scholarly journals On the Dynamics of a Gravitational Dipole

2021 ◽  
Vol 17 (3) ◽  
pp. 247-261
Author(s):  
A. P. Markeev ◽  
Keyword(s):  
Nonlinear Oscillations ◽  
Initial Conditions ◽  
Nonlinear Approximation ◽  
Center Of Mass ◽  
Kam Theory ◽  
Computer Algorithms ◽  
Periodic Motions ◽  
Small Eccentricity ◽  
Formal Stability ◽  
Nominal Mode

The main purpose of this paper is to investigate nonlinear oscillations of the gravitational dipole in a neighborhood of its nominal mode. The orbit of the center of mass is assumed to be circular or elliptic with small eccentricity. Consideration is given both to planar and arbitrary spatial deviations of the gravitational dipole from its position corresponding to the nominal mode. The analysis is based on the classical Lyapunov and Poincaré methods and the methods of Kolmogorov – Arnold – Moser (KAM) theory. The necessary calculations are performed using computer algorithms. An analytic representation is given for conditionally periodic oscillations. Special attention is paid to the problem of the existence of periodic motions of the gravitational dipole and their Lyapunov stability, formal stability (stability in an arbitrarily high, but finite, nonlinear approximation) and stability for most (in the sense of Lebesgue measure) initial conditions.

scholarly journals On Nonlinear Oscillations and Stability of Coupled Pendulums in the Case of a Multiple Resonance

2020 ◽  
Vol 16 (4) ◽  
pp. 607-623
Author(s):  
A.P. Markeev ◽  
◽  
T.N. Chekhovskaya ◽  
Keyword(s):  
Periodic Motion ◽  
Nonlinear Oscillations ◽  
Initial Conditions ◽  
Nonlinear Approximation ◽  
Natural Oscillations ◽  
Periodic Action ◽  
Straight Line Passing ◽  
Multiple Resonance ◽  
Horizontal Beam ◽  
Existence And Stability

The points of suspension of two identical pendulums moving in a homogeneous gravitational field are located on a horizontal beam performing harmonic oscillations of small amplitude along a fixed horizontal straight line passing through the points of suspension of the pendulums. The pendulums are connected to each other by a spring of low stiffness. It is assumed that the partial frequency of small oscillations of each pendulum is exactly equal to the frequency of horizontal oscillations of the beam. This implies that a multiple resonance occurs in this problem, when the frequency of external periodic action on the system is equal simultaneously to two its frequencies of small (linear) natural oscillations. This paper solves the nonlinear problem of the existence and stability of periodic motions of pendulums with a period equal to the period of oscillations of the beam. The study uses the classical methods due to Lyapunov and Poincaré, KAM (Kolmogorov, Arnold and Moser) theory, and algorithms of computer algebra. The existence and uniqueness of the periodic motion of pendulums are shown, its analytic representation as a series is obtained, and its stability is investigated. For sufficiently small oscillation amplitudes of the beam, depending on the value of the dimensionless parameter which characterizes the stiffness of the spring connecting the pendulums, the found periodic motion is either Lyapunov unstable or stable for most (in the sense of Lebesgue measure) initial conditions or formally stable (stable in an arbitrarily large, but finite, nonlinear approximation).

scholarly journals Dynamics of Space Particles and Spacecrafts Passing by the Atmosphere of the Earth

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Vivian Martins Gomes ◽  
Antonio Fernando Bertachini de Almeida Prado ◽  
Justyna Golebiewska
Keyword(s):  
Initial Conditions ◽  
Equations Of Motion ◽  
Center Of Mass ◽  
The Sun ◽  
Three Body Problem ◽  
The Earth ◽  
Before And After ◽  
Three Body ◽  
Body Energy ◽  
Spacecraft Position

The present research studies the motion of a particle or a spacecraft that comes from an orbit around the Sun, which can be elliptic or hyperbolic, and that makes a passage close enough to the Earth such that it crosses its atmosphere. The idea is to measure the Sun-particle two-body energy before and after this passage in order to verify its variation as a function of the periapsis distance, angle of approach, and velocity at the periapsis of the particle. The full system is formed by the Sun, the Earth, and the particle or the spacecraft. The Sun and the Earth are in circular orbits around their center of mass and the motion is planar for all the bodies involved. The equations of motion consider the restricted circular planar three-body problem with the addition of the atmospheric drag. The initial conditions of the particle or spacecraft (position and velocity) are given at the periapsis of its trajectory around the Earth.

scholarly journals Infinitely Many Coexisting Attractors in No-Equilibrium Chaotic System

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Qiang Lai ◽  
Paul Didier Kamdem Kuate ◽  
Huiqin Pei ◽  
Hilaire Fotsin
Keyword(s):  
Adaptive Control ◽  
Chaotic System ◽  
Physical Meaning ◽  
Control Method ◽  
Initial Conditions ◽  
Periodic Motions ◽  
Hidden Attractors ◽  
Coexisting Attractors ◽  
Dynamic Behaviors

This paper proposes a new no-equilibrium chaotic system that has the ability to yield infinitely many coexisting hidden attractors. Dynamic behaviors of the system with respect to the parameters and initial conditions are numerically studied. It shows that the system has chaotic, quasiperiodic, and periodic motions for different parameters and coexists with a large number of hidden attractors for different initial conditions. The circuit and microcontroller implementations of the system are given for illustrating its physical meaning. Also, the synchronization conditions of the system are established based on the adaptive control method.

CHAOS IN THE REORIENTATION PROCESS OF A DUAL-SPIN SPACECRAFT WITH TIME-DEPENDENT MOMENTS OF INERTIA

2000 ◽  
Vol 10 (05) ◽  
pp. 997-1018 ◽  
Author(s):  
M. IÑARREA ◽  
V. LANCHARES
Keyword(s):  
Chaotic Behavior ◽  
Initial Conditions ◽  
Center Of Mass ◽  
Melnikov Method ◽  
Moments Of Inertia ◽  
Suitable Parameter ◽  
Nutation Angle ◽  
Principal Axes ◽  
Reorientation Process ◽  
Subharmonic Resonances

We study the spin-up dynamics of a dual-spin spacecraft containing one axisymmetric rotor which is parallel to one of the principal axes of the spacecraft. It will be supposed that one of the moments of inertia of the platform is a periodic function of time and that the center of mass of the spacecraft is not modified. Under these assumptions, it is shown that in the absence of external torques and spinning rotors the system possesses chaotic behavior in the sense that it exhibits Smale's horseshoes. We prove this statement by means of the Melnikov method. The presence of chaotic behavior results in a random spin-up operation. This randomness is visualized by means of maps of the initial conditions with final nutation angle close to zero. This phenomenon is well described by a suitable parameter that measures the amount of randomness of the process. Finally, we relate this parameter with the Melnikov function in the absence of the spinning rotor and with the presence of subharmonic resonances.

scholarly journals Decentralized Bioinspired Non-Discrete Model for Autonomous Swarm Aggregation Dynamics

2020 ◽  
Vol 10 (3) ◽  
pp. 1067
Author(s):  
Panagiotis Oikonomou ◽  
Stylianos Pappas
Keyword(s):  
Learning Strategy ◽  
Polynomial Regression ◽  
Initial Conditions ◽  
Cost Minimization ◽  
Center Of Mass ◽  
Scalar Fields ◽  
Discrete State ◽  
Robotic Swarms ◽  
Discrete Mathematical Model ◽  
Basin Hopping

In this paper a microscopic, non-discrete, mathematical model based on stigmergy for predicting the nodal aggregation dynamics of decentralized, autonomous robotic swarms is proposed. The model departs from conventional applications of stigmergy in bioinspired path-finding optimization, serving as a dynamic aggregation algorithm for nodes with limited or no ability to perform discrete logical operations, aiding in agent miniaturization. Time-continuous simulations were developed and carried out where nodal aggregation efficiency was evaluated using the following metrics: time to aggregation equilibrium, agent spatial distribution within aggregate (including average inter-nodal distance, center of mass of aggregate deviation from target), and deviation from target agent number. The system was optimized using cost minimization of the above factors through generating a random set of cost datapoints with varying initial conditions (number of aggregates, agents, field dimensions, and other specific agent parameters) where the best-fit scalar field was obtained using a random forest ensemble learning strategy and polynomial regression. The scalar cost field global minimum was obtained through basin-hopping with L-BFGS-B local minimization on the scalar fields obtained through both methods. The proposed optimized model describes the physical properties that non-digital agents must possess so that the proposed aggregation behavior emerges, in order to avoid discrete state algorithms aiming towards developing agents independent of digital components aiding to their miniaturization.

scholarly journals Solitary Rossby Waves in the Lower Tropical Troposphere

2013 ◽  
Vol 2013 ◽  
pp. 1-8
Author(s):  
Andre Lenouo ◽  
Francois Kamga Nkankam
Keyword(s):  
West Africa ◽  
Large Scale ◽  
Rossby Waves ◽  
Initial Conditions ◽  
Nonlinear Approximation ◽  
Tropical Storm ◽  
Weakly Nonlinear ◽  
African Easterly Jet ◽  
Tropical Troposphere ◽  
Cascade Method

Weakly nonlinear approximation is used to study the theoretical comportment of large-scale disturbances around the intertropical midtropospheric jet. We show here that the Korteweg de Vries (KdV) theory is appropriated to describe the structure of the streamlines around the African easterly jet (AEJ) region. The introduction of the additional velocity of the soliton C1 permits to search the stage where the configuration of the wave structures is going to emerge out of specified initial conditions and this is the direct and inverse cascade method. It was also shown that the configurations of disturbances can be influenced by this parameter so that we can look if the disturbances are in the control or not of their dispersive effects. This permits to explain the evolution of initial conditions of the Tropical Storm (TS) Debby over West Africa from 20 to 24 August 2006.

Inertially Coupled Galloping of Iced Conductors

1992 ◽  
Vol 59 (1) ◽  
pp. 140-145 ◽  
Author(s):  
P. Yu ◽  
A. H. Shah ◽  
N. Popplewell
Keyword(s):  
Periodic Solutions ◽  
Cross Section ◽  
Mixed Mode ◽  
Bifurcation Theory ◽  
Degrees Of Freedom ◽  
Center Of Mass ◽  
Asymptotic Solutions ◽  
Periodic Motions ◽  
Two Degrees Of Freedom ◽  
First Time

This paper is concerned with the galloping of iced conductors modeled as a two-degrees-of-freedom system. It is assumed that a realistic cross-section of a conductor has eccentricity; that is, its center of mass and elastic axis do not coincide. Bifurcation theory leads to explicit asymptotic solutions not only for the periodic solutions but also for the nonresonant, quasi-periodic motions. Critical boundaries, where bifurcations occur, are described explicitly for the first time. It is shown that an interesting mixed-mode phenomenon, which cannot happen in cocentric cases, may exist even for nonresonance.

Frequency analysis of nonlinear oscillations via the global error minimization

2016 ◽  
Vol 0 (0) ◽  
Author(s):  
M Kalami Yazdi ◽  
P Hosseini Tehrani
Keyword(s):  
Variational Approach ◽  
Nonlinear Oscillations ◽  
Initial Conditions ◽  
Order Approximation ◽  
Nonlinear Problems ◽  
Global Error ◽  
Error Minimization ◽  
Free Oscillations ◽  
First Order ◽  
First Order Approximation

AbstractThe capacity and effectiveness of a modified variational approach, namely global error minimization (GEM) is illustrated in this study. For this purpose, the free oscillations of a rod rocking on a cylindrical surface and the Duffing-harmonic oscillator are treated. In order to validate and exhibit the merit of the method, the obtained result is compared with both of the exact frequency and the outcome of other well-known analytical methods. The corollary reveals that the first order approximation leads to an acceptable relative error, specially for large initial conditions. The procedure can be promisingly exerted to the conservative nonlinear problems.

DISCRETIZATION CHAOS IN THE SWITCHING CONTROL SYSTEM WITH FINITE SWITCHING VALUES

1995 ◽  
Vol 05 (06) ◽  
pp. 1749-1755 ◽  
Author(s):  
XING HUO YU
Keyword(s):  
Control System ◽  
Initial Conditions ◽  
Switching Control ◽  
Second Order ◽  
Periodic Motions ◽  
Theoretical Investigations ◽  
Simulation Results

A case study is presented to demonstrate the discretization chaos in the switching control system with only finite switching values. It is proved that for the second order oscillator, with several classes of sampling periods, discrete switching control enables periodic motions around the desired equilibrium. The pattern of the discretized system is determined by the initial conditions as well as the sampling periods. Simulation results are presented to confirm the theoretical investigations.

Sign in / Sign up

Export Citation Format

Share Document