Dynamical Indicator of Human Body’s Physical Endurance

Physical endurance is the time span between the beginning of physical activity by an individual and the termination because of exhaustion. Physical endurance involves a multifaceted behaviour which can be understood by complexities. Everyone performs physical activity in order to sustain-life. However, the number of activities done are largely subject to personal choice and varies from person to person as well as for a given person over time. Physical activity like meditation/exercises are positively related to physical fitness. One needs to understand relation between physical activity, exercise, physical fitness and health. These activities can be partitioned mutually exclusively into many different ways. This paper categorizes daily physical activity into three broad subdivisions based on amount of body movements taking place are: (i) light, (ii) moderate and (iii) high intensity. These three characterizations are considered to be mutually exclusive and sum up to total energy spent by an individual. The behavior of the three factors physical activity, heart and energy generated is analyzed with the help of Fast Lyapunov indicator (FLI), Dynamic Lyapunov indicator (DLI), Small alignment index (SALI). FLI’s increase for chaotic orbits for values of R=20, Q=70 for the case of high intensity exercises and to linearly regular orbits for values of R=5, Q=8 and R=10, Q=12 in the cases of light and moderate exercises respectively. SALI’s alters through non-zero value for R=20, Q=70 while it tends to zero for values of R=5, Q=8 and R=10, Q=12. DLI’s the largest Eigen values form a definite pattern/curve for n=2000 for values of R=5, Q=8 and n=100 for R=10, Q=12 respectively as the motion stays regular plus dispersed randomly as the motion is chaotic for n=60 and for R=20, Q=70.

Short-period effects of the planetary perturbations on the Sun–Earth Lagrangian point L3

Context. The Lagrangian point L3 of the Sun–Earth system, and its Lyapunov orbits, have been proposed to perform station-keeping, although L3 is only rigorously defined for the extremely simplified model represented by the reduced Sun–Earth–spacecraft system. As in L3 the planetary perturbations (mainly from Jupiter and Venus) are stronger than Earth’s attraction, it is necessary to understand whether or not the dynamics close to L3 persist under such a strong perturbation, allowing for a definition of dynamical substitutes for models that are more realistic than the circular restricted three-body problem. Aims. In this paper we address the problem of the existence of motions that remain close to L3 for a time-span which is relevant for space missions in a model of the Solar System compatible with the precision of JPL digital ephemerides. Methods. First, we computed analytically the main short-period effects of planetary perturbations in a simplified model of the Solar System with the orbits of all the planets co-planar and circular. We then applied the Fast Lyapunov Indicator method in order to find dynamical substitutes that exist for time-spans of hundreds of years in the model of the Solar System that is used to produce the modern ephemerides. Results. We find that the original system is conjugate by a canonical transformation to an averaged system that has an equilibrium close to L3: even if Venus and Jupiter each move the position of this equilibrium by about 218 and 176 km, respectively, in opposite directions, in the model where both the planets are included, their effects almost perfectly compensate for one another, leaving a displacement of about 40 km only. This equilibrium is then mapped in the original system to a quasi-periodic dynamical substitute; the contributions of each planet to the amplitude of this quasi-periodic libration around L3 do not compensate for one another, and sum to about 10 000 km. The Fast Lyapunov Indicator method allowed us to find orbits of any amplitude bigger than this one (up to 0.03 AU) for time-spans of hundreds of years in the model of the Solar System that is used to produce the modern ephemerides. Conclusions. Using a combination of the Hamiltonian averaging method with a new implementation of the Fast Lyapunov Indicator method we find orbits useful for astrodynamics originating at the Sun–Earth Lagrangian point L3 for a realistic model of the Solar System. In particular, this usage of the chaos indicator provides an innovative application of dynamical systems theory to astrodynamics, where the short-period perturbations represent a relevant part of the model.

Puzzling out the coexistence of terrestrial planets and giant exoplanets

Aims. Hundreds of giant planets have been discovered so far and the quest of exo-Earths in giant planet systems has become intriguing. In this work, we aim to address the question of the possible long-term coexistence of a terrestrial companion on an orbit interior to a giant planet, and explore the extent of the stability regions for both non-resonant and resonant configurations. Methods. Our study focuses on the restricted three-body problem, where an inner terrestrial planet (massless body) moves under the gravitational attraction of a star and an outer massive planet on a circular or elliptic orbit. Using the detrended fast Lyapunov indicator as a chaotic indicator, we constructed maps of dynamical stability by varying both the eccentricity of the outer giant planet and the semi-major axis of the inner terrestrial planet, and identify the boundaries of the stability domains. Guided by the computation of families of periodic orbits, the phase space is unravelled by meticulously chosen stable periodic orbits, which buttress the stability domains. Results. We provide all possible stability domains for coplanar symmetric configurations and show that a terrestrial planet, either in mean-motion resonance or not, can coexist with a giant planet, when the latter moves on either a circular or an (even highly) eccentric orbit. New families of symmetric and asymmetric periodic orbits are presented for the 2/1 resonance. It is shown that an inner terrestrial planet can survive long time spans with a giant eccentric outer planet on resonant symmetric orbits, even when both orbits are highly eccentric. For 22 detected single-planet systems consisting of a giant planet with high eccentricity, we discuss the possible existence of a terrestrial planet. This study is particularly suitable for the research of companions among the detected systems with giant planets, and could assist with refining observational data.

Application of Indicators for Chaos in Chaotic Circuit Systems

Lyapunov exponent (LE), fast Lyapunov indicator (FLI), relative finite-time Lyapunov indicator (RLI), smaller alignment index (SALI), and generalized alignment index (GALI) are some of the available methods in most conservative systems. This study focuses on the effects of the above indicators on dissipative chaotic circuit systems such as the Lorenz system and a hyperchaotic model. Numerical experiments show that the performances of the chaos indicators in the hyperchaotic system are almost similar to those in the Lorenz system. These indicators clearly provide transition from chaotic to regular motion. However, FLI, RLI, SALI, and GALI cannot describe transition from chaos to hyperchaos. These indicators are also applied to study a new four-dimensional chaotic circuit system. The basic dynamic behaviors and structures are investigated analytically and numerically. The dynamic qualitative properties of individual orbits are observed using an oscilloscope. Moreover, the entire set of LE about the parameter is found to have three threshold values. Comparisons show that all chaos indicators are able to capture chaotic and periodic motion in chaotic circuit systems, but SALI displays significantly different behavior in several periodic orbits. SALI drops exponentially to zero for “morphologically regular” orbits that are actually unstable and sensitive to perturbation. This conclusion can also be confirmed by GALI.

Unveiling the Influence of Dark Matter in Axially Symmetric Galaxies

We investigate the regular or chaotic nature of orbits of stars moving in the meridional plane (R,z) of an axially symmetric galactic model with a dense, massive spherical nucleus and a dark matter halo component. In particular, we study the influence of the fractional portion of the dark matter, by computing in each case the percentage of chaotic orbits, as well as the percentages of orbits of the main regular resonant families. In an attempt to distinguish between regular and chaotic motion, we use the fast Lyapunov indicator method to extensive samples of orbits obtained by integrating numerically the equations of motion as well as the variational equations. Furthermore, a technique which is based mainly on the field of spectral dynamics that utilises the Fourier transform of the time series of each coordinate is used for identifying the various families of regular orbits and also to recognise the secondary resonances that bifurcate from them. Two cases are studied in our work: (i) the case where we have a disk galaxy model and (ii) the case where our model represents an elliptical galaxy. A comparison with early related work is also made.

COMPARATIVE STUDY OF VARIATIONAL CHAOS INDICATORS AND ODEs' NUMERICAL INTEGRATORS

The reader can find in the literature a lot of different techniques to study the dynamics of a given system and also, many suitable numerical integrators to compute them. Notwithstanding the recent work of [Maffione et al., 2011b] for mappings, a detailed comparison among the widespread indicators of chaos in a general system is still lacking. Such a comparison could lead to select the most efficient algorithms given a certain dynamical problem. Furthermore, in order to choose the appropriate numerical integrators to compute them, more comparative studies among numerical integrators are also needed. This work deals with both problems. We first extend the work of [Maffione et al., 2011b] for mappings to the 2D [Hénon & Heiles, 1964] potential, and compare several variational indicators of chaos: the Lyapunov Indicator (LI); the Mean Exponential Growth Factor of Nearby Orbits (MEGNO); the Smaller Alignment Index (SALI) and its generalized version, the Generalized Alignment Index (GALI); the Fast Lyapunov Indicator (FLI) and its variant, the Orthogonal Fast Lyapunov Indicator (OFLI); the Spectral Distance (D) and the Dynamical Spectra of Stretching Numbers (SSNs). We also include in the record the Relative Lyapunov Indicator (RLI), which is not a variational indicator as the others. Then, we test a numerical technique to integrate Ordinary Differential Equations (ODEs) based on the Taylor method implemented by [Jorba & Zou, 2005] (called taylor), and we compare its performance with other two well-known efficient integrators: the [Prince & Dormand, 1981] implementation of a Runge–Kutta of order 7–8 (DOPRI8) and a Bulirsch–Stöer implementation. These tests are run under two very different systems from the complexity of their equations point of view: a triaxial galactic potential model and a perturbed 3D quartic oscillator. We first show that a combination of the FLI/OFLI, the MEGNO and the GALI 2N succeeds in describing in detail most of the dynamical characteristics of a general Hamiltonian system. In the second part, we show that the precision of taylor is better than that of the other integrators tested, but it is not well suited to integrate systems of equations which include the variational ones, like in the computing of almost all the preceeding indicators of chaos. The result which induces us to draw this conclusion is that the computing times spent by taylor are far greater than the times consumed by the DOPRI8 and the Bulirsch–Stöer integrators in such cases. On the other hand, the package is very efficient when we only need to integrate the equations of motion (both in precision and speed), for instance to study the chaotic diffusion. We also notice that taylor attains a greater precision on the coordinates than either the DOPRI8 or the Bulirsch–Stöer.

Dynamics of particles in slowly rotating black holes with dipolar halos

In general, the model of galaxy assumes a central huge black hole surrounded by a massive halo, disk or ring. In this paper, we investigate the gravitational field structure of a slowly rotating black hole with a dipolar halo, and the dynamics and chaos of test particles moving in it. Using Poincaré sections and fast Lyapunov indicator (FLI) in general relativity, we investigate chaos under different dynamical parameters, and find that the FLI is suitable for detecting chaos and even resonant orbits.