Nonintegrability of the problem of a spherical top rolling on a vibrating plane

This paper investigates the rolling motion of a spherical top with an axisymmetric mass distribution on a smooth horizontal plane performing periodic vertical oscillations. For the system under consideration, equations of motion and conservation laws are obtained. It is shown that the system admits two equilibrium points corresponding to uniform rotations of the top about the vertical symmetry axis. The equilibrium point is stable when the center of mass is located below the geometric center, and is unstable when the center of mass is located above it. The equations of motion are reduced to a system with one and a half degrees of freedom. The reduced system is represented as a small perturbation of the problem of the Lagrange top motion. Using Melnikov’s method, it is shown that the stable and unstable branches of the separatrix intersect transversally with each other. This suggests that the problem is nonintegrable. Results of computer simulation of the top dynamics near the unstable equilibrium point are presented.

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A Four-Zone Model and Nonlinear Dynamic Analysis of Solution Multiplicity of Buoyancy Ventilation in Underground Building

Complexity ◽

10.1155/2020/8658797 ◽

2020 ◽

Vol 2020 ◽

pp. 1-24

Author(s):

Yuxing Wang ◽

Chunyu Wei

Keyword(s):

Differential Equations ◽

Equilibrium Point ◽

Equilibrium Points ◽

Underground Structure ◽

Unstable Equilibrium ◽

Phase Portraits ◽

Zone Model ◽

Unstable Equilibrium Point ◽

Index 2 ◽

The Stability

The solution multiplicity of natural ventilation in buildings is very important to personnel safety and ventilation design. In this paper, a four-zone model of buoyancy ventilation in typical underground building is proposed. The underground structure is divided to four zones, a differential equation is established in each zone, and therefore, there are four differential equations in the underground structure. By solving and analyzing the equilibrium points and characteristic roots of the differential equations, we analyze the stability of three scenarios and obtain the criterions to determine the stability and existence of solutions for two scenarios. According to these criterions, the multiple steady states of buoyancy ventilation in any four-zone underground buildings for different stack height ratios and the strength ratios of the heat sources can be obtained. These criteria can be used to design buoyancy ventilation or natural exhaust ventilation systems in underground buildings. Compared with the two-zone model in (Liu et al. 2020), the results of the proposed four-zone model are more consistent with CFD results in (Liu et al. 2018). In addition, the results of proposed four-zone model are more specific and more detailed in the unstable equilibrium point interval. We find that the unstable equilibrium point interval is divided into two different subintervals corresponding to the saddle point of index 2 and the saddle focal equilibrium point of index 2, respectively. Finally, the phase portraits and vector field diagrams for the two scenarios are given.

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High-Security Image Encryption Based on a Novel Simple Fractional-Order Memristive Chaotic System with a Single Unstable Equilibrium Point

Electronics ◽

10.3390/electronics10243130 ◽

2021 ◽

Vol 10 (24) ◽

pp. 3130

Author(s):

Zain-Aldeen S. A. Rahman ◽

Basil H. Jasim ◽

Yasir I. A. Al-Yasir ◽

Raed A. Abd-Alhameed

Keyword(s):

Equilibrium Point ◽

Image Encryption ◽

Fractional Order ◽

Chaotic System ◽

Chaotic Systems ◽

Equilibrium Points ◽

Unstable Equilibrium ◽

Phase Portraits ◽

Time Efficiency ◽

Unstable Equilibrium Point

Fractional-order chaotic systems have more complex dynamics than integer-order chaotic systems. Thus, investigating fractional chaotic systems for the creation of image cryptosystems has been popular recently. In this article, a fractional-order memristor has been developed, tested, numerically analyzed, electronically realized, and digitally implemented. Consequently, a novel simple three-dimensional (3D) fractional-order memristive chaotic system with a single unstable equilibrium point is proposed based on this memristor. This fractional-order memristor is connected in parallel with a parallel capacitor and inductor for constructing the novel fractional-order memristive chaotic system. The system’s nonlinear dynamic characteristics have been studied both analytically and numerically. To demonstrate the chaos behavior in this new system, various methods such as equilibrium points, phase portraits of chaotic attractor, bifurcation diagrams, and Lyapunov exponent are investigated. Furthermore, the proposed fractional-order memristive chaotic system was implemented using a microcontroller (Arduino Due) to demonstrate its digital applicability in real-world applications. Then, in the application field of these systems, based on the chaotic behavior of the memristive model, an encryption approach is applied for grayscale original image encryption. To increase the encryption algorithm pirate anti-attack robustness, every pixel value is included in the secret key. The state variable’s initial conditions, the parameters, and the fractional-order derivative values of the memristive chaotic system are used for contracting the keyspace of that applied cryptosystem. In order to prove the security strength of the employed encryption approach, the cryptanalysis metric tests are shown in detail through histogram analysis, keyspace analysis, key sensitivity, correlation coefficients, entropy analysis, time efficiency analysis, and comparisons with the same fieldwork. Finally, images with different sizes have been encrypted and decrypted, in order to verify the capability of the employed encryption approach for encrypting different sizes of images. The common cryptanalysis metrics values are obtained as keyspace = 2648, NPCR = 0.99866, UACI = 0.49963, H(s) = 7.9993, and time efficiency = 0.3 s. The obtained numerical simulation results and the security metrics investigations demonstrate the accuracy, high-level security, and time efficiency of the used cryptosystem which exhibits high robustness against different types of pirate attacks.

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TANGLED SEPARATRICES ON TWO DEGREES OF FREEDOM SWING EQUATION SYSTEM

International Journal of Bifurcation and Chaos ◽

10.1142/s021812740100398x ◽

2001 ◽

Vol 11 (12) ◽

pp. 3033-3058

Author(s):

YOSHITAKA HASEGAWA ◽

YOSHISUKE UEDA

Keyword(s):

Equilibrium Point ◽

Degrees Of Freedom ◽

Invariant Manifolds ◽

Equilibrium Points ◽

Sufficient Conditions ◽

Liapunov Function ◽

Equation System ◽

Stable Equilibrium Point ◽

Two Degrees Of Freedom ◽

Swing Equation

One of the applications of two degrees of freedom swing equation system is the transient stability problem of an electrical power system. Using the Liapunov function method, there are many reports on the sufficient conditions of normal operations. However the basin structure of the stable equilibrium point, which corresponds to the normal operating state, is not entirely known. In this paper we will approach this problem by investigating the structure of the separatrix of the corresponding conservative system. This separatrix decomposes the phase space into regions of bound motions and divergent motions. This boundary concerns the invariant manifolds of the closed orbits on the center-manifolds of the saddle-center equilibrium points, which appear under the conservative condition. By numerical simulations, we will confirm in this report that there are transverse homoclinic intersections on these manifolds, whose cross-sectional view is different from a familiar homoclinic structure due to the asymmetric feature of the potential. Such circumstances are not observed, for example, on the well-known Hénon–Heiles system whose saddle-center equilibrium point has a trivial homoclinic loop.

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Classical n-body system in geometrical and volume variables: I. Three-body case

International Journal of Modern Physics A ◽

10.1142/s0217751x21501402 ◽

2021 ◽

pp. 2150140

Author(s):

A. M. Escobar-Ruiz ◽

R. Linares ◽

Alexander V. Turbiner ◽

Willard Miller

Keyword(s):

Kinetic Energy ◽

Degrees Of Freedom ◽

Equations Of Motion ◽

Center Of Mass ◽

Symmetric Polynomial ◽

Body System ◽

Generalized Coordinates ◽

New Variables ◽

Three Body ◽

Mass Dependent

We consider the classical three-body system with [Formula: see text] degrees of freedom [Formula: see text] at zero total angular momentum. The study is restricted to potentials [Formula: see text] that depend solely on relative (mutual) distances [Formula: see text] between bodies. Following the proposal by J. L. Lagrange, in the center-of-mass frame we introduce the relative distances (complemented by angles) as generalized coordinates and show that the kinetic energy does not depend on [Formula: see text], confirming results by Murnaghan (1936) at [Formula: see text] and van Kampen–Wintner (1937) at [Formula: see text], where it corresponds to a 3D solid body. Realizing [Formula: see text]-symmetry [Formula: see text], we introduce new variables [Formula: see text], which allows us to make the tensor of inertia nonsingular for binary collisions. In these variables the kinetic energy is a polynomial function in the [Formula: see text]-phase space. The three-body positions form a triangle (of interaction) and the kinetic energy is [Formula: see text]-permutationally invariant with respect to interchange of body positions and masses (as well as with respect to interchange of edges of the triangle and masses). For equal masses, we use lowest order symmetric polynomial invariants of [Formula: see text] to define new generalized coordinates, they are called the geometrical variables. Two of them of the lowest order (sum of squares of sides of triangle and square of the area) are called volume variables. It is shown that for potentials which depend on geometrical variables only (i) and those which depend on mass-dependent volume variables alone (ii), the Hamilton’s equations of motion can be considered as being relatively simple. We study three examples in some detail: (I) three-body Newton gravity in [Formula: see text], (II) three-body choreography in [Formula: see text] on the algebraic lemniscate by Fujiwara et al., where the problem becomes one-dimensional in the geometrical variables and (III) the (an)harmonic oscillator.

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Behavior of Small Variable Mass Particle in Electromagnetic Copenhagen Problem

Sultan Qaboos University Journal for Science [SQUJS] ◽

10.24200/squjs.vol25iss1pp61-77 ◽

2020 ◽

Vol 25 (1) ◽

pp. 61

Author(s):

Abdullah A. Ansari

Keyword(s):

Test Particle ◽

Equilibrium Points ◽

Equations Of Motion ◽

Center Of Mass ◽

Variable Mass ◽

Circular Path ◽

Third Body ◽

Common Center ◽

The Stability ◽

Mass Test

The paper presents the behavior of the motion properties of the variable mass test particle (third body), moving under the influence of the two equal primaries having electromagnetic dipoles. These primaries move on the same circular path around their common center of mass in the same plane. We have determined the equations of motion of the test particle whose mass varies according to Jean's law. Using the system of equations of motion we have evaluated the locations of equilibrium points, their movements and basins of the attracting domain. Finally, we examine the stability of these equilibrium points, all of which are unstable.

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Motion of the Infinitesimal Variable Mass in the Generalized Circular Restricted Three-Body Problem under the Effect of Asteroids Belt

Advances in Astronomy ◽

10.1155/2020/6684728 ◽

2020 ◽

Vol 2020 ◽

pp. 1-10

Author(s):

Ferdaous Bouaziz-Kellil

Keyword(s):

Body Problem ◽

Equilibrium Points ◽

Equations Of Motion ◽

Center Of Mass ◽

Variable Mass ◽

Three Body Problem ◽

Common Center ◽

The Stability ◽

Effects Of Radiation ◽

Three Body

The present paper deals with the study of the motion’s properties of the infinitesimal variable mass body moving in the same orbital plan as two massive bodies (considered as primaries). It is assumed that the massive bodies have radiating effects, have oblate shapes, and are moving in circular orbits around their common center of mass. Using the procedures established by Singh and Abouelmagd, we determined the equations of motion of the infinitesimal body for which we assumed that under the effects of radiation and oblateness of the primaries, its mass varies following Jean’s law. We evaluated analytically and numerically the locations of equilibrium points and examined the stability of these equilibrium points. Finally, we found that all the points are unstable.

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The motion of a lunar orbiter

Symposium - International Astronomical Union ◽

10.1017/s0074180900105686 ◽

1966 ◽

Vol 25 ◽

pp. 373

Author(s):

Y. Kozai

Keyword(s):

Degrees Of Freedom ◽

Dimensional Space ◽

Artificial Satellite ◽

Equations Of Motion ◽

Triaxial Ellipsoid ◽

The Moon ◽

Secular Motion ◽

The Earth ◽

Close Satellite ◽

The Mean

The motion of an artificial satellite around the Moon is much more complicated than that around the Earth, since the shape of the Moon is a triaxial ellipsoid and the effect of the Earth on the motion is very important even for a very close satellite.The differential equations of motion of the satellite are written in canonical form of three degrees of freedom with time depending Hamiltonian. By eliminating short-periodic terms depending on the mean longitude of the satellite and by assuming that the Earth is moving on the lunar equator, however, the equations are reduced to those of two degrees of freedom with an energy integral.Since the mean motion of the Earth around the Moon is more rapid than the secular motion of the argument of pericentre of the satellite by a factor of one order, the terms depending on the longitude of the Earth can be eliminated, and the degree of freedom is reduced to one.Then the motion can be discussed by drawing equi-energy curves in two-dimensional space. According to these figures satellites with high inclination have large possibilities of falling down to the lunar surface even if the initial eccentricities are very small.The principal properties of the motion are not changed even if plausible values ofJ3andJ4of the Moon are included.This paper has been published in Publ. astr. Soc.Japan15, 301, 1963.

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The Influence of Loading Position in A Priori High Stress Detection using Mode Superposition

Reports in Mechanical Engineering ◽

10.31181/rme200101093s ◽

2020 ◽

Vol 1 (1) ◽

pp. 93-102

Author(s):

Carsten Strzalka ◽

◽

Manfred Zehn ◽

Keyword(s):

Finite Element ◽

Degrees Of Freedom ◽

Equations Of Motion ◽

A Priori ◽

High Stress ◽

Von Mises Stress ◽

Detailed Knowledge ◽

Variable Loading ◽

Numerical Effort ◽

Von Mises

For the analysis of structural components, the finite element method (FEM) has become the most widely applied tool for numerical stress- and subsequent durability analyses. In industrial application advanced FE-models result in high numbers of degrees of freedom, making dynamic analyses time-consuming and expensive. As detailed finite element models are necessary for accurate stress results, the resulting data and connected numerical effort from dynamic stress analysis can be high. For the reduction of that effort, sophisticated methods have been developed to limit numerical calculations and processing of data to only small fractions of the global model. Therefore, detailed knowledge of the position of a component’s highly stressed areas is of great advantage for any present or subsequent analysis steps. In this paper an efficient method for the a priori detection of highly stressed areas of force-excited components is presented, based on modal stress superposition. As the component’s dynamic response and corresponding stress is always a function of its excitation, special attention is paid to the influence of the loading position. Based on the frequency domain solution of the modally decoupled equations of motion, a coefficient for a priori weighted superposition of modal von Mises stress fields is developed and validated on a simply supported cantilever beam structure with variable loading positions. The proposed approach is then applied to a simplified industrial model of a twist beam rear axle.

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Dynamics of an Eco-Epidemic Predator–Prey Model Involving Fractional Derivatives with Power-Law and Mittag–Leffler Kernel

Symmetry ◽

10.3390/sym13050785 ◽

2021 ◽

Vol 13 (5) ◽

pp. 785

Author(s):

Hasan S. Panigoro ◽

Agus Suryanto ◽

Wuryansari Muharini Kusumawinahyu ◽

Isnani Darti

Keyword(s):

Hopf Bifurcation ◽

Fractional Derivative ◽

Equilibrium Point ◽

Power Law ◽

Equilibrium Points ◽

Essential Difference ◽

Predator Prey ◽

Predator Prey Model ◽

Prey Model ◽

Infected Prey

In this paper, we consider a fractional-order eco-epidemic model based on the Rosenzweig–MacArthur predator–prey model. The model is derived by assuming that the prey may be infected by a disease. In order to take the memory effect into account, we apply two fractional differential operators, namely the Caputo fractional derivative (operator with power-law kernel) and the Atangana–Baleanu fractional derivative in the Caputo (ABC) sense (operator with Mittag–Leffler kernel). We take the same order of the fractional derivative in all equations for both senses to maintain the symmetry aspect. The existence and uniqueness of solutions of both eco-epidemic models (i.e., in the Caputo sense and in ABC sense) are established. Both models have the same equilibrium points, namely the trivial (origin) equilibrium point, the extinction of infected prey and predator point, the infected prey free point, the predator-free point and the co-existence point. For a model in the Caputo sense, we also show the non-negativity and boundedness of solution, perform the local and global stability analysis and establish the conditions for the existence of Hopf bifurcation. It is found that the trivial equilibrium point is a saddle point while other equilibrium points are conditionally asymptotically stable. The numerical simulations show that the solutions of the model in the Caputo sense strongly agree with analytical results. Furthermore, it is indicated numerically that the model in the ABC sense has quite similar dynamics as the model in the Caputo sense. The essential difference between the two models is the convergence rate to reach the stable equilibrium point. When a Hopf bifurcation occurs, the bifurcation points and the diameter of the limit cycles of both models are different. Moreover, we also observe a bistability phenomenon which disappears via Hopf bifurcation.

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Research of Trajectory Optimization Approaches in Synthesized Optimal Control

Symmetry ◽

10.3390/sym13020336 ◽

2021 ◽

Vol 13 (2) ◽

pp. 336

Author(s):

Askhat Diveev ◽

Elizaveta Shmalko

Keyword(s):

Optimal Control ◽

Equilibrium Point ◽

Trajectory Optimization ◽

Stable Equilibrium ◽

Equilibrium Points ◽

Control Object ◽

Symbolic Regression ◽

Optimal Location ◽

Stable Equilibrium Point ◽

Stabilization System

This article presents a study devoted to the emerging method of synthesized optimal control. This is a new type of control based on changing the position of a stable equilibrium point. The object stabilization system forces the object to move towards the equilibrium point, and by changing its position over time, it is possible to bring the object to the desired terminal state with the optimal value of the quality criterion. The implementation of such control requires the construction of two control contours. The first contour ensures the stability of the control object relative to some point in the state space. Methods of symbolic regression are applied for numerical synthesis of a stabilization system. The second contour provides optimal control of the stable equilibrium point position. The present paper provides a study of various approaches to find the optimal location of equilibrium points. A new problem statement with the search of function for optimal location of the equilibrium points in the second stage of the synthesized optimal control approach is formulated. Symbolic regression methods of solving the stated problem are discussed. In the presented numerical example, a piece-wise linear function is applied to approximate the location of equilibrium points.

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