Maximum Lyapunov Characteristic Exponent Research Articles

Rotorcraft stability analysis using Lyapunov characteristic exponents estimated from multibody dynamics

Stability analysis of complex, nonlinear dynamical systems is a challenge. The use of Lyapunov Characteristic Exponents through a Jacobian-less method is proposed as a means to identify the Maximum Lyapunov Characteristic Exponent, namely the fundamental stability indicator of a generic problem, solely from time series obtained through general-purpose multibody dynamics simulations of complex rotorcraft aeromechanics models. The method is first applied to a relatively simple scenario concerning the identification of ground resonance. Then, its application to more complex models is addressed by studying the aeroelastic stability and identifying the whirl flutter of the XV-15 tiltrotor using a comprehensive aeroelastic model.

Confined chaos and the chaotic angular motion of Atlas, a Saturn’s inner satellite

ABSTRACT The dynamics of the Solar system exhibit inherent chaos and instability. Mathematical tools, such as the maximum Lyapunov characteristic exponent (LCE) and Lyapunov time (TL), play a crucial role in providing a qualitative understanding of chaos within celestial objects, such as asteroids and moonlets. Celestial bodies with relatively small Lyapunov times have garnered significant research interest due to their stable orbits, a phenomenon referred to as stable or confined chaos. Notable examples include Saturn’s satellites: Atlas, with a Lyapunov time on the order of 10 yr, Prometheus, and Pandora. This work aims to study the chaotic behaviour of the Atlas satellite and its relatively small TL. We present a three-dimensional model approach designed to isolate the radial contribution from the LCE and assess its influence within the LCE. Our investigation focuses on the Saturn system, comprising Saturn itself, along with its satellites Atlas, Prometheus, Pandora, and Mimas. To estimate the radial contribution of the LCE, we find the projection of the radial vector of a ghost Atlas (a slightly displaced Atlas) onto the Atlas radial vector, which allows us to calculate the difference between the radial vectors. This methodology enables us to estimate the radial contribution of the LCE and calculate the Lyapunov time. Remarkably, our results demonstrate that orbits remain confined even for integration times exceeding TL. Furthermore, we investigate the temporal behaviour of Atlas’ angular position in its orbit, potentially shedding light on chaotic angular dynamics.

1:1 Ground-track resonance in a uniformly rotating 4th degree and order gravitational field

Using a gravitational field truncated at the 4th degree and order, the 1:1 ground-track resonance is studied. To address the main properties of this resonance, a 1-degree of freedom (1-DOF) system is firstly studied. Equilibrium points (EPs), stability and resonance width are obtained. Different from previous studies, the inclusion of non-spherical terms higher than degree and order 2 introduces new phenomena. For a further study about this resonance, a 2-DOF model which includes a main resonance term (the 1-DOF system) and a perturbing resonance term is studied. With the aid of Poincaré sections, the generation of chaos in the phase space is studied in detail by addressing the overlap process of these two resonances with arbitrary combinations of eccentricity (e) and inclination (i). Retrograde orbits, near circular orbits and near polar orbits are found to have better stability against the perturbation of the second resonance. The situations of complete chaos are estimated in the e-i plane. By applying the maximum Lyapunov Characteristic Exponent (LCE), chaos is characterized quantitatively and similar conclusions can be achieved. This study is applied to three asteroids 1996 HW1, Vesta and Betulia, but the conclusions are not restricted to them.

Revealing the Character of Orbits in a Binary System Consisting of a Primary Galaxy and a Satellite Companion

AbstractIn this article, we present a galactic gravitational model of three degrees of freedom (3D), in order to study and reveal the character of the orbits of the stars, in a binary stellar system composed of a primary quiet or active galaxy and a small satellite companion galaxy. Our main dynamical analysis will be focused on the behaviour of the primary galaxy. We investigate in detail the regular or chaotic nature of motion, in two different cases: (i) the time-independent model in both 2D and 3D dynamical systems and (ii) the time-evolving 3D model. For the description of the structure of the 2D system, we use the classical method of the Poincaré (x, px), y = 0, py < 0 phase plane. In order to study the structure of the phase space of the 3D system, we take sections in the plane y = 0 of the 3D orbits, whose initial conditions differ from the plane parent periodic orbits, only by the z component. The set of the four-dimensional points in the (x, px, z, pz) phase space is projected on the (z, pz) plane. The maximum Lyapunov characteristic exponent is used in order to make an estimation of the chaoticity of our galactic system, in both 2D and 3D dynamical models. Our numerical calculations indicate that the percentage of the chaotic orbits increases when the primary galaxy has a dense and massive nucleus. The presence of the dense galactic core also increases the stellar velocities near the center of the galaxy. Moreover, for small values of the distance R between the two bodies, low-energy stars display chaotic motion, near the central region of the galaxy, while for larger values of the distance R, the motion in active galaxies is entirely regular for low-energy stars. Our simulations suggest that in galaxies with a satellite companion, the chaotic nature of motion is not only a result of the galactic interaction between the primary galaxy and its companion, but also a result caused by the presence of the dense nucleus in the core of the primary galaxy. Theoretical arguments are presented in order to support and interpret the numerically derived outcomes. Furthermore, we follow the 3D evolution of the primary galaxy, when mass is transported adiabatically from the disk to the nucleus. Our numerical results are in satisfactory agreement with observational data obtained from the M51-type binary stellar systems. A comparison between the present research and similar and earlier work is also made.

ANALYSIS OF ROUND OFF ERRORS WITH REVERSIBILITY TEST AS A DYNAMICAL INDICATOR

We compare the divergence of orbits and the reversibility error for discrete time dynamical systems. These two quantities are used to explore the behavior of the global error induced by round off in the computation of orbits. The similarity of results found for any system we have analyzed suggests the use of the reversibility error, whose computation is straightforward since it does not require the knowledge of the exact orbit, as a dynamical indicator. The statistics of fluctuations induced by round off for an ensemble of initial conditions has been compared with the results obtained in the case of random perturbations. Significant differences are observed in the case of regular orbits due to the correlations of round off error, whereas the results obtained for the chaotic case are nearly the same. Both the reversibility error and the orbit divergence computed for the same number of iterations on the whole phase space provide an insight on the local dynamical properties with a detail comparable with other dynamical indicators based on variational methods such as the finite time maximum Lyapunov characteristic exponent, the mean exponential growth factor of nearby orbits and the smaller alignment index. For 2D symplectic maps, the differentiation between regular and chaotic regions is well full-filled. For 4D symplectic maps, the structure of the resonance web as well as the nearby weakly chaotic regions are accurately described.

Fractal analysis: Annual rainfall in Chennai

The annual rainfall data of Chennai is analyzed using the Fractal Construction Technique. According to Mandelbrot the dimension of any line including nautical lines may not be Euclidean but Fractional, Mandelbrot, 1982. This fractional dimension leads to a repetitive appearance of any pattern. Climate which is usually periodic by nature can be analyzed through this technique. Efforts are on to search the fractal geometry of climate and to predict its periodicity on different temporal scales. This paper estimates the various parameters like Lyapunov exponent, Maximum Lyapunov characteristic exponent, Lyapunov time, Kaplan-Yorke dimension for the annual rainfall of Chennai.

Peculiar trajectories around the Lagrangian equilateral points

In the present work we analyse the behaviour of a particle under the gravitational influence of two massive bodies and a particular dissipative force. The circular restricted three body problem, which describes the motion of this particle, has five equilibrium points in the frame which rotates with the same angular velocity as the massive bodies: two equilateral stable points ( L 4 , L 5 ) and three colinear unstable points ( L 1 , L 2 , L 3 ). A particular solution for this problem is a stable orbital libration, called a tadpole orbit, around the equilateral points. The inclusion of a particular dissipative force can alter this configuration. We investigated the orbital behaviour of a particle initially located near L 4 or L 5 under the perturbation of a satellite and the Poynting–Robertson drag. This is an example of breakdown of quasi-periodic motion about an elliptic point of an area-preserving map under the action of dissipation. Our results show that the effect of this dissipative force is more pronounced when the mass of the satellite and/or the size of the particle decrease, leading to chaotic, although confined, orbits. From the maximum Lyapunov Characteristic Exponent a final value of γ was computed after a time span of 10 6 orbital periods of the satellite. This result enables us to obtain a critical value of log γ beyond which the orbit of the particle will be unstable, leaving the tadpole behaviour. For particles initially located near L 4 , the critical value of log γ is −4.07 and for those particles located near L 5 the critical value of log γ is −3.96.

Dynamical Influences on the Orbits of Prometheus and Pandora

We present the results of a numerical and analytical study of the orbits of the Saturnian satellites Prometheus and Pandora, including the perturbing effects of other moons. The full equations of motion have been integrated numerically between 1980 and 2010, taking into account Saturn's oblateness up to terms in J6. Included in the simulations are the effects of the eight major satellites of Saturn, together with the co-orbital satellites, Janus and Epimetheus. The results show that the anticorrelation in the temporal variation of the mean longitudes of Prometheus and Pandora, demonstrated in previously published two-satellite simulations, survives the addition of the other satellites to the model. Chaos is apparent through sensitivity to initial conditions and a positive value for the maximum Lyapunov characteristic exponent. There is evidence that the other satellites also contribute to the chaotic motion on a longer timescale. The effects of the nearby Mimas 3 : 2 resonance on the orbit of Pandora are clearly detectable. We find evidence that Janus and Epimetheus have a secondary role in the dynamical evolution of Prometheus and Pandora and discuss possible mechanisms. We show from theory that two independent sets of second-order resonances due to Epimetheus sweep across the orbits of Prometheus and Pandora every 4 years, when the orbit of Epimetheus switches position with its co-orbital companion. However, no effects related to this have so far been identified. Comparison of integrated results with extrapolations of current published ephemerides suggest uncertainties on the order of 104 km in the down-track positions of Prometheus and Pandora, equivalent to 4° in mean longitude, during the Cassini tour.

Assessing Chaos in Time Series: Statistical Aspects and Perspectives

Chaos theory offers to time series analysis new perspectives as well as concepts and ideas that have a through contribution to statistics. On the other hand, statistical methodology has shown to play a crucial role for the comprehension of nonlinear and chaotic phenomena. One peculiar feature of chaotic systems is sensitivity to initial conditions, which is responsible of the unpredictability we experience in such phenomena. One of the most popular quantity that measures this property is the maximum Lyapunov characteristic exponent (MLCE). In this paper we discuss from a statistical perspective the problems arising in estimating both the MLCE and its generalizations in time series, issues that have recently deserved attention in the community of time series analysts. We also present a method based on resampling in order to assign confidence interval to the estimates of the MLCE. It is shown that in addition to answering the question of the presence of chaos, these methods give relevant contributions to the characterization of many other aspects of nonlinear time series.

Maximum Lyapunov Exponents for Chaotic Rotation of Natural Planetary Satellites

Based on the Chirikov approach [1, 2] within the context of the theory of separatrix mappings, we suggest and substantiate a simple method for estimating the maximum Lyapunov characteristic exponent (MLCE) of motion in a chaotic layer in the neighborhood of nonlinear resonance separatrix of a Hamiltonian system under an asymmetric periodic perturbation. For a number of natural planetary satellites, using this method, the estimates are made of the MLCE of chaotic rotation (relative to the center of mass) in the main chaotic layer, i.e., in the chaotic layer in the neighborhood of a synchronous resonance separatrix. The value of the MLCE is determined by an orbital eccentricity and by the parameter of the satellite's dynamic asymmetry. The quantity inverse to the MLCE gives a typical time of predictable rotational motion.

A COMPARISON BETWEEN METHODS TO COMPUTE LYAPUNOV EXPONENTS

Lyapunov characteristic exponents measure the rate of exponential divergence between neighboring trajectories in the phase space. For a given autonomous dynamical system, the maximum Lyapunov characteristic exponent (hereafter LCE) is computed from the solution of the variational equations of the system. There are many dynamical systems in which the formulation and solution of the variational equations is a cumbersome task. In those cases an alternative procedure, first introduced by Benettin et al., is to replace the variational solution by computing two neighbor trajectories (the test particle and its shadow) and calculating the mutual distance. In this paper, we deal with a comparison between these two different techniques for the calculation of LCE: the variational method and the two-particle method. We point out a problem that can appear when the two-particle method is used, which can lead to a false estimation of a positive LCE. The explanation of this phenomenon can be analyzed in two different situations: (1) for relatively large initial separations the two-particle method is not a good approximation to the solution of the variational equations, and (2) for small initial separations the two-particle method have problems related to the machine precision, even when the separation can be many order of magnitudes larger than the machine precision. We show some examples of false estimates of the LCE that have already appeared in the literature using the two-particle method, and finally we present some suggestions to be taken into account when this method has to be used.

Restricting Chaos to a Small Region in the Phase Space

The idea of restricting chaos in dissipative systems to a small region in the phase space is proposed. The possibility of realization of this idea is demonstrated by applying a simple method summed up from computer simulations successfully to three different dynamical systems. It is found that not only does the trajectory of the controlled system occupy a region smaller than that of the uncontrolled chaotic system, the corresponding attractor of the Poincaré map is also smaller than that of the uncontrolled system. In addition, but also the maximum Lyapunov characteristic exponent of the system is greatly lowered.

Relaxation properties and ergodicity breaking in nonlinear Hamiltonian dynamics.

The time evolution toward equipartition of energy is numerically investigated in nonlinear Hamiltonian systems with a large number N of degrees of freedom. The relaxation process is studied by computing the spectral entropy \ensuremath{\eta}; for a wide class of initial conditions, it follows a stretched exponential law \ensuremath{\eta}(t)\ensuremath{\sim}exp[-(t/${\ensuremath{\tau}}_{0}$${)]}^{\ensuremath{\xi}}$, \ensuremath{\xi}1, up to times t${\ensuremath{\tau}}_{R}$, and \ensuremath{\eta}(t)=const for t\ensuremath{\ge}${\ensuremath{\tau}}_{R}$. By taking advantage of this fact, a good definition of a relaxation time becomes possible. Below a critical value ${\ensuremath{\varepsilon}}_{c}$ (model dependent) of the energy density \ensuremath{\varepsilon}, the relaxation time ${\ensuremath{\tau}}_{R}$ is found to follow a scaling with \ensuremath{\varepsilon}, which is compatible with a ``Nekhoroshev-like'' law, i.e., ${\ensuremath{\tau}}_{R}$=${\ensuremath{\tau}}_{0}$exp(${\ensuremath{\varepsilon}}_{0}$/\ensuremath{\varepsilon}${)}^{\ensuremath{\gamma}}$, for both the Fermi-Pasta-Ulam (FPU) \ensuremath{\beta} model and the classical lattice ${\ensuremath{\varphi}}^{4}$ model; a remarkable difference with respect to Nekhoroshev's theorem (where the exponent \ensuremath{\gamma} scales as 1/${N}^{2}$) is the N independence of numerical experiment results.An important consequence of this fact is the existence of nonequilibrium states of arbitrary lifetimes also at large N values. On the other hand, at high-energy densities (\ensuremath{\varepsilon}g${\ensuremath{\varepsilon}}_{c}$) ${\ensuremath{\tau}}_{R}$ is almost independent of \ensuremath{\varepsilon}. The maximum Lyapunov characteristic exponent ${\ensuremath{\lambda}}_{1}$ is measured in both models as a function of the energy density. The striking result is a change in the scaling ${\ensuremath{\lambda}}_{1}$(\ensuremath{\varepsilon}) occurring at the same \ensuremath{\varepsilon} (=${\ensuremath{\varepsilon}}_{c}$) at which ${\ensuremath{\tau}}_{R}$ has its crossover. At \ensuremath{\varepsilon}g${\ensuremath{\varepsilon}}_{c}$, the scaling ${\ensuremath{\lambda}}_{1}$\ensuremath{\sim}${\ensuremath{\varepsilon}}^{2/3}$ is found for both the FPU and the ${\ensuremath{\varphi}}^{4}$ models; this is in agreement with a random matrix approximation of the tangent dynamics, which means that the dynamics itself mimics a random process. Below ${\ensuremath{\varepsilon}}_{c}$, steeper and model-dependent scalings are found. Hence, evidence for the existence of a rather sharp transition of the phase-space structure is provided, and a strong stochasticity threshold (SST) can be defined. A preliminary result, suggesting the stability of the SST in the limit of large N, is also reported.

Numerical evidence of a sharp order window in a Hamiltonian system

A detailed numerical analysis for a small range of the nonlinearity parameter exhibits the existence of a sharp order window for the n = 3 discrete self-trapping equation. The analysis performed by using maximum Lyapunov characteristic exponent, power spectra, Poincaré maps and correlation exponents gives a clear-cut evidence of a biperiodic dynamics on bidimensional tori.

Scaling law for the maximum Lyapunov characteristic exponent of infinite product of random matrices

The authors present a simple derivation for the scaling behaviour of the maximum Lyapunov characteristic exponent lambda of infinite product of symplectic random matrices. The considered random matrices depend on a parameter epsilon and lambda =0 for epsilon =0. They obtain lambda varies as epsilon beta with either beta =1/2 or beta =2/3 depending on the probability distribution of the matrix elements. The results are in agreement with a previous numerical simulation.

Characterisation of intermittency in chaotic systems

The authors discuss the characterisation of intermittency in chaotic dynamical systems by means of the time fluctuations of the response to a slight perturbation on the trajectory. A set of exponents is introduced which generalise the maximum Lyapunov characteristic exponent. The link with the statistical mechanics formalism is emphasised and they show that the exponents are connected to a free energy formally defined for chaotic systems. They perform some analytical computations in simple cases and give a few numerical examples.

Scaling Behavior of Chaotic Flows

It is shown that in the turbulent regime of systems with period-doubling subharmonic bifurcations, the maximum Lyapunov characteristic exponent behaves like $\overline{\ensuremath{\lambda}}={\overline{\ensuremath{\lambda}}}_{0}{(\mathcal{r}\ensuremath{-}{\mathcal{r}}_{c})}^{t}$, with $t$ a universal exponent which is calculated to be $t=0.4498069\dots{}$. This result is in agreement with the available data on $\overline{\ensuremath{\lambda}}$ for a number of dynamical systems.