical techniques developed by Froeschlé et al. (Science 289 (5487): 2108–2110, 2000) and by Lega et al. (Physica D 182: 179–187, 2003) we have studied diffusion and stochastic properties of an a priori unstable 4D symplectic map. We have found two different kinds of diffusion that coexist for values of the perturbation below the critical value for the Chirikov overlapping of resonances. A fast diffusion along some resonant lines that exist already in the unperturbed case and a slow diffusion occurring in regions of the phase space far from such resonances. The latter one, although the system does not satisfy the Nekhoroshev hypothesis, decreases faster than a power law and possibly exponentially. We compare the diffusion coefficient to the indicators of stochasticity like the Lyapunov exponent and filling factor showing their behavior for chaotic orbits in regions of the Arnold web where the secondary resonances appear, or where they overlap.     

Phase Mixing in Unperturbed and Perturbed Hamiltonian Systems

This paper summarises a numerical investigation of phase mixing in time-independent Hamiltonian systems that admit a coexistence of regular and chaotic phase space regions, allowing also for low amplitude perturbations idealised as periodic driving, friction, and/or white and coloured noise. The evolution of initially localised ensembles of orbits was probed through lower order moments and coarse-grained distribution functions. In the absence of time-dependent perturbations, regular ensembles disperse initially as a power law in time and only exhibit a coarse-grained approach towards an invariant equilibrium over comparatively long times. Chaotic ensembles generally diverge exponentially fast on a time scale related to a typical finite time Lyapunov exponent, but can exhibit complex behaviour if they are impacted by the effects of cantori or the Arnold web. Viewed over somewhat longer times, chaotic ensembles typical converge exponentially towards an invariant or near-invariant equilibrium. This, however, need not correspond to a true equilibrium, which may only be approached over very long time scales. Time-dependent perturbations can dramatically increase the efficiency of phase mixing, both by accelerating the approach towards a near-equilibrium and by facilitating diffusion through cantori or along the Arnold web so as to accelerate the approach towards a true equilibrium. The efficacy of such perturbations typically scales logarithmically in amplitude, but is comparatively insensitive to most other details, a conclusion which reinforces the interpretation that the perturbations act via a resonant coupling.     

On the Structure of Symplectic Mappings. The Fast Lyapunov Indicator: a Very Sensitive Tool

It is already known (Froeschlé, Lega and Gonczi, 1997) that the Fast Lyapunov Indicator (FLI), that is the computation on a relatively short time of the largest Lyapunov indicator, allows to discriminate between ordered and weak chaotic motion. We have found that, under certain conditions, the FLI also discriminates between resonant and non-resonant orbits, not only for two-dimensional symplectic mappings but also for higher dimensional ones. Using this indicator, we present an example of the Arnold web detection for four and six-dimensional symplectic maps. We show that this method allows to detect the global transition of the system from an exponentially stable Nekhoroshevs like regime to the diffusive Chirikovs one.     

Spatial Structure of Regular and Chaotic Orbits in Self-Consistent Models of Galactic Satellites

In several previous papers we had investigated the orbits of the stars that make up galactic satellites, finding that many of them were chaotic. Most of the models studied in those works were not self-consistent, the single exception being the Heggie and Ramamani (1995) models; nevertheless, these ones are built from a distribution function that depends on the energy (actually, the Jacobi integral) only, what makes them rather special. Here we built up two self-consistent models of galactic satellites, freezed theirs potential in order to have smooth and stationary fields, and investigated the spatial structure of orbits whose initial positions and velocities were those of the bodies in the self-consistent models. We distinguished between partially chaotic (only one non-zero Lyapunov exponent) and fully chaotic (two non-zero Lyapunov exponents) orbits and showed that, as could be expected from the fact that the former obey an additional local isolating integral, besides the global Jacobi integral, they have different spatial distributions. Moreover, since Lyapunov exponents are computed over finite time intervals, their values reflect the properties of the part of the chaotic sea they are navigating during those intervals and, as a result, when the chaotic orbits are separated in groups of low- and high-valued exponents, significant differences can also be recognized between their spatial distributions. The structure of the satellites can, therefore, be understood as a superposition of several separate subsystems, with different degrees of concentration and trixiality, that can be recognized from the analysis of the Lyapunov exponents of their orbits.     

Planetary dynamics in the system α Centauri: The stability diagrams

The stability of the motion of a hypothetical planet in the binary system ?? Cen A?CB has been investigated. The analysis has been performed within the framework of a planar (restricted and full) three-body problem for the case of prograde orbits. Based on a representative set of initial data, we have obtained the Lyapunov spectra of the motion of a triple system with a single planet. Chaotic domains have been identified in the pericenter distance-eccentricity plane of initial conditions for the planet through a statistical analysis of the data obtained. We have studied the correspondence of these chaotic domains to the domains of initial conditions that lead to the planet??s encounter with one of the binary??s stars or to the escape of the planet from the system. We show that the stability criterion based on the maximum Lyapunov exponent gives a more clear-cut boundary of the instability domains than does the encounterescape criterion at the same integration time. The typical Lyapunov time of chaotic motion is ??500 yr for unstable outer orbits and ??60 yr for unstable inner ones. The domain of chaos expands significantly as the initial orbital eccentricity of the planet increases. The chaos-order boundary has a fractal structure due to the presence of orbital resonances.     

Adiabatic chaos in the Prometheus–Pandora system

The chaotic orbital motion of Prometheus and Pandora, the 16th and 17th satellites of Saturn, is studied. Chaos in their orbital motion, as found by Goldreich & Rappaport and Renner & Sicardy, is due to interaction of resonances in the resonance multiplet corresponding to the 121:118 commensurability of the mean motions of the satellites. It is shown rigorously that the system moves in adiabatic regime. The Lyapunov time (the 'time horizon of predictability' of the motion) is calculated analytically and compared to the available numerical–experimental estimates. For this purpose, a method of analytical estimation of the maximum Lyapunov exponent in the perturbed pendulum model of non-linear resonance is applied. The method is based on the separatrix map theory. An analytical estimate of the width of the chaotic layer is made as well, based on the same theory. The ranges of chaotic diffusion in the mean motion are shown to be almost twice as big compared to previous estimates for both satellites.     

Instability and Diffusion in the Elliptic Restricted Three-body Problem

The importance of the stability characteristics of the planar elliptic restricted three-body problem is that they offer insight about the general dynamical mechanisms causing instability in celestial mechanics. To analyze these concerns, elliptic–elliptic and hyperbolic–elliptic resonance orbits (periodic solutions with lower period) are numerically discovered by use of Newton's differential correction method. We find indications of stability for the elliptic–elliptic resonance orbits because slightly perturbed orbits define a corresponding two-dimensional invariant manifold on the Poincaré surface-section. For the resonance orbit of the hyperbolic–elliptic type, we show numerically that its stable and unstable manifolds intersect transversally in phase-space to induce instability. Then, we find indications that there are orbits which jump from one resonance zone to the next before escaping to infinity. This phenomenon is related to the so-called Arnold diffusion. This revised version was published online in August 2006 with corrections to the Cover Date.     

Origin of chaos in the Prometheus-Pandora system

We demonstrate that the chaotic orbits of Prometheus and Pandora are due to interactions associated with the 121:118 mean motion resonance. Differential precession splits this resonance into a quartet of components equally spaced in frequency. Libration widths of the individual components exceed the splitting, resulting in resonance overlap which causes the chaos. Mean motions of Prometheus and Pandora wander chaotically in zones of width 1.8 and 3.1 deg yr⁻¹, respectively. A model with 1.5 degrees of freedom captures the essential features of the chaotic dynamics. We use it to show that the Lyapunov exponent of 0.3 yr⁻¹ arises because the critical argument of the dominant member of the resonant quartet makes approximately two separatrix crossings every 6.2 year precessional cycle.     

Spatial Structure of Regular and Chaotic Orbits in A Self-Consistent Triaxial Stellar System

We created a triaxial stellar system through the cold dissipationless collapse of 100,000 particles whose evolution was followed with a multipolar code. Once an equilibrium system had been obtained, the multipolar expansion was freezed and smoothed in order to get a stationary smooth potential. The resulting model was self-consistent and the orbits and Lyapunov exponents could then be computed for a randomly selected sample of 3472 of the bodies that make up the system. More than half of the orbits (52.7 % ) turned out to be chaotic. Regular orbits were then classified using the frequency analysis automatic code of Carpintero and Aguilar (1998, MNRAS 298(1), 1–21). We present plots of the distributions of the different kinds of orbits projected on the symmetry planes of the system. We distinguish chaotic orbits with only one non-zero Lyapunov exponent from those with two non-zero exponents and show that their spatial distributions differ, that of the former being more similar to the one of the regular orbits. Most of the regular orbits are boxes and boxlets, but the minor axis tubes play an important role filling in the wasp waists of the boxes and helping to give a lentil shape to the system. We see no problem in building stable triaxial models with substantial amounts of chaotic orbits; the difficulties found by other authors may be due not to a physical cause but to a limitation of Schwarzschild’s method.     

On the Stochasticity of the Asteroid Belt

The study of the stochasticity of the asteroid belt requires the analysis of a large number of orbits. We detect the dynamical character of a set of 5 400 asteroids using the Fast Lyapunov Indicator, a method of analysis closely related to the computation of the Lyapunov Characteristic Exponents, but cheaper in computational time. For both regular and chaotic orbits we try to associate the motion to the underlying resonances network. For it we consider different methods of classification of rational numbers proposed by number theory, and we choose the one which seems to be strictly related to the dynamical behaviour of a system. This revised version was published online in July 2006 with corrections to the Cover Date.     

Frequency analysis of a dynamical system

Frequency analysis is a new method for analyzing the stability of orbits in a conservative dynamical system. It was first devised in order to study the stability of the solar system (Laskar, Icarus, 88, 1990). It is a powerful method for analyzing weakly chaotic motion in hamiltonian systems or symplectic maps. For regular motions, it yields an analytical representation of the solutions. In cases of 2 degrees of freedom system with monotonous torsion, precise numerical criterions for the destruction of KAM tori can be found. For a 4D symplectic map, plotting the frequency map in the frequency plane provides a clear representation of the global dynamics and describes the actual Arnold web of the system.     

On the Relationship Between Fast Lyapunov Indicator and Periodic Orbits for Continuous Flows

It is already known (Froeschlé et al., 1997a) that the fast Lyapunov indicator (hereafter FLI), i.e. the computation on a relatively short time of a quantity related to the largest Lyapunov indicator, allows us to discriminate between ordered and weak chaotic motion. Using the FLI many results have been obtained on the standard map taken as a model problem. On this model we are not only able to discriminate between a short time weak chaotic motion and an ordered one, but also among regular motion between non resonant and resonant orbits. Moreover, periodic orbits are characterised by constant FLI values which appear to be related to the order of periodic orbits (Lega and Froeschlé, 2001). In the present paper we extend all these results to the case of continuous dynamical systems (the Hénon and Heiles system and the restricted three-body problem). Especially for the periodic orbits we need to introduce a new value: the orthogonal FLI in order to fully recover the results obtained for mappings.     

on the relationship between fast lyapunov indicator and periodic orbits for symplectic mappings

The computation on a relatively short time of a quantity, related to the largest Lyapunov Characteristic Exponent, called Fast Lyapunov Indicator allows to discriminate between ordered and weak chaotic motion and also, under certain conditions, between resonant and non resonant regular orbits. The aim of this paper is to study numerically the relationship between the Fast Lyapunov Indicator values and the order of periodic orbits. Using the two-dimensional standard map as a model problem we have found that the Fast Lyapunov Indicator increases as the logarithm of the order of periodic orbits up to a given order. For higher order the Fast Lyapunov Indicator grows linearly with the order of the periodic orbits. We provide a simple model to explain the relationship that we have found between the values of the Fast Lyapunov Indicator, the order of the periodic orbits and also the minimum number of iterations needed to obtain the Fast Lyapunov Indicator values.     

The Shannon entropy as a measure of diffusion in multidimensional dynamical systems

In the present work, we introduce two new estimators of chaotic diffusion based on the Shannon entropy. Using theoretical, heuristic and numerical arguments, we show that the entropy, S, provides a measure of the diffusion extent of a given small initial ensemble of orbits, while an indicator related with the time derivative of the entropy, \(S'\), estimates the diffusion rate. We show that in the limiting case of near ergodicity, after an appropriate normalization, \(S'\) coincides with the standard homogeneous diffusion coefficient. The very first application of this formulation to a 4D symplectic map and to the Arnold Hamiltonian reveals very successful and encouraging results.     

Conditional Entropy

In this paper we show that the Conditional Entropy of nearby orbits may be a useful tool to explore the phase space associated to a given Hamiltonian. The arc length parameter along the orbits, instead of the time, is used as a random variable to compute the entropy. In the first part of this work we summarise the main analytical results to support this tool while, in the second part, we present numerical evidence that this technique is able to localise (stable) periodic and quasiperiodic orbits, 'aperiodic' orbits (chaotic motion) and unstable periodic orbits (the 'source' of chaotic motion). Besides, we show that this technique provides a measure of chaos which is similar to that given by the largest Lyapunov Characteristic Number. It is important to remark that this method is very simple to compute and does not require long time integrations, just realistic physical times. This revised version was published online in July 2006 with corrections to the Cover Date.     

Chaos in Relativity and Cosmology

Chaos appears in various problems of Relativity and Cosmology. Here we discuss (a) the Mixmaster Universe model, and (b) the motions around two fixed black holes. (a) The Mixmaster equations have a general solution (i.e. a solution depending on 6 arbitrary constants) of Painlevé type, but there is a second general solution which is not Painlevé. Thus the system does not pass the Painlevé test, and cannot be integrable. The Mixmaster model is not ergodic and does not have any periodic orbits. This is due to the fact that the sum of the three variables of the system (α + β + γ) has only one maximum for τ = τ_m and decreases continuously for larger and for smaller τ. The various Kasner periods increase exponentially for large τ. Thus the Lyapunov Characteristic Number (LCN) is zero. The "finite time LCN" is positive for finite τ and tends to zero when τ → ∞. Chaos is introduced mainly near the maximum of (α + β + γ). No appreciable chaos is introduced at the successive Kasner periods, or eras. We conclude that in the Belinskii-Khalatnikov time, τ, the Mixmaster model has the basic characteristics of a chaotic scattering problem. (b) In the case of two fixed black holes M₁ and M₂ the orbits of photons are separated into three types: orbits falling into M₁ (type I), or M₂ (type II), or escaping to infinity (type III). Chaos appears because between any two orbits of different types there are orbits of the third type. This is a typical chaotic scattering problem. The various types of orbits are separated by orbits asymptotic to 3 simple unstable orbits. In the case of particles of nonzero rest mass we have intervals where some periodic orbits are stable. Near such orbits we have order. The transition from order to chaos is made through an infinite sequence of period doubling bifurcations. The bifurcation ratio is the same as in classical conservative systems. This revised version was published online in July 2006 with corrections to the Cover Date.     

Chaotic Orbits in Galactic Satellites

In several previous papers we had investigated the orbits of the stars that make up galactic satellites and found that many of those orbits were chaotic. In those investigations we made extensive use of the frequency analysis method of Carpintero and Aguilar (1998) to classify the orbits, because that method is much faster than the use of Lyapunov exponents, allows the classification of the regular orbits and our initial comparison of both methods had shown excellent agreement between their results. More recently, we have found some problems with the use of frequency analysis in rotating systems, so that here we present a new investigation of orbits inside galactic satellites using exclusively Lyapunov exponents. Some of our previous conclusions are confirmed, while others are altered. Besides, the Lyapunov times that are now obtained show that the time scales of the chaotic processes are shorter than, or comparable to, other time scales characteristic of galactic satellites.     

Dynamics of the Jupiter Trojans with Saturn’s perturbation in the present configuration of the two planets

The dynamics of the two Jupiter triangular libration points perturbed by Saturn is studied in this paper. Unlike some previous works that studied the same problem via the pure numerical approach, this study is done in a semianalytic way. Using a literal solution, we are able to explain the asymmetry of two orbits around the two libration points with symmetric initial conditions. The literal solution consists of many frequencies. The amplitudes of each frequency are the same for both libration points, but the initial phase angles are different. This difference causes a temporary spatial asymmetry in the motions around the two points, but this asymmetry gradually disappears when the time goes to infinity. The results show that the two Jupiter triangular libration points should have symmetric spatial stable regions in the present status of Jupiter and Saturn. As a test of the literal solution, we study the resonances that have been extensively studied in Robutel and Gabern (Mon Not R Astron Soc 372:1463–1482, 2006). The resonance structures predicted by our analytic theory agree well with those found in Robutel and Gabern (Mon Not R Astron Soc 372:1463–1482, 2006) via a numerical approach. Two kinds of chaotic orbits are discussed. They have different behaviors in the frequency map. The first kind of chaotic orbits (inner chaotic orbits) is of small to moderate amplitudes, while the second kind of chaotic orbits (outer chaotic orbits) is of relatively larger amplitudes. Using analytical theory, we qualitatively explain the transition process from the inner chaotic orbits to the outer chaotic orbits with increasing amplitudes. A critical value of the diffusion rate is given to separate them in the frequency map. In a forthcoming paper, we will study the same problem but keep the planets in migration. The time asymmetry, which is unimportant in this paper, may cause an observable difference in the two Jupiter Trojan groups during a very fast planet migration process.     

Long-term behavior of the motion of Pluto over 5.5 billion years

The motion of Pluto is said to be chaotic in the sense that the maximum Lyapunov exponent is positive: the Lyapunov time (the inverse of the Lyapunov exponent) is about 20 million years. So far the longest integration up to now, over 845 million years (42 Lyapunov times), does not show any indication of a gross instability in the motion of Pluto. We carried out the numerical integration of Pluto over the age of the solar system (5.5 billion years ≈ 280 Lyapunov times). This integration also did not give any indication of chaotic evolution of Pluto. The divergences of Keplerian elements of a nearby trajectory at first grow linearly with the time and then start to increase exponentially. The exponential divergences stop at about 420 million years. The divergences in the semi-major axis and the mean anomaly ( equivalently the longitude and the distance) saturate. The divergences of the other four elements, the eccentricity, the inclination, the argument of perihelion, and the longitude of node still grow slowly after the stop of the exponential increase and finally saturate.     

Stable Chaos in High-Order Jovian Resonances

In a previous publication (Tsiganis et al. 2000, Icarus146, 240-252), we argued that the occurrence of stable chaos in the 12/7 mean motion resonance with Jupiter is related to the fact that there do not exist families of periodic orbits in the planar elliptic restricted problem and in the 3-D circular problem corresponding to this resonance. In the present paper we show that nonexistence of resonant periodic orbits, both for the planar and for the 3-D problem, also occurs in other jovian resonances—namely the 11/4, 22/9, 13/6, and 18/7—where cases of real asteroids on stable-chaotic orbits have been identified. This property may provide a “protection mechanism”, leading to semiconfinement of chaotic orbits and extremely slow migration in the space of proper elements, so that diffusion is practically unrelated to the value of the Lyapunov time, T_L, of chaotic orbits. However, we show that, in more complicated dynamical models, the long-term evolution of chaotic orbits initiated in the vicinity of these resonances may also be governed by secular resonances. Finally, we find that stable-chaotic orbits have a characteristic spectrum of autocorrelation times: for the action conjugate to the critical argument the autocorrelation time is of the order of the Lyapunov time, while for the eccentricity- and inclination-related actions the autocorrelation time may be longer than 10³T_L. This behavior is consistent with the trajectory being sticky around a manifold of lower-than-full dimensionality in phase space (e.g., a 4-D submanifold of the 5-D energy manifold in a three-degrees-of-freedom autonomus Hamiltonian system) and reflects the inability of these “flawed” resonances to modify secular motion significantly, at least for times of the order of 200 Myr.